SPEAKER: Andrea Agazzi (University of Pisa)

TITLE: Random Splitting of Fluid Models: Ergodicity, Convergence and Lyapunov exponents

ABSTRACT: We consider a family of processes obtained by decomposing the deterministic dynamics associated with some fluid models (e.g. Lorenz 96, 2d Galerkin-Navier-Stokes) into fundamental building blocks - i.e., minimal vector fields preserving some fundamental aspects of the original dynamics - and by sequentially following each vector field for a random amount of time. We characterize some ergodic properties of these stochastic dynamical systems and discuss their convergence to the original deterministic flow in the small noise regime. Finally, we show that the top Lyapunov exponent of these models is positive. This is joint work with Jonathan Mattingly and Omar Melikechi.

SPEAKER: Gianmarco Bet (University of Florence)

TITLE: Detecting a late changepoint in the preferential attachment model

ABSTRACT: Motivated by the problem of detecting a change in the evolution of a network, we consider the preferential attachment random graph model with a time-dependent attachment function. Our goal is to detect whether the attachment mechanism changed over time, based on a single snapshot of the network at time $n$ and without directly observable information about the dynamics. We cast this question as a hypothesis testing problem, where the null hypothesis is a preferential attachment model with a constant affine attachment parameter \delta_0, and the alternative hypothesis is a preferential attachment model where the affine attachment parameter changes from $\delta_0$ to $\delta_1$ at an unknown changepoint time $\tau_n$. For our analysis we assume that the changepoint occurs close to the observation time of the network, which corresponds to the relevant scenario where we aim to detect the changepoint shortly after it has happened. We present two tests based on the number of vertices with minimal degree. The first test assumes knowledge of $\delta_0$, while the second does not. We show that these are asymptotically powerful when $n-\tau_n \gg n^{1/2}$. Furthermore, we prove that the test statistics for both tests are asymptotically normal, allowing for accurate calibration of the tests. If time allows, we will present numerical experiments to illustrate the finite sample test properties.

SPEAKER: Ilya Losev (University of Cambridge, UK)

TITLE: How long are the arms in Dielectric-Breakdown Model?

ABSTRACT: In this talk we will discuss the recent progress on such random growth models as Diffusion Limited Aggregation (DLA) and Dielectric-Breakdown Model (DBM) in 2 and 3 dimensions. These models are believed to exhibit non-equilibrium growth, producing irregular fractal patterns. The main questions about these processes include finding their scaling limits and fractal dimensions. However, almost nothing is known rigorously. The main result about these models is due to Kesten, who gave a non-trivial lower bound on the fractal dimension of DLA clusters. The main tool in his proof was the famous Beurling's estimate. We generalize this result to DBM and give a new proof of Kesten's Theorem. Our proof does not rely on Beurling's estimate. Instead, we exploit the connection between DBM growth properties and multifractal spectrum of the harmonic measure.

SPEAKER: Giuseppe Cannizzaro (University of Warwick, UK)

TITLE: Weak coupling scaling of critical SPDEs

ABSTRACT: The study of stochastic PDEs has known tremendous advances in recent years and, thanks to Hairer's theory of regularity structures and Gubinelli and Perkowski's paracontrolled approach, (local) existence and uniqueness of solutions of subcritical SPDEs is by now well-understood. The goal of this talk is to move beyond the aforementioned theories and present novel tools to derive the scaling limit (in the so-called weak coupling scaling) for some stationary SPDEs at the critical dimension. Our techniques are inspired by the resolvent method developed by Landim, Olla, Yau, Varadhan, and many others, in the context of particle systems in the supercritical dimension. Time allowing, we will explain how it is possible to use our techniques to study a much wider class of statistical mechanics models at criticality such as (self-)interacting diffusions in random environment.

SPEAKER: Alexandre Stauffer (King's College London and University of Bath)

TITLE: Mixing time of random walk on dynamical random cluster

ABSTRACT: We consider a random walk jumping on a dynamic graph; that is, a graph that changes at the same time as the walker moves. Previous works considered the case where the graph changes via dynamical percolation, in which the edges of the graph switch between two states, open and closed, and the walker is only allowed to cross open edges. In dynamical percolation, edges change their state independently of one another. In this work, we consider a graph dynamics with unbounded dependences: Glauber dynamics on the random cluster model. We derive tight bounds on the mixing time when the density of open edges is small enough. For the proof, we construct a non-Markovian coupling using a multiscale analysis of the environment. This is based on joint work with Andrea Lelli.

SPEAKER: Michle Salvi (University Tor Vergata, Rome)

TITLE: From the Uniform to the Minimum Spanning Tree

ABSTRACT: A spanning tree of a graph G is a connected subset of G without cycles. The Uniform Spanning Tree (UST) is obtained by choosing one of the possible spanning trees of G at random. The Minimum Spanning Tree (MST) is realised instead by putting random weights on the edges of G and then selecting the spanning tree with the smallest weight. These two models exhibit markedly different behaviours: for example, their diameter on the complete graph with n nodes transitions from n^1/2 for the UST to n^1/3 for the MST. What lies in between? We introduce a model of Random Spanning Trees in Random Environment (RSTRE) designed to interpolate between UST and MST. In particular, when the environment disorder is sufficiently low, the RSTRE on the complete graph has a diameter of n^1/2 as the UST. Conversely, when the disorder is high, the diameter behaves like n^1/3 as for the MST. We conjecture a smooth transition between these two values for intermediate levels of disorder. This talk is based on joint work with Rongfeng Sun and Luca Makowiec (NUS Singapore).

SPEAKER: Daniel Parisi (Sapienza University of Rome)

TITLE: Entropy and mixing time of non-local Markov chains

ABSTRACT: We discuss the convergence to the stationary distribution for non-local Markov chains on general spin systems on arbitrary graphs. We show that the relative entropy functional of the corresponding Gibbs measure satisfies the block factorization of entropy, an inequality that controls the entropy on a given region V in terms of a weighted sum of the entropies on blocks A ⊂ V when each A is given an arbitrary nonnegative weight α_A. This inequality generalizes the approximate tensorization of entropy and provides a natural extension of the classical Shearer inequality satisfied by the Shannon entropy. As a consequence of block factorization, we obtain optimal bounds on the mixing time of a large class of sampling algorithms for the ferromagnetic Ising/Potts models, including non-local Markov chains such as the heat-bath block dynamics and the Swendsen-Wang dynamics. The methods also apply to spin systems with hard constraints such as q-colorings and the hard-core gas model. First, we consider spin systems on the d-dimensional lattice Z^d satisfying strong spatial mixing. Then we extend our analysis to spin systems on an arbitrary graph satisfying spectral independence. Finally, we show that the existence of a contractive coupling for any local Markov chain implies spectral independence.

SPEAKER: Federico Capannoli (Leiden University)

TITLE: Meeting, Coalescence and Consensus on random directed graphs.

ABSTRACT: We consider Markovian dynamics on a typical realization of the so-called Directed Configuration Model (DCM), which is a random directed graph with prescribed in- and out-degrees. In this random geometry, we study the meeting time of two random walks starting at stationarity, the coalescence time for a system of coalescent random walks, and the consensus time of the voter model. Indeed, it is known that the latter three quantities are related to each other when the underlying sequence of graphs satisfies certain mean field conditions. We provide a complete characterization of the distribution of meeting, coalescence and consensus time on a typical random graph as a function of a single quantity θ. More precisely we show that, for a typical large graph from the DCM ensemble, the distribution of the meeting time is well-approximated by an exponential random variable. Furthermore, we provide the precise first-order approximation of its expectation, showing that the latter is linear in the size of the graph, and the explicit preconstant θ depends on some easy statistics of the degree sequence. As a consequence, we can analyze the effect of the degree sequence on the expected meeting time and, via some explicit examples, how its regularity/variability play crucial roles in the information diffusion. This is based on a joint work with Luca Avena (University of Florence), Rajat Subhra Hazra (Leiden University) and Matteo Quattropani (Sapienza, University of Rome).

SPEAKER: Wei Wu (NYU Shangai)

TITLE: Massless phases for the Villain model in d>=3

ABSTRACT: The XY and the Villain models are models which exhibit the celebrated Kosterlitz-Thouless phase transitions in two dimensions. The spin wave conjecture, originally proposed by Dyson and by Mermin and Wagner, predicts that at low temperature, spin correlations of these models are closely related to Gaussian free fields. I will review the historical background and discuss some recent progress on this conjecture in d>=3. Based on the joint work with Paul Dario (CNRS).

SPEAKER: Lorenzo Taggi (Università La Sapienza)

TITLE: Sistemi di marce aleatorie cicliche interagenti

ABSTRACT: Consideriamo un sistema di marce aleatorie cicliche (``random walk loop soup") in presenza di autointerazioni e di interazioni mutuali. Tale modello dipende da un parametro, la temperatura inversa, il quale favorisce la lunghezza totale dei cicli, e presenta una transizione di fase. Esso non è solamente interessante di per sé per le sue proprietà matematiche, ma è anche una riformulazione di importanti modelli della meccanica statistica, quali il modello di Spin O(N) (una generalizzazione dei modelli di Ising, XY e di Heisenberg, corrispondenti rispettivamente ai casi N=1, N=2, e N=3), del modello di doppio dimero, del modello delle permutazioni aleatorie, e presenta inoltre delle analogie con il gas di Bose. Una delle domande principali riguarda la descrizione della dimensione tipica dei cicli e del fenomeno della transizione di fase, il quale presenta caratteristiche diverse in ogni dimensione del reticolo. Il seminario presenterà un'introduzione al modello e illustrerà le domande principali.

SPEAKER: Federico Sau (Università di Trieste)

TITLE: Spectral gap of the symmetric inclusion process.

ABSTRACT: In this talk, we consider the symmetric inclusion process on a general finite graph. Our main result establishes universal upper and lower bounds for the spectral gap of this interacting particle system in terms of the spectral gap of the random walk on the same graph. In the regime in which the gamma-like reversible measures of the particle system are log-concave, our bounds match, yielding a version for the symmetric inclusion process of the celebrated Aldous' spectral gap conjecture --- originally formulated for the interchange process and proved by Caputo, Liggett and Richthammer (JAMS 2010). Finally, by means of duality techniques, we draw analogous conclusions for an interacting diffusion-like unbounded conservative spin system known as Brownian energy process, which may be interpreted as a spatial version of the Wright-Fisher diffusion with mutation. Based on a joint work with Seonwoo Kim (SNU, South Korea).

SPEAKER: Davide Gabrielli (Università dell'Aquila)

TITLE: HIDDEN TEMPERATURE IN THE KIPNIS-MARCHIORO-PRESUTTI MODEL

ABSTRACT: Stationary non equilibrium states (SNS) have a rich and complex structure. The large deviations rate functionals for the empirical measure of a few one dimensional SNS of stochastic interacting systems have been computed, among which the boundary driven exclusion process and Kipnis-Marchioro-Presutti (KMP) model. The corresponding rate functionals are not local due to the presence of long range correlations. We show for the KMP model that this can be explained by introducing new variables that can be interpreted naturally as the temperatures of the oscillators that are exchanging the energies. When two oscillators exchange energy they thermalize at the same time. We deduce that the invariant measure of the boundary driven KMP model is a mixture of inhomogeneous products of exponential distributions, the law of the mixture is the invariant measure of the auxiliary temperature process. This is a joint work with Anna De Masi and Pablo Ferrari. We show moreover that a similar representation of the invariant measure holds also for the boundary driven harmonic model; in this case the hidden variables are distributed according to the order statistics of uniform random variables. This is a joint work with G. Carinci, C. Franceschini, C. Giardinà and D. Tsagkarogiannis.

SPEAKER: Gerard Letac (Paul Sabatier University - Toulouse III)

TITLE: From the Matsumoto Yor observation to stationary measures for a discrete KdV model

ABSTRACT: See Notiziario Scientifico (the text contains too many formulas)

SPEAKER: Nikolay Barashkov (University of Helsinki)

TITLE: Gluing for $\phi^4_3$ on cylinders

ABSTRACT: The $\phi^4_3$ model is a 3-dimensional non-Gaussian Euclidean QFT. Showing existence of such a measure was one of the highlights of the constructive QFT programme in the '70s. In this talk I will describe joint work with Trishen Gunaratnam in analysing how $\phi^4_3$ models glue together on cylinders.

SPEAKER: Fraydoun Rezakhanlou (Berkeley University)

TITLE: Kinetic Theory for Laguerre Tessellations

ABSTRACT: In this talk I will discuss a family of Gibbsian measures on the set of Laguerre tessellations. These measures may be used to provide a systematic approach for constructing Gibbsian solutions to Hamilton-Jacobi PDEs by exploring the Eularian description of the shock dynamics. Such solutions depend on kernels satisfying kinetic-like equations reminiscent of the Smoluchowski model for coagulating and fragmenting particles.

SPEAKER: Lionel Levine (Cornell University)

TITLE: Universality Conjectures For Activated Random Walk

ABSTRACT: Activated Random Walk is a particle system displaying avalanches on all scales. How universal are these avalanches? I’ll narrate five interlocking conjectures aimed at different aspects of this question: infinite-volume limits, cutoff, incompressibility, rotational symmetry, and hyperuniformity. Joint work with Feng Liang and with Vittoria Silvestri.

SPEAKER: Marta Leocata (SNS, Pisa)

TITLE: Modeling an example of green transition

ABSTRACT: The work presented is part of a collaboration between mathematicians and philosophers of ethics, politics, and society aiming to understand mechanisms of green energy transition where some kind of interaction between many subjects and collective behaviors seem to play a role. We have identified as a first example the case of solar photovoltaic, with the purpose of explaining the time series of Italy and of different regions. We use tools from continuous-time Markov chains, mean field limits, optimal control, and mean field games. Then, we specify our analysis on the case of domestic installations. In order to explain them we propose a Markovian model of interacting particle systems to describe the decision of a significant proportion of the general public in Italy on the installation of solar photovoltaics (PVs) and compare our models to real data. We assume that the adoption decision follows a well precise pattern. First, the general public develops a certain level of sensitivity to climate change and environmental issues. Then it plans to install the PVs because she/he has a sufficient amount of information on the benefits of PVs. Finally, it installs PVs because the economic benefits out-weights the costs. In conclusion, we propose some original policies, not based not on financial incentives. This work is in collaboration with Franco Flandoli (SNS), Giulia Livieri (SNS), Silvia Morlacchi (SNS), Fausto Corvino (University of Gothenburg) and Alberto Pirni (SSSA).

SPEAKER: Claudio Landim (IMPA, Rio de Janeiro)

TITLE: Gamma-expansion of the level two large deviations rate functionals

ABSTRACT:

We present a general method, based on tools used to prove the metastable behaviour of Markov chains, to derive a full expansion of its level two large deviations rate functional, expressing it as In=I(0)+∑1≤p≤q(1/θ(p)n)I(p), where I(p) represent rate functionals independent of n and θ(p)n sequences such that θ(1)n→∞, θ(p)n/θ(p+1)n→0 for 1≤p

SPEAKER: Markus Fischer (Università di Padova)

TITLE: On correlated equilibria and mean field games

ABSTRACT: Mean field games are limit models for symmetric N-player games, as the number of players N tends to infinity. The prelimit models are usually solved in terms of Nash equilibria. A generalization of the notion of Nash equilibrium, due to Robert Aumann (1974, 1987), is that of correlated equilibrium. In a simple discrete setting, we will discuss correlated equilibria for mean field games and their connection with the underlying N-player games. We first consider equilibria in restricted strategies (Markov open-loop), where control actions depend only on time and a player's own state. In this case, N-player correlated equilibria are seen to converge to the mean field game limit and, conversely, correlated mean field game solutions induce approximate N-player correlated equilibria. We then discuss the problem of constructing approximate equilibria when deviating players have access to the aggregate system state. We also give an explicit example of a correlated mean field game solution not of Nash-type. Results (with L. Campi and Federico Cannerozzi) on a related notion of equilibrium in a diffusion-type setting will be mentioned as well. Joint work with Ofelia Bonesini (Imperial College London) and Luciano Campi (University of Milan "La Statale")

SPEAKER: Luisa Andreis (Politecnico di Milano)

TITLE: Rare events in sparse random graphs

ABSTRACT: Rare events for dense random graphs are well described using the theory of large deviations and graphons. When graphs are sparse the picture is less clear, objects that describe globally the graphs and their limits, as graphons do for dense graphs, have not been defined yet. Nevertheless we do have information on how these graphs look like when explored locally from a vertex and this, under some assumptions, gives also information on global properties. In this talk we will give an overview on what is known on rare events in this regime, focusing in particular on large deviation statements on connected components in inhomogeneous random graphs and on links with coagulation processes. This talk is based on joint works with Wolfgang König (WIAS and TU Berlin), Tejas Iyer, Heide Langhammer, Elena Magnanini and Robert Patterson (WIAS).

SPEAKER: Daniel Ueltschi (University of Warwick)

TITLE: Quantum spin chains, loop representations, dimerisation

ABSTRACT: In contrast to their classical counterparts, one-dimensional quantum spin systems are interesting, they have intriguing behaviour, and they are difficult to study. I will describe a family of systems with nearest-neighbour interactions and O(n) symmetry. Its ground state phase diagram is expected to have a rich structure. For large spins it is possible to prove the occurrence of dimerisation in an open domain. The proof involves a loop representation and the method of cluster expansions. Joint work with Jakob Bjornberg, Peter Muhlbacher, and Bruno Nachtergaele.

SPEAKER: Lorenzo Dello Schiavo (ISTA - Vienna)

TITLE: Scaling Limits of Random Walks, Harmonic Profiles, and Stationary Non-Equilibrium States in Lipschitz Domains

ABSTRACT: We consider the open symmetric exclusion (SEP) and inclusion (SIP) processes on a bounded Lipschitz domain Ω, with both fast and slow boundary. For the random walks on Ω dual to SEP/SIP we establish a functional-CLT-type convergence to the Brownian motion on Ω with either Neumann (slow boundary), Dirichlet (fast boundary), or Robin (at criticality) boundary conditions. We further show the discrete-to-continuum convergence of the corresponding harmonic profiles. As a consequence, we rigorously derive the hydrodynamic and hydrostatic limits for SEP/SIP on Ω, and analyze their stationary non-equilibrium fluctuations. Based on joint work arXiv:2112.14196 with Lorenzo Portinale (IAM Bonn) and Federico Sau (ISTA)

SPEAKER: Giacomo Greco (EURANDOM, Eindhoven)

TITLE: Travelling through turnpikes in the Kinetic Schrödinger Problem

ABSTRACT: The Schrödinger problem consists in finding the most likely evolution of a system of i.i.d. particles, conditionally on their initial and final configurations. In the last decade this problem has become more and more popular, thanks to its connections with stochastic optimal control theory and optimal transport. In this talk we are going to introduce a Schrödinger problem where the particles follow an (underdamped) Langevin dynamics (i.e. their density satisfies the kinetic Fokker-Planck equation) and where we are given only a partial observation of the initial and final configurations. Then, we will investigate the long-time behaviour of this kinetic system, proving that its most likely evolution is exactly the one that spends most of the time exponentially close to the equilibrium configuration (this property is commonly known in stochastic control theory as the turnpike property). Based on the paper Electron. J. Probab. 27: 1-32 (2022). DOI: 10.1214/22-EJP850

SPEAKER: Alberto Chiarini (Università: di Padova)

TITLE: On the cost of covering a fraction of a macroscopic body by a simple random walk.

ABSTRACT: In this talk we aim at establishing large deviation estimates for the probability that a simple random walk on the Euclidean lattice (d>2) covers a substantial fraction of a macroscopic body. It turns out that, when such rare event happens, the random walk is locally well approximated by random interlacements with a specific intensity, which can be used as a pivotal tool to obtain precise exponential rates. Random interlacements have been introduced by Sznitman in 2007 in order to describe the local picture left by the trace of a random walk on a large discrete torus when it runs up to times proportional to the volume of the torus, and has been since a popular object of study. In the first part of the talk we introduce random interlacements and give a brief account of some results surrounding this object. In the second part of the talk we study the event that random interlacements cover a substantial fraction of a macroscopic body. This allows to obtain an upper bound on the probability of the corresponding event for the random walk. Finally, by constructing a near-optimal strategy for the random walk to cover a macroscopic body, we discuss a matching large deviation lower bound. The talk is based on ongoing work with M. Nitzschner (NYU Courant).

SPEAKER: Mauro Mariani (HSE University, Moscow)

TITLE: Long time asymptotic of action functionals

ABSTRACT: I will provide a self-contained variational approach to state some classical and new results in the framework of Aubry-Mather theory. More precisely, I will discuss the expansion by Gamma-convergence of the action functional of classical mechanics, as the time-scale diverges. Joint work with Carlo Orrieri.

SPEAKER: Matteo Quattropani (Sapienza Università di Roma)

TITLE: Mixing of the Averaging process on graphs

ABSTRACT: The Averaging process (a.k.a. repeated averages) is a mass redistribution model over the vertex set of a graph. Given a graph G, the process starts with a non-negative mass associated to each vertex. The edges of G are equipped with Poissonian clocks: when an edge rings, the masses at the two extremes of the edge are equally redistributed on these two vertices. Clearly, as time grows to infinity the state of the system will converge (in some sense) to a flat configuration in which all the vertices have the same mass. The process has been introduced to the probabilistic community by Aldous and Lanoue [1] in 2012, and recently received some attention thanks to the work of Chatterjee, Diaconis, Sly and Zhang [2], where the authors show an abrupt convergence to equilibrium (measured in L^1 distance) in the case in which the underlying graph is complete (and of diverging size). In this talk, I will present some recent results obtained in collaboration with F. Sau (IST Austria) [3,4] and P. Caputo (Roma Tre) [4]. In [3] we show that if the underlying graph is “finite dimensional” (e.g., a finite box of Z^d), then the convergence to equilibrium is smooth (i.e., without cutoff) when measured in L^p with p \in [1,2]. On the other hand, in [4] we show that a cutoff phenomenon (for the L^1 and L^2 distance to equilibrium) takes place when the underlying graph is the hypercube or the complete bipartite graph.

[1] David Aldous, and Daniel Lanoue. A lecture on the averaging process. Probab. Surv., 9:90-102, 2012. [2] Sourav Chatterjee, Persi Diaconis, Allan Sly, and Lingfu Zhang. A phase transition for repeated averages. Ann. Probab. 50(1):1-17, 2022. [3] Matteo Quattropani and Federico Sau. Mixing of the Averaging process and its discrete dual on finite-dimensional geometries. Ann. Appl. Probab. (to appear). [4] Pietro Caputo, Matteo Quattropani and Federico Sau. Cutoff for the Averaging process on the hypercube and complete bipartite graphs. (to appear).

Stop to seminars due to COVID

SPEAKER: Lorenzo Dello Schiavo (IST Austria)

TITLE: Conformally invariant random fields, quantum Liouville measures, and random Paneitz operators on Riemannian manifolds of even dimension

ABSTRACT: On large classes of closed even-dimensional Riemannian manifolds M, we construct and study the Copolyharmonic Gaussian Field, i.e. a conformally invariant log-correlated Gaussian field of distributions on M. This random field is defined as the unique centered Gaussian field with covariance kernel given as the resolvent kernel of Graham-Jenne-Mason-Sparling (GJMS) operators of maximal order. The corresponding Gaussian Multiplicative Chaos is a generalization to the 2m-dimensional case of the celebrated Liouville Quantum Gravity measure in dimension two. We study the associated Liouville Brownian motion and random GJMS operator, the higher-dimensional analogues of the 2d Liouville Brownian Motion and of the random Laplacian. Finally, we study the Polyakov-Liouville on the space of distributions on M induced by the copolyharmonic Gaussian field, providing explicit conditions for its finiteness and computing the conformal anomaly. (arXiv:2105.13925, joint work with Ronan Herry, Eva Kopfer, Karl-Theodor Sturm)

SPEAKER: Francesco Caravenna (Università di Milano-Bicocca)

TITLE: On the two-dimensional KPZ and Stochastic Heat Equation

ABSTRACT: We consider the Kardar-Parisi-Zhang equation (KPZ) and the multiplicative Stochastic Heat Equation (SHE) in two space dimensions, driven by with space-time white noise. These singular PDEs are "critical" and lack a solution theory, so it is standard to consider regularized versions of these equations - e.g. convolving the noise with a smooth mollifier - and to investigate the behavior of the regularized solutions when the regularization is removed. Based on joint works with Rongfeng Sun and Nikos Zygouras, we show that these regularized solutions undergo a phase transition as the noise strength is varied on a logarithmic scale, with an explicit critical point. In the sub-critical regime, the regularized solutions of both KPZ and SHE exhibit so-called Edwards-Wilkinson fluctuations, i.e. they converge to the solution of the *additive* Stochastic Heat Equation (after centering and rescaling), with a non-trivial constant on the noise. We finally discuss the critical regime, where many questions are open.

SPEAKER: Antonio Galves (Universidade de Sao Paulo)

TITLE: Estimating the interaction graph of stochastic neural dynamics

ABSTRACT: We address the question of statistical model selection for a class of stochastic models of biological neural nets.Models in this class are systems of interacting chains with memory of variable length. Each chain describes the activity of a single neuron, indicating whether it spikes or not at a given time. The spiking probability of a given neuron depends on the time evolution of its presynaptic neurons since its last spike time. When a neuron spikes, its potential is reset to a resting level and postsynaptic current pulses are generated, modifying the membrane potential of all its postsynaptic neurons. The relationship between a neuron and its pre- and postsynaptic neurons defines an oriented graph, the interaction graph of the model. The goal is to estimate this graph based on the observation of the spike activity of a finite set of neurons over a finite time. We provide explicit exponential upper bounds for the probabilities of under- and overestimating the interaction graph restricted to the observed set and obtain the strong consistency of the estimator. Our result does not require stationarity nor uniqueness of the invariant measure of the process. Joint work with A. Duarte, E. Locherbach and G. Ost.

SPEAKER: Maria Gordina (University of Connecticut)

TITLE: Ergodicity for Langevin dynamics with singular potentials

ABSTRACT: Abstract: We discuss Langevin dynamics of N particles on R^d interacting through a singular repulsive potential, such as the Lennard-Jones potential, and show that the system converges to the unique invariant Gibbs measure exponentially fast in a weighted total variation distance. The proof relies on an explicit construction of a Lyapunov function using a modified Gamma calculus. In contrast to previous results for such systems, our results imply geometric convergence to equilibrium starting from an essentially optimal family of initial distributions. This is based on join work with F.Baudoin and D.Herzog.

SPEAKER: Lionel Levine (Cornell University)

TITLE: Random walks with local memory

ABSTRACT: The theme of this talk is walks in a random environment of "signposts" altered by the walker. I'll focus on three related examples: 1. Rotor walk on Z^2. Your initial signposts are independent with the uniform distribution on {North,East,South,West}. At each step you rotate the signpost at your current location clockwise 90 degrees and then follow it to a nearest neighbor. Priezzhev et al. conjectured that in n such steps you will visit order n^{2/3} distinct sites. I'll outline an elementary proof of a lower bound of this order. The upper bound, which is still open, is related to a famous question about the path of a light ray in a grid of randomly oriented mirrors. This part is joint work with Laura Florescu and Yuval Peres. 2. p-rotor walk on Z. In this walk you flip the signpost at your current location with probability 1-p and then follow it. I'll explain why your scaling limit will be a Brownian motion perturbed at its extrema. This part is joint work with Wilfried Huss and Ecaterina Sava-Huss. 3. p-rotor walk on Z^2. Rotate the signpost at your current location clockwise with probability p and counterclockwise with probability 1-p, and then follow it. This walk “organizes” its environment by destroying cycles of signposts. A native environment -- stationary in time, from your perspective as the walker -- is an orientation of the uniform spanning forest, plus one additional edge. This part is joint work with Swee Hong Chan, Lila Greco, and Peter Li: https://arxiv.org/abs/1809.04710

SPEAKER: Gonzalo Panizo Garcia (IMCA, Lima, Perù)

TITLE: A self-interacting random walk

ABSTRACT: In 2011, Benjamini, Kozma and Schapira introduced a “balanced excited random walk” in the 4-dimensional lattice. In 2016, a similar model was studied by Peres, Schapira and Sousi in the 3-dimensional lattice. Here we generalize their constructions in the d-dimensional lattice, in the following way: if the walk visits a site for the first time, it makes a simple random walk step in the first d_1 dimensions, whereas if the site has been already visited, it makes a simple random walk step in the last d_2 coordinates. Both BKS and PSS proved transience in the non-overlapping case d=d_1+d_2, with d_1=d_2=2 (BKS) and d_1=1, d_2=3 (PSS). In this talk some result for the overlapping case (d_1+d_2>d) will be presented, in particular for d=4, d_1=2 and d_2=3. (Joint work with D. Camarena and A. Ramirez).

SPEAKER: Hlafo Alfie Mimun (Sapienza)

TITLE: Percolation in the Miller-Abrahams random resistor network

ABSTRACT: The Miller-Abrahams random resistor network is used to study electron transport in amorphous solids. This resistor network is given by the complete random graph built on a marked homogeneous Poisson point process on R^d and each edge {x,y} is associated to a filament with conductance depending on the temperature, the distance between the points x,y and their associated marks. In this talk we consider the subgraph containing only edges with lower bounded conductances and, using the method of randomized algorithms developed by Duminil-Copin et al. and the renormalization argument proposed by Grimmett and Marstrand, we analyze the connection probabilities and the left-right crossings in appropriate regimes. These percolation properties are key ingredients for understanding the asymptotic behavior at low temperature of the effective conductivity of the Miller-Abrahams random resistor network. Joint work with Alessandra Faggionato (Sapienza University, Rome).

SPEAKER: Lorenzo Taggi (WIAS Berlino)

TITLE: Absorbing-state phase transition in Activated Random Walk and Oil and Water

ABSTRACT: We consider two interacting particle systems, Activated Random Walk and Oil and Water, which belong to the so-called class of Abelian networks. In these systems particles of two different types are present and the dynamics conserves the number of particles. This conservation law introduces hard-core constraints and long-range correlations which make the rigorous mathematical analysis particularly challenging. Activated Random Walk undergoes an absorbing-state phase transition, namely if the particle density is small enough, the activity dies out with time, while if the particle density is large enough, the activity is sustained. In recent years significant effort has been made to provide a rigorous proof of the occurrence of such a phase transition and a non-trivial characterisation of the critical particle density. Oil and Water shows a substantially different behaviour from Activated Random Walks and other Abelian networks: the activity dies out with time at all densities for any graph which is vertex-transitive. This talk reviews the main questions, results and conjectures about these models and presents joint works with E. Candellero and A. Stauffer.

SPEAKER: Emilio De Santis (Sapienza)

TITLE: Tecniche di Percolazione per lo studio di Sistemi Interagenti

ABSTRACT: Nel seminario saranno presentati alcuni risultati riguardanti i sistemi interagenti ed in particolare il modello di Ising. I modelli interagenti saranno studiati sia dal punto di vista delle misure di equilibrio sia dal punto di vista dei processi stocastici e della simulazione perfetta. Nel campo delle misure d’equilibrio le tecniche di percolazione permettono di dare condizioni suﬃcienti per l’unicità/non-unicità della misura. Nell'ambito dei processi stocastici Markoviani l’approccio tramite la percolazione fornisce stime sul Gap-Spettale e soprattutto l’esistenza e la costruzione di algoritmi di simulazione perfetta per sistemi a memoria inﬁnita o interazioni a lungo raggio.

SPEAKER: Antonio Galves (Instituto de Matematica e Estadistica Universidade de Sao Paulo, Brasile)

TITLE: Chains with memory of variable length as neurobiological models

SPEAKER: Otavio Menezes (University of Lisbon)

TITLE: Relative entropy and scaling limits of interacting particle systems (lecture I)

ABSTRACT: The relative entropy method was developed by H.T. Yau in the 90’s to study the $

SPEAKER: Otavio Menezes (University of Lisbon)

TITLE: Relative entropy and scaling limits of interacting particle systems (lecture I)

ABSTRACT: The relative entropy method was developed by H.T. Yau in the 90’s to study the hydrodynamics of the Ginzburg-Landau model, and then adapted to several different dynamics. In this course (2 lectures, 2h per lecture) we present the Relative Entropy Method of Yau in the context of general continuous time Markov chains, as well as recent progress in the setting of exclusion and Glauber dynamics.

SPEAKER: Gerard Letac (IMT, University Paul Sabatier, Toulouse, France)

TITLE: Multivariate Reciprocal Inverse Gaussian Distributions: the Surprising Integrals of Supersymmetry

ABSTRACT: [The abstract contains formulas in latex , please see the Notiziario Settimanale]

SPEAKER: Otavio Menezes (University of Lisbon)

TITLE: Relative entropy and scaling limits of interacting particle systems

ABSTRACT: We obtain product approximations to the law of particle systems with exclusion and Glauber dynamics in finite volume, by establishing a bound on the relative entropy between the law of the system and the product measure. As applications of the entropy estimate we obtain the scaling limits of the density fluctuation fields close of equilibrium and bounds on the speed of convergence of the hydrodynamic limit. Joint work with Milton Jara.

SPEAKER: Kirone Mallick (Institut de Physique Théorique, Parigi)

TITLE: Continuous-time Quantum Walks

ABSTRACT: Quantum analogs of classical random walks have been defined in quantum information theory as a useful concept to implement algorithms. Due to interference effects, statistical properties of quantum walks can drastically differ from their classical counterparts, leading to much faster computations. In this talk, we shall discuss various statistical properties of continuous-time quantum walks on a lattice, such as: survival properties of quantum particles in the presence of traps (i.e. a quantum generalization of the Donsker-Varadhan stretched exponential law), the growth of a quantum population in the presence of a source, quantum return probabilities and Loschmidt echoes.

SPEAKER: Mauro Mariani, National Research University Higher School of Economics, Moscow

TITLE: Scaling limit of a continuous model of active particles

ABSTRACT: A free energy functional arising from kinetic mean field models of interacting particles is considered. We study the variational limit, in the regime of long time and strong interaction. While the density of particles only features a weak limit in the phase space (say position and velocity), the projection of such a density on the spacial coordinates has a meaningful limit, which satisfies a hydrodynamic equation. We characterizes the tensors appearing in the hydrodynamic limit, in terms of the interaction and velocity field of the original model.

SPEAKER: Lorenzo Dello Schiavo (Università di Bonn)

TITLE: The Dirichlet-Ferguson diffusion

ABSTRACT: We define, via Dirichlet forms' theory, a geometric diffusion process on the L^2-Wasserstein space over a closed Riemannian manifold. The process is associated with the Dirichlet form induced by the L^2-Wasserstein gradient and by the Dirichlet-Ferguson random measure with intensity the Riemannian volume measure on the base manifold. We discuss the closability of the form via an integration-by-parts formula, which allows explicit computations for the generator and a specification of the process via a measure-valued SPDE. We comment how the construction is related to previous work of von Renesse-Sturm on the Wasserstein Diffusion and of Konarovskyi-von Renesse on the Modified Massive Arratia Flow.

SPEAKER: Marco Romito (Università di Pisa)

TITLE: Fluctuations for point vortices

ABSTRACT: The first part of the presentation is a short review of a statistical mechanics model of point vortices for the 2D Euler equations and their mean field limit. In the second part we outline a proof of Gaussian fluctuations from the mean field limit. The result holds on the torus, on the sphere and on bounded domains. This is a work in collaboration with Francesco Grotto (Scuola Normale Superiore, Pisa).

SPEAKER: Fabio Lucio Toninelli (Università di Lyon 1)

TITLE: The dimer model: equilibrium and non-equilibrium aspects (corso di dottorato)

ABSTRACT: This course focuses on various mathematical aspects of lattice dimer models. These are very classical two-dimensional statistical mechanics models, that are exactly solvable in some sense (Kasteleyn, 1961): partition function and correlations can be computed in determinantal form. Recently there has been a new wave of interest in dimer models, both in probability, combinatorics and mathematical physics. One reason is that these models, as well as other two-dimensional critical models, exhibit conformal invariance properties. Another interesting aspect is that they allow to obtain very nice Markov dynamics of two-dimensional interfaces, whose large-scale dynamical behavior can be studied.

SPEAKER: Claudio Landim (IMPA, Rio de Janeiro, Brazil)

TITLE: Homogenization for diffusion processes, part 3.

ABSTRACT: We discuss homogenization for diffusion processes in stationary random environment and several characterizations of the homogenized diffusion coefficient.

SPEAKER: Claudio Landim (IMPA, Rio de Janeiro, Brazil)

TITLE: Homogenization for diffusion processes, part 2.

ABSTRACT: We discuss central limit theorems for martigales and homogenization for random walks in random environment.

SPEAKER: Claudio Landim (IMPA, Rio de Janeiro, Brazil)

TITLE: Homogenization for diffusion processes

ABSTRACT: We present general tools to prove the central limit theorem for addive functionals of Markov processes and discuss in some detail the application to diffusions in periodic or random enevironment.

SPEAKER: Raphael Chetrite (CNRS, Nice, France)

TITLE: On Gibbs-Shannon Entropy

ABSTRACT: This talk will focus on the question of the physical contents of the Gibbs-Shannon entropy outside equilibrium. It will be based on the article Gavrilov-Chetrite-Bechhoeffer, Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs-Shannon form, PNAS 2017.

SPEAKER: Andrey Piatnitski (Arctic University of Norway, Russian Academy of Sciences)

TITLE: Stochastic homogenization of zero order convolution type operators

ABSTRACT: The talk will focus on homogenization problem for a family of zero order non-local convolution type operators that satisfy proper moment and ellipticity conditions. Under the assumptions that the coefficients of these operators are statistically homogeneous and ergodic we show that the family converges to a second order elliptic differential operator with constant coefficients.

SPEAKER: Tomasz Rychlik (Istituto di Matematica dell'Accademia Polacca delle Scienze)

TITLE: Effects of prior selection in nonparametric Bayes problems

ABSTRACT: We consider an arbitrary family of stochastically ordered distribution functions dependent on parameters from an interval in the real line. We compare distribution functions and moments of random variables with distribution functions being the mixtures of the members of the above stochastically ordered family. In particular, we determine lower and upper bounds on the values of one mixture distribution function under the condition that the other mixture distribution is arbitrarily fixed. We also evaluate the differences between the expectations of mixture variables in various scale units. The results are illustrated by some examples.

SPEAKER: Giacomo di Gesù (University of Wien)

TITLE: Small noise asymptotics for stochastic Allen-Cahn equations in finite volume

ABSTRACT: We consider lattice approximations of the Allen-Cahn equation on the torus perturbed by small space-time white noise and discuss metastable transition times between the two stable phases.

SPEAKER: Frederic Patras (University of Nice)

TITLE: Revisiting chaos expansions and Wick products

ABSTRACT: Recently, the interest in the structure of cumulants and Wick products for non-Gaussian variables has been revived, since they both play important roles in M. Hairer's theory of regularity structures. We present a new approach $ (group actions, Hopf algebras) that allow to understand the theory of Wick products as a deformation, a presentation from which many of their properties follow easily. We also discuss how these ideas appear in the theory of regularity structu$ Tapia, L. Zambotti.

SPEAKER: Federico Camia (New York - Abu Dhabi University)

TITLE: Limit Theorems and Random Fractal Curves in Statistical Mechanics

ABSTRACT: Rigorous statistical mechanics deals with stochastic systems that have a large number of components and for which geometry often plays an important role. The main goal is to understand their average behavior and deviations from that. Statistical mechanics has many applications to physics and other fields, but in this talk I will only focus on some aspects of its mathematical theory which combine discrete probability, stochastic processes and complex analysis. For concreteness, I will discuss two specific (two-dimensional) models, percolation and the Ising model, both with a long history. After introducing the two models, I will present an approach to the study of two-dimensional systems that leads to a special family of random fractal curves and that has produced, in the last twenty years, deep results and major breakthroughs.

SPEAKER: Alessandra Occelli (Università di Bonn)

TITLE: On time correlations for last passage percolation models

ABSTRACT: We study time correlations of last passage percolation (LPP), a model in the Kardar-Parisi-Zhang universality class, with three different geometries: step, flat and stationary. We prove the convergence of the covariances of the LPP at two different times to a limiting expression given in terms of Airy processes. Furthermore, we prove the behaviour of the covariances when the two times are close to each other, conjectured in a work of Ferrari and Spohn.

SPEAKER: Alberto Chiarini (ETH, Zurigo)

TITLE: Invariance principle for the degenerate dynamic random conductance model.

ABSTRACT: After the brilliant result of Papanicolau and Varadhan (1979) in the case of bounded stationary and ergodic environments, there has been a recent upsurge in the research of quenched homogenization in random media. In particular, to identify the optimal conditions that a general stationary and ergodic environment must satisfy in order to obtain the convergence to a non-degenerate Brownian motion, is still an open problem. In this talk, we study a continuous-time random walk on Z^d in an environment of dynamic random conductances. We assume that the law of the conductances is ergodic and stationary with respect to space-time shifts. We prove a quenched invariance principle for the random walk under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by the celebrated Moser’s iteration scheme. This is joint work with S. Andres, J-D. Deuschel and M. Slowik.

SPEAKER: Mauro Maurelli (WIAS, Berlin)

TITLE: Enhanced Sanov theorem for Brownian rough paths and an application to interacting particles

ABSTRACT: We establish a Sanov type large deviation principle for an ensemble of interacting Brownian rough paths. As application a large deviations principle for the (k-layer, enhanced) empirical measure of weakly interacting diffusions is obtained. This in turn impli$

SPEAKER: Yaakov Malinovsky (University of Maryland, Baltimore County)

TITLE: Nested Group Testing Procedures and Generalized GT Problem

ABSTRACT: Group testing has its origins in the identication of syphilis in the US army during World War II. It is a useful method that has broad applications in medicine, engineering, and even in airport security control. Consider a finite population of N units, where unit i has a probability p to be defective. A group test is a simultaneous test on an arbitrary group of units with two possible outcomes: all units are good or at least one of the units is defective. The goal is to construct a procedure which classifies all units in a given population, with as small as possible expected number of tests. In this talk I shall review previously known results in the group testing literature and present new results characterizing optimality of commonly used nested group testing procedures. In the second part of the talk, the generalized group testing problem (where unit i has a probability p_{i} to be defective) will be discussed as well.

SPEAKER: Lorenzo Dello Schiavo (Institut fur Angewandte Mathematik - University of Bonn)

TITLE: Two characterizations of Dirichlet-Ferguson measures

ABSTRACT: We consider the Dirichlet-Ferguson (DF) measure, a random probability on a locally compact Polish space X introduced by Ferguson in [1]. The measure has ever since found many applications, widely ranging from Bayesian non-parametrics to population genetics and stochastic dynamics of infinite particle systems. Firstly, we compute the characteristic functional of DF measures (addressing, if time permits, connections of these measures with Lie algebra theory and Polya Enumeration Theory). Secondly, we prove a characterization of DF measures via a Mecke-type integral identity. Profiting of connections between DF measures and Poisson measures on configuration spaces, we argue how DF measures may be regarded as 'canonical' measures on the space P(X) of Borel probability measures on X. Partly based on joint work with E. W. Lytvynov, University of Swansea, Wales, UK. [1] Ferguson, T. S., Ann. Stat. 1(2), pp. 209-230, 1973.

SPEAKER: Luca Scarpa ( Department of Mathematics University College London, UK)

TITLE: Well-posedness of semilinear SPDEs with singular drift: a variational approach

ABSTRACT: Well-posedness is proved for singular semilinear SPDEs on a smooth bounded domain D in R^n. The linear part is associated to a coercive linear maximal monotone operator on L^2(D) while the drift is represented by a multivalued maximal monotone graph everywhere defined on R, on which no growth nor smoothness conditions are required. Moreover, the noise is given by a cylindrical Wiener process on a Hilbert space U, with a stochastic integrand taking values in the Hilbert-Schmidt operators from U to L^2(D): classical Lipschitz-continuity hypotheses for the diffusion coefficient are assumed. The proof consists in approximating the equation, finding uniform estimates both pathwise and in expectation on the approximated solutions, and then passing to the limit using compactness and lower semicontinuity results. Finally, possible generalizations are discussed. This study is based on a joint work with Carlo Marinelli (University College London).

SPEAKER: Elena Di Bernardino (Parigi)

TITLE: A test of Gaussianity based on the Euler characteristic of excursion sets

ABSTRACT: We deal with a stationary isotropic random field X:R^d→R and we assume it is partially observed through some level functionals. We aim at providing a methodology for a test of Gaussianity based on this information. More precisely, the level functionals are given by the Euler characteristic of the excursion sets above a finite number of levels. On the one hand, we study the properties of these level functionals under the hypothesis that the random field X is Gaussian. In particular, we focus on the mapping that associates to any level u the expected Euler characteristic of the excursion set above level u. On the other hand, we study the same level functionals under alternative distributions of X, such as chi-square, harmonic oscillator and shot noise. In order to validate our methodology, a part of the work consists in numerical experimentations. We generate Monte-Carlo samples of Gaussian and non-Gaussian random fields and compare, from a statistical point of view, their level functionals. Goodness-of-fit p−values are displayed for both cases. Simulations are performed in one dimensional case (d=1) and in two dimensional case (d=2), using R.

SPEAKER: Vittoria Silvestri (Cambdrige)

TITLE: The reset time of Internal DLA

ABSTRACT: Consider Internal DLA on cylinder graphs of the form GxZ. How does a large cluster typically look like? How long does it take for the process to forget its initial profile? In this talk I will address these questions, explaining how the answer depends on the mixing properties of the base graph G. Joint work with Lionel Levine.

SPEAKER: Ehud Lehrer (Tel Aviv University)

TITLE: Non-additive probability, integration and decision making

ABSTRACT: Frequently a decision maker (DM) does not have a full information about the underlying uncertainty. For instance, the information about the probability of some events might be missing, or the probability distribution might fail to be additive. In this talk I will present non-additive probabilities, describe a few methods of integration with respect to these probabilities and how one may use these schemes in decision making.

SPEAKER: T.G. Kurtz, University of Wisconsin-Madison.

TITLE: Stochastic equations for processes built from bounded generators

ABSTRACT: The generator for a pure jump process with bounded jump rate is a bounded operator on the space of measurable functions. For any such process, it is simple to write a stochastic equation driven by a Poisson random measure. Uniqueness for both the stochastic equation and the corresponding martingale problem is immediate, and consequently, the martingale problem and the stochastic equation are equivalent in the sense that they uniquely characterize the same process. A variety of Markov processes, including many interacting particle models, have generators which are at least formally given by infinite sums of bounded generators. In considerable generality, we can write stochastic equations that are equivalent to these generators in the sense that every solution of the stochastic equation is a solution of the martingale problem and every solution of the martingale problem determines a weak solution of the stochastic equation. It follows that uniqueness for one approach is equivalent to uniqueness for the other.

SPEAKER: C. Landim (IMPA, CNRS Rouen)

TITLE: Static large deviations for a reaction-diffusion model

ABSTRACT: We consider the superposition of a symmetric simple exclusion dynamics, speeded-up in time, with a spin-flip dynamics in a one-dimensional interval with periodic boundary conditions. We prove the large deviations principle for the empirical measure under the stationary state. We deduce from this result that the stationary state is concentrated on the stationary solutions of the hydrodynamic equation which are stable.

SPEAKER: M. Piccioni (Sapienza)

TITLE: How it happened (to me) to meet ?

ABSTRACT: The function ? was defined by Minkowski with the purpose of mapping quadratic irrationals bijectively onto the rationals of (0,1) (in addition, it maps the rationals onto the dyadic rationals). It has remarkable self-similarity properties and it maps (increasingly) th$ sion is also obtained. This is a joint work with Gerard Letac.

SPEAKER: M. Campanino (Bologna)

TITLE: Recurrence properties for random walks on a two-dimensional random graph.

ABSTRACT: Recurrcnce properties for random walks on a two-dimensional random graph. In [1] a random walk on a bi-dimensional random graph was studied. This model had been previously introduced in the physical literature. and studied numerically. It is established in [1] that, in contrast with the corresponding periodic graph, this random walk is transient with probability one with respect to the random environment. Subsequently several papers appeared on related models ([3], [4] [5] [6]). The results of [1] have been used in [7] to study a model of corner percolation on Z2. In [2] transience was obtained on a more general random environment and a transition from recurrence to transience has been proved to occur. At the moment work is going on in collaboration with G. Bosi to study the problem on a honeycomb random graph. [1] M. Campanino, D. Petritis. Random walks on randomly oriented lattices. Markov Process. Relat. Fields 9, 391-412 (2003). [2] M. Campanino, D. Petritis. Type transition of simple random walks on randomly directed regular lattices, J. Appl. Prob. 51, 1065-1080 (2014). [3] B. De Loynes. Marche aleatoire sur un di-graphe et frontière de Martin. C. R. Acad. Sci. Paris 350, 87-90 (2012). [4] A. Devuldier, F. Pene. Random walk in random environment in a two-dimensional stratified medium with orientation. Electron. J. Prob. 18, no. 18 (2013). [5] Guillotin-Plantard, A, Le Ny. A functional limit theorem for a 2D-random walk with dependent marginals. Electron. Commun. Prob. 13, 337-351 (2008). [6] Guillotin-Plantard, A, Le Ny. Transient random walks on 2D-oriented lattice. Theory Prob. Appl. 52, 699-711 (2008). [7] G. Pete. Corner percolation on Z2 and the square root of 17. Ann. Prob. 36, 1711-1747 (2008).

SPEAKER: Yosi Rinott, The Hebrew University and LUISS.

TITLE: Differential Privacy applied to common methods of dissemination of frequency tables

ABSTRACT: When official data are to be disseminated to the public, the agency that releases the data must guarantee $ It is not clear how to define and measure privacy. I will explain a notion developed in computer science known as Differen$ issues that arise in applying it to the dissemination of frequency tables.

SPEAKER: Mauro Piccioni (Sapienza)

TITLE: Conditional iid structures in exchangeable multivariate distributions: from Marshall-Olkin to 'load sharing' $

ABSTRACT: The discovery of a conditional iid structure allows the understanding of the dependence contained in a d-v$

SPEAKER: O. Blondel (Lyon, CNRS)

TITLE: More random walks on random walks

ABSTRACT: We consider a Poissonian distribution of particles performing independent simple random walks. Simultaneously, on top of this system, a random walker evolves with a drift to the right when it is on top of (at least) a particle, to the left when it is on an empty site. We obtain a LLN, CLT and large deviation bounds in high and low density. Joint work with Marcelo Hilario, Renato dos Santos, Vladas Sidoravicius, Augusto Teixeira.

SPEAKER: W. Woess (TU Graz, Austria)

TITLE: Multidimensional reflected random walk - some results and many questions

ABSTRACT: Let (Y_n,V_n) be i.i.d. distributed, with the components r and s-dimensional, respectively. Reﬂected random walk starting at a point x of the positive r-dimensional orthant is deﬁned recursively by X_0 = x, X_n = |X_{n−1}−Y_n|, where |(a_1,...,a_r)| = (|a_1|,...,|a_r|). In R^s, consider the ordinary sum S_n = V_1 +···+V_n . We are interested in (topological) recurrence of the process (X_n,v+S_n) starting at (x,v). While this is quite well understood for refelcted random walk with r=1, in higher dimension (r \geq 2) or with some non-reﬂected coordinates (s \in {1,2}) we have a few basic results and various open questions with some partial answers. This is work with Judith Kloas, with input from Marc Peigne' and Wojciech Cygan.

SPEAKER: Lorenzo Bertini (Univ. La Sapienza, Roma)

TITLE: Grandi deviazioni rispetto al moto per curvatura media

ABSTRACT: Si consideri l'equazione di Allen-Cahn in dimensione d=2 o d=3. Effettuando un riscalamento diffusivo, per dati iniziali opportuni, la dinamica limite dell'interfaccia tra le due fasi stabili e' descritta dal moto per curvatura media. Verra' introdotta una perturbazione stocastica di tale equazione e analizzata la corrispondente asintotica di grandi deviazioni nel limite di interfacce concentrate. Il corrispondente funzionale di tasso e' analogo a quello ottenuto analizzando la convergenza variazionale dei funzionali d'azione. La dimostrazione della stima di grandi deviazioni utilizza strumenti di teoria geometrica della misura.

SPEAKER: Valentina Cammarota (Sapienza Università di Roma)

TITLE: On the critical values of random spherical harmonics.

ABSTRACT: Abstract: We study the limiting distribution, in the high energy limit, of critical points and extrema of random spherical harmonics. In particular, we first derive the density functions of extrema and saddles and then we provide analytic expressions for the variances. Our arguments require a careful investigation of the validity of the Kac-Rice formula in nonstandard circumstances, entailing degeneracies of covariance matrices for first and second derivatives of the processes being analyzed. It is well known that after proper rescaling random spherical harmonics converge to Berry's random plane waves; in the second part of the talk we focus on the spatial distribution of critical points of random plane waves. Based on joint works with Dmitry Beliaev, Domenico Marinucci and Igor Wigman.

SPEAKER: Sokol Ndreca (University Center of Belo Horizonte)

TITLE: On a Problem of Kendall

ABSTRACT: Abstract: In this talk we consider a stochastic point process $i + \xi_i$, where $i\in \mathbb{N}$ and the $\xi_i's$ are i.i.d exponential random variables with standard deviation $\sigma$. Some properties of this process are investigated. We then study a discrete time single server queueing system with this process as arrival process and deterministic unit service time. We obtain a functional equation of the bivariate probability generating function of the stationary distribution for the system. The functional equation is quite singular, does not admit simple solution. We find the solution of such equation on a subset of its set of definition. Finally we prove that the stationary distribution of the system decays super-exponentially fast in the quarter plane. The queueing model, motivated by air and railway traffic, has been proposed by Kendall and others some five decades ago, but no solution of it has been found so far. This is a joint work with Gianluca Guadagni, Carlo Lancia and Benedetto Scoppola.

SPEAKER: D. Marinucci (Tor Vergata, Roma)

TITLE: A Quantitative Central Limit Theorem for the Euler-Poincaré Characteristic of Random Spherical Eigenfunctions

ABSTRACT: We establish here a Quantitative Central Limit Theorem (in Wasserstein distance) for the Euler-Poincaré Characteristic of excursion sets of random spherical eigenfunctions in dimension 2. Our proof is based upon a decomposition of the Euler-Poincaré Characteristic into different Wiener-chaos components: we prove that its asymptotic behaviour is dominated by a single term, corresponding to the chaotic component of order two. As a consequence, we show how the asymptotic dependence on the threshold level u is fully degenerate, i.e. the Euler-Poincaré Characteristic converges to a single random variable times a deterministic function of the threshold. This deterministic function has a zero at the origin, where the variance is thus asymptotically of smaller order. Our results can be written as an asymptotic second-order Gaussian Kinematic Formula for the excursion sets of Gaussian spherical harmonics. Based on a joint work with Valentina Cammarota.

SPEAKER: Mauro Mariani (Università La Sapienza)

TITLE: Metastable regimes of diffusion processes

ABSTRACT: I will discuss some old and new results concerning a classical example featuring a metastable behavior: finite-dimensional diffusion processes in the vanishing noise limit. Sharp estimates have been introduced during the '60s. More recently, similar ideas appeared in the context of potential theory and calculus of variations.

SPEAKER: Antonio Lijoi (Università di Pavia)

TITLE: Bayesian nonparametric inference with heterogeneous data

ABSTRACT: The talk provides an overview of some recent work on random probability measure vectors and their role in Bayesian statistics. Indeed, dependent nonparametric priors are useful tools for drawing inferences on data that arise from different, though related, studies or experiments and for which the usual exchangeability assumption is not satisfied. The presentation will focus on models based on completely random measures, or suitable transformations thereof, that have an additive, a hierarchical and a nested structure. Some of their distributional properties relevant for prediction will be discussed. These theoretical results are, then, used for devising Markov chain Monte Carlo algorithms that will be implemented within some illustrative examples for analyzing data in the contexts of species sampling problems and survival analysis.

SPEAKER: Pietro Caputo (Università di Roma Tre)

TITLE: Random walk on sparse random directed graphs

ABSTRACT: random walk on a finite graph exhibits cutoff if its distance from stationarity remains close to the initial value for a certain number of iterations and then abruptly drops to near zero on a much shorter time scale. Originally discovered in the context $

SPEAKER: Carlo Orrieri, Dipartimento di Matematica, Sapienza Università di Roma

TITLE: Controlled Vlasov-type dynamics

ABSTRACT: The aim of the talk is to connect the optimal control of Vlasov-type PDEs with large systems of controlled interacting particles/agents. The main difference towards the classical mean-field theory, where the particle are freely interacting with each other, is the presence of a central planner influencing the dynamics. In this situation it is not obvious that the mean field limit should commute with the optimization. In order to develop a rigorous limit theory for this problem we employ the well-known concept of Gamma convergence. The talk is based on a work in project with M. Fornasier, S. Lisini and G. Savarè

SPEAKER: Marc Peigné, Université de Tours, France

TITLE: Random dynamical systems: contraction and recurrence properties

ABSTRAC Consider a proper metric space X and a sequence (Fn) of i.i.d. random continuous mappings from X to X. It induces the stochastic dynamical system (SDS) Xn = Fn ∘ ∘ ∘ F1(x) starting at x ∈ X. We study existence and uniqueness of invariant measures, under some assumptions of contraction on the Fn, as well as recurrence and ergodicity of this process. We will consider two main examples: the case where the Fn are affine maps of the real line and the case where Xn is the reflected random walk on the positive real line.

SPEAKER: Pham Thi Da Cam, Université de Tours, France

TITLE: The survival probability of a critical multi-type branching process in i.i.d. random environment

ABSTRACT: We consider Galton Watson branching processes of unique type and multi-type in fixed and in random environment. The main target is to observe the asymptotic behaviour of the survival probability of the population. In particular, we utilise the generating function method thanks to the recursive structure of the process in deterministic case and condition to the generating function of offspring distribution in random case.

SPEAKER: C. Lefèvre, Université Libre de Bruxelles, Département de Mathématique

TITLE: Epidemic Risk and Insurance Coverage

ABSTRACT: This paper aims to apply simple actuarial methods to build an insurance plan protecting against an epidemic risk in a population. The studied model is an extended SIR epidemic in which the removal and infection rates may depend on the number of registered removals. The costs due to the epidemic are measured through the expected epidemic size and infectivity time. The premiums received during the epidemic outbreak are measured through the expected susceptibility time. Using martingale arguments, a method by recursion is developed to calculate the cost components and the corresponding premium levels in this extended epidemic model. Some numerical examples illustrate the effect of removals and the premium calculation in an insurance plan. This is a joint work with P. Picard (ISFA, Lyon) and M. Simon (ULB).

SPEAKER: Abram Kagan (Dept. of Mathematics, Univ. of Maryland, College Park and Dept. of Probability and Statistics, Charles University, Prague)

TITLE: On quantifying dependence between random elements

ABSTRACT: In the introduction some relatively recent results on the maximum correlation coefficient will be presented in two setups. The first deals with partial sums of independent identically distributed random variables, and the second with the components of a bivariate Gaussian stationary process. The main part of the talk is devoted to a discussion of the properties required from a reasonable numerical measure of dependence. My own contribution is in the construction of a calibrated scale of models starting at independence and going away from it in a specified direction.

SPEAKER: G. Letac (Institut Mathematique de Toulouse, Université Paul Sabatier, Toulouse)

TITLE: The Wishart distribution which never was

ABSTRACT: Replacing the integer parameter in the family of chi square distributions by a continuous parameter leads to the family of the gamma distributions. A similar phenomena occurs with the non central chi square. What happens in higher dimensions, where the natural extension of the chi square family is the Wishart family? The extension to a continuous parameter is clarified by the Gyndikin theorem (1975), with a beautiful short proof by Shanbhag (1989). The case of the non central Wishart is really difficult, and its explicit solution has been conjectured by E. Mayerhoffer (2010). We prove this conjecture by a detailed analysis of the zonal polynomials of symmetric matrices and of the convolution of measures concentrated on singular semi positive matrices (Joint work with H. Massam).

SPEAKER: Eleonora Perversi (Dipartimento di Matematica, Universita' di Pavia)

TITLE: Inequality and risk aversion in economies open to altruistic attitudes

ABSTRACT: In recent years there has been a great interest in the phenomenon of economic inequality, especially in relation to its connection with other important aspects of an advanced economy, for example attitude towards risk, growth, financial developments, and so on. In this talk, based on a joint work with Eugenio Regazzini, I will introduce a model for the surplus/deficit distribution, which points out a relationship between agents' risk aversion and inequality. More precisely, on the one hand, the long-time surplus/deficit distribution turns out to be a weak Pareto law whose exponent is given by an affine transformation of the agents relative risk aversion index, supposed to be the same for every agent. On the other hand, it is proved that concentration in a weak Pareto law can be measured through a function of its exponent. This way, a link is established between inequality and risk aversion. Finally, some feasible actions of economic policies suitable for the control of inequality are derived.

SPEAKER: Paul Chleboun (University of Warwick)

TITLE: Large deviations of the empirical current in zero-range processes on a ring.

ABSTRACT: We examine atypical current fluctuations in totally asymmetric zero-range processes in one dimension with periodic boundary conditions. The zero-range processes is a stochastic lattice gas in which each lattice site can be occupied by, a-priori, an unbounded number of particles. Particles move to their neighbour at a rate which only depends on the occupation of the departure site. For large systems, by calculating the Jensen-Varadhan action functional, we are able to find the time dependent optimal profiles which realise currents below the typical value. Under certain conditions on the jump rates, we demonstrate that these systems can exhibit a dynamical phase transition, in which above a critical non-typically current the optimal macroscopic density profile is given by a traveling wave with a shock and anti-shock pair. While rare events below the critical current are realised by a condensate, whereby a non-zero fraction of all the particles accumulate on a single site in the thermodynamic limit. This gives rise to a non-convex rate function for the empirical current, which in turn leads to a breakdown of the equivalence between the conditioned dynamics and the s-ensemble cloning methods typically used in simulations to sample these rare events.

SPEAKER: Alexandre Boritchev (Universite' de Lyon)

TITLE: 1D and multi-d Burgers Turbulence as a model case for the Kolmogorov Theory

Abstract: The Kolmogorov 1941 theory (K41) is, in a way, the starting point for all models of turbulence. In particular, K41 and corrections to it provide estimates of small-scale quantities such as increments and energy spectrum for a 3D turbulent flow. However, because of the well-known difficulties involved in studying 3D turbulent flows, there are no rigorous results confirming or infirming those predictions. Here, we consider a well-known simplified model for 3D turbulence: Burgulence, or turbulence for the 1D or multi-dimensional potential Burgers equation. In the space-periodic case with a stochastic white in time and smooth in space forcing term, we give sharp estimates for small-scale quantities such as increments and energy spectrum.

SPEAKER: Radko Mesiar (STU Bratislava)

TITLE: On the role of ultramodularity (and Schur concavity) in the construction of binary copulas

ABSTRACT: We discuss and stress the role of ultramodularity in special types of constructions of binary copulas. After recalling of some known ultamodularity-based results, we focus on a the so-called D-product of a copula and its dual. We show that for each copula D which is ultramodular and Schur concave on the left upper triangle of the unit square, this D-product of an arbitrary copula and its dual is again a copula. Several examples and counterexamples are given. Finally, some of our results are generalized to the case of semicopulas and quasi-copulas. Work in collaboration with E. P. Klement, A. Kolesarova and S. Saminger-Platz

Speaker: A. Erdely (Universidad Nacional Autonoma de Mexico)

Title: Modeling complex dependence using gluing and vine copulas

Abstract: Bivariate dependence may be of such complexity that no single family of known parametric copulas is able to give an acceptable goodnes of fit. The gluing copula approach may be of good help in decomposing a complex non monotone dependence into easier to model piecewise monotone dependencies. This is good news for the vine copula approach since bivariate copulas are building blocks of such approach in higher dimensions. An application to real data in economics and geophysics will be discussed.

Speaker: Oriane Blondel (Lione)

Title: Random walk on the East model (and other environments with spectral gap)

Abstract: The East model is a one-dimensional interacting particle system with non attractive spin-flip dynamics. In the physics literature, it is a key example of a model with glassy features. Here we take this model as a random environment and investigate the b$ Joint work with Luca Avena and Alessandra Faggionato.

Speaker: Vittoria Silvestri (Cambridge)

Title: Title: Fluctuation results for Hastings-Levitov planar growth

Abstract: In 1998 the physicists Hastings and Levitov introduced a family of continuum models to describe a range of physical phenomena of planar aggregation/diffusion. These consist of growing random clusters on the complex plane, which are built by iterated composition of random conformal maps. It was shown by Norris and Turner (2012) that in the case of i.i.d. maps the limiting shape of these clusters is a disc: in this talk I will show that the fluctuations around this shape are given by a random holomorphic Gaussian field F on {|z| > 1}, of which I will provide an explicit construction. When the cluster is allowed to grow indefinitely, I will show that the boundary values of F converge to a distribution-valued Fractional Gaussian Field on the unit circle, which is log-correlated, and critical in a sense that I will explain.

Speaker: Pierre Mathieu (Marsiglia)

Title: Random walks on (hyperbolic) groups.

Abstract: the first part of the talk will be an introduction to the general theory of random walks on groups with some classical results on entropy, rate of escape ... . For hyperbolic groups, these probabilistic objects have geometric counterparts in terms of Gromov boundary, quasi-conformal measures ... I will then discuss fluctuation results, in particular a recent work with A. Sisto on deviation inequalities for random walks on acylindrically hyperbolic groups.

Speaker: Yan Shu (Parigi)

Title: Hamilton Jacobi equations on graphs and applications

Abstract: We introduce a notion of gradient and an infimal-convolution operator that extend properties of solutions of Hamilton Jacobi equations to more general spaces, in particular to graphs. As a main application, the hypercontractivity of this class of in infimal-convolution operators is connected to some discrete version of the log-Sobolev inequality and to a discrete version of Talagrand's transport inequality.

Speaker: Max Fathi (Parigi)

Title: A gradient flow approach to large deviations for diffusion processes

Abstract: In the 80s, De Giorgi introduced the notion of abstract gradient flows, which allowed to define a notion of solutions to ordinary differential equations of the form x' = −grad F(x) on metric spaces (rather than Riemannian manifolds for the usual definition). In 2005, Ambrosio, Gigli and Savare showed that when we consider the space of probability measures on R d endowed with the Wasserstein metric, this notion allows to give an alternate formulation for Fokker-Planck equations. These equations are the PDEs whose solutions are the flow of marginals of solutions of stochastic differential equations of the form dX = −grad H(X)dt + dB . In this talk, I will explain how we can use this notion to study large deviations for sequences of SDEs. The main result is that proving a large deviation principle is equivalent to studying the limit of a sequence of functionals that appear in the abstract gradient flow formulation for Fokker-Planck equations. As an application, I will show how to obtain large deviations from the hydrodynamic scaling limit for a system of interacting continuous spins in a random environment.

Speaker: Andrea Puglisi (Sapienza Università di Roma)

Title: Dynamics of an intruder in a granular fluid: from dilute to dense experiment.

Abstract: A single experimental setup is the occasion for a tour into many issues in non-equilibrium statistical mechanics. In the experiment, a rotating intruder performs a Brownian-like dynamics under the influence of collisions with a shaken granular media. Dissipations in the form of inelastic collisions and dry tangential friction make the system inherently out of equilibrium. When the granular medium is diluted, the results agree with a Boltzmann-Lorentz description which, in the large mass limit, is well approximated by a Fokker-Planck equation for a Ornstein-Uhlenbeck process with Coulomb friction. When the intruder is shaped so as to break the symmetry under rotation-inversion, an average drift ('ratchet' or 'motor' effect) is observed, with properties depending on the dominant dissipation: friction or inelastic collisions. When the density of the surrounding medium increases, non-Markovian effects appear. The first consequence is a violation of the Einstein relation which -near equilibrium- describes the linear response to a small force provided by an external motor. The analysis of a generalized fluctuation-dissipation relation explains the nature of the violation: a joint effect of dissipation and coupling with the dense fluid. When the density is increased further and the jamming transition is approached, anomalous diffusion appears in the form of transient cage effects (subdiffusion) and -more surprisingly- superfdiffusion at large times.

Speaker: E. Kosygina (Baruch College and the CUNY Graduate Center)

Title: On the connection between homogenization of some stochastic Hamilton-Jacobi-Bellman equations and large deviations of diffusions with random drift in a random potential

Abstract: The connection mentioned in the title is well-known (see, for example, A.-S. Sznitman, Brownian motion in a Poissonian potential, 1993, PTRF). The goal of the talk is to review some necessary facts from large deviations theory and explain that this connection is actually an equivalence.

Speaker: E. Kosygina (Baruch College and the CUNY Graduate Center)

Title: Excited random walks in Markovian cookie environments on Z

Abstract: We consider a nearest-neighbor random walk on Z whose probability ω(x, n) to jump to the right from site x depends not only on x but also on the number of prior visits n to x. The collection (ω(x, n)) is sometimes called the “cookie environment” due to the following informal interpretation. Upon each visit to a site the walker eats a cookie from the cookie stack at that site and chooses the probability to jump to the right according to the “flavour” of the cookie eaten. Assume that the cookie stacks are i.i.d. and that only the first M cookies in each stack may have “flavour”. All other cookies are assumed to be “plain”, i.e. after their consumption the walker makes unbiased steps to one of its neighbours. The “flavours” of the first M cookies within the stack can be different and dependent. We discuss recurrence/transience, ballisticity, and limit theorems for such walks. The talk is based on joint works with Dolgopyat (University of Maryland), Mountford, Zerner.

Speaker: Stefano Olla (CEREMADE, Paris-Dauphine)

Title: Diffusione e superdiffusione dell'energia in catene di oscillatori

Abstract: In catene unidimensonali tipo FPU la conduttività termica e' infinita e ci si aspetta una superdiffusione dell'energia. In una catena di oscillatori armonici con collisioni stocastiche conservano l'energia e il momento, dimostriamo che il profilo di temperatura evolve macroscopicamente seguendo una equazione di diffusione frazionaria.

Speaker: Stella Brassesco (Instituto Venezolano de Investigaciones Científicas)

Title: The winding number of planar Brownian motion

Abstract: From an explicit formula for the joint density of the radial part and the winding number of a planar Brownian motion, we obtain asymptotic expansions (as t tends to infinity) for the density of the winding number. In particular, this expansion yields corrections of any order (in inverse powers of log(t)) to Spitzer's law (Trans. Am. Math. Soc, 1958) and to a local limit theorem proved by Delbaen, Kowalski and Nikehbali (Int. Math. Research Notices, 2014).

Speaker: Larry Goldstein (University of Southern California)

Title: Normal approximation for recovery of structured unknowns in high dimension: Steining the Steiner formula

Abstract: Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications. In particular, the discrete probability distribution L(VC) given by the sequence v{0},...,v{d} of conic intrinsic volumes of a closed convex cone C in Rd summarizes key information about the success of convex programs used to solve for sparse vectors, and other structured unknowns such as low rank matrices, in high dimensional regularized inverse problems. The concentration of VC implies the existence of phase transitions for the probability of recovery of the unknown in the number of observations. Additional information about the probability of recovery success is provided by a normal approximation for VC. Such central limit theorems can be shown by first considering the squared length GC of the projection of a Gaussian vector on the cone C. Applying a second order Poincar´e inequality, proved using Stein’s method, then produces a non-asymptotic total variation bound to the normal for L(GC). A conic version of the classical Steiner formula in convex geometry translates finite sample bounds and a normal limit for GC to that for VC. Joint with Ivan Nourdin and Giovanni Peccati.

Speaker: Kirone Mallick (CEA, Saclay)

Title: Fluctuations and large deviations in non equilibrium systems

Abstract: Studies of non-equilibrium fluctuations have held a center stage during the past few decades of the development of non-equilibrium statistical mechanics. Indeed, macroscopic fluctuations are supposed to display some universal behaviour, regardless of the precise microscopic dynamics of the process under study. In this talk, we shall consider a few examples of non-equilibrium fluctuations: the particle current in an interacting particle system, the diffusion of a tracer in a one-dimensional gas of particles with excluded mutual passage and the melting of an Ising quadrant. These different problems will be analyzed quantitatively with the help of the macroscopic fluctuation theory.

Speaker: MARCO RIBEZZI CRIVELLARI ((Universita' di Barcellona)

Title: Experimental measurements of entropy production at the nano-scale: an application for the 'fluctuation theorems'

Abstract: The study of non-equilibrium systems has led to several mathematically rigorous and general results on the statistics of entropy production in non-equilibrium systems. These results are generally known under the name of 'fluctuation theorems' and include the Gallavotti-Cohen theorem or the Jarzynski equality. My research focuses on performing experiments in which such results can be tested in real physical systems. I am interested in assessing their range of validity but even more interested in exploiting them to develop new measurement techniques. I will present some recent experiments, in which focused laser beams (optical tweezers) are used to perform thermodynamic transformations on single DNA molecules. As a first example I will show how to use fluctuation theorems to measure the free energy change across the transformation. As a second example I will discuss how, in specific cases, fluctuation theorems can be used to measure the full entropy production in a nano-scale system starting from a partial measurement through what we call an 'inference' procedure. This last example provides a new and general application of fluctuation theorems which we are only beginning to explore.

Speaker: ALESSANDRA BIANCHI

Title: A random walk in a Levy random environment

Speaker: FABRIZIO LEISEN

Title: Dependent Vectors of Random Probability Measures

Abstract: The definition of vectors of dependent random probability measures is a topic of interest in applications to Bayesian statistics. Indeed, they can be used for identifying the de Finetti mixing measure in the representation of the law of a partially exchangeable array of random elements taking values in a separable and complete metric space. In this talk we describe the construction of vectors of random probability measures based on the normalization of an exchangeable vector of completely random measures that are jointly infinitely divisible. The dependence can be achieved in many ways and in this talk will be shown some recent constructions based on Levy Copulas or the stick breaking representation. Finally, the dependence structure of the vectors is studied through some quantities of interest. This talk is a review of papers in collaboration with Antonio Lijoi, Federico Bassetti, Roberto Casarin, Weixuan Zhu and Dario Spano

Speaker: JULIEN SOHIER (Eindhoven - The Netherlands)

Title: A comparison between different cycle decompositions for Metropolis dynamics

Abstract: In the last decades the problem of metastability has been attacked on rigorous grounds via many different approaches and techniques which are briefly reviewed in this paper. It is then useful to understand connections between different point of views. In view of this we consider irreducible, aperiodic and reversible Markov chains with exponentially small transition probabilities in the framework of Metropolis dynamics. We compare two different cycle decompositions and prove their equivalence.

Speaker: CLAUDIO LANDIM (IMPA Rio de Janeiro, CNRS Rouen)

Title: Zero-temperature limit of the Kawasaki dynamics for the Ising lattice gas in a large two-dimensional torus

Abstract:We consider the Kawasaki dynamics at inverse temperature beta for the Ising lattice gas on a two-dimensional square of length 2L+1 with periodic boundary conditions. We assume that initially the particles form a square of length n, which may increase, as well as L, with beta. We show that in a proper time scale L^2, theta_beta particles form almost always a square and that this square evolves as a Brownian motion when the temperature vanishes.

Speaker: PAUL CHLEBOUN (Warwick University)

Title: Large deviations and metastability in a size-dependent zero-range process.

Abstract: We discuss a general approach to understand phase separation and metastability in stochastic particle systems that exhibit a condensation transition. Condensation occurs when, above some critical density, a finite fraction of all the particles in the system accumulate on a single lattice site. We present a detailed analysis of a particular size-dependent zero-range process which was introduced as a toy model for clustering in granular media. This model also captures all the relevant details of more generic condensing zero-range processes close to the critical point. Results on the equivalence of ensembles and metastability are based on large deviation principles for the maximum of triangular arrays of independent random variables conditioned on their sum. We derive the saddle point structure of the associated free energy landscape, which implies different mechanisms for the dynamics of the condensate depending on the system parameters. These results lead us to an interesting conjecture on the stationary dynamics of the condensate in the thermodynamic limit.

Speaker: LUCA AVENA, WIAS, Berlin (Germany)

Title: A local CLT for some convolution equations with applications to self-avoiding walks

Abstract: We discuss a fixed point method to obtain a local central limit theorem for distributions defined by certain renewal type equations. The main motivation for investigating these equations stems from applications to lace expansions, in particular to high dimensional weakly self-avoiding walks. As an application we introduce and treat such self-avoiding walks in continuous space. The error bounds obtained are sharper that the ones obtained by other methods. Joint work with E. Bolthausen and C. Ritzmann.

Speaker: GUSTAVO POSTA (Politecnico di Milano, Università La Sapienza)

Title: Decadimento dell'entropia per sistemi markoviani interagenti

Abstract: Verrà illustrata una tecnica per ottenere disuguaglianze funzionali che implicano il decadimento dell'entropia per alcune catene di Markov a tempo continuo. La tecnica è ispirata alle idee di Bakry ed Emery in questo contesto. Si otterranno stime del tempo di mixing, uniformi nel volume, per alcuni sistemi di particelle interagenti.

Speaker: LORENZO CARVELLI (Università La Sapienza)

Titile: Restart di algoritmi meta-euristici per problemi di ottimizzazione combinatoria

Abstract: La soluzione di problemi di ottimizzazione combinatoria di grande dimensione, come ad esempio il problema del commesso viaggiatore con migliaia di città, richiede spesso l'uso di algoritmi meta-euristici. Questi algoritmi sono di tipo generale e non dipendono dal particolare problema. Tra essi ci sono il Simulated Annealing, gli algoritmi genetici e quelli Ant Colony. Per diminuire le possibilità che alla fine dell'esecuzione l'algoritmo fornisca una soluzione subottimale, viene spesso utilizzata la tecnica del "restart" che consiste nell'inizializzare periodicamente l'algoritmo in modo random. Nonostante il restart sia molto utilizzato in pratica, ci sono pochi studi teorici al riguardo. La scelta del tempo di restart viene quindi effettuata sulla base di criteri empirici. In questo seminario, dapprima illustrerò alcuni risultati teorici del mio lavoro di tesi sulle condizioni in cui il restart migliora le possibilità di successo dell'algoritmo meta-euristico e sulla scelta del tempo di restart. I risultati teorici non sono purtroppo utili nella pratica in quanto basati sulla conoscenza della funzione di sopravvivenza del tempo necessario a trovare la soluzione tramite l'algoritmo meta-euristico. Successivamente descriverò una nuova procedura iterativa per l'ottimizzazione in tempo reale del restart dimostrandone la convergenza. La procedura da me proposta non è basata sulla conoscenza della soluzione del problema. Questi due fatti permettono di applicarla con successo. Applicherò infine la procedura ad un algoritmo Ant Colony per trovare la soluzione di alcune istanze del problema del commesso viaggiatore con centinaia o migliaia di città.

Speaker: FABIO CAMILLI (SBAI, Università La Sapienza)

Title: Equazioni di Hamilton-Jacobi su networks

Abstract: Si discuteranno alcuni recenti risultati riguardanti le equazioni di Hamilton-Jacobi definite su un network. Per equazioni ellittiche su networks si evidenzierà il ruolo chiave delle condizioni di transizione di tipo Kirchhoff sui nodi interni, ma si mostrerà che tale approccio non si adatta alle equazioni di Hamilton-Jacobi. Si introdurrà invece un'appropriata definizione di soluzione viscosità per caratterizzare la soluzione significativa del problema.

Speaker: PIETRO CAPUTO (Università Roma Tre)

Title: Large deviations in sparse random graphs: a local weak convergence approach

Abstract: Consider the Erdös-Renyi random graph on n vertices where each edge is present independently with probability p=c/n, with c>0 fixed. For large n, a typical realization locally behaves like the Galton-Watson tree with Poisson offspring distribution with mean c. We discuss large deviations from this typical behavior, within the framework of the local weak convergence introduced by Benjamini-Schramm and Aldous-Steele. The associated rate function is expressed in terms of an entropy functional on unimodular measures and takes finite values only at measures supported by trees. Along the way, we present a new configuration model which allows one to sample uniform random graphs with a given finite neighborhood distribution, provided the latter is supported by trees. We also present a new class of unimodular random trees, which generalizes the Galton-Watson tree with given degree distribution to the case of neighborhoods of arbitrary finite depth. This is joint work with Charles Bordenave.

Speaker: ALESSANDRA FAGGIONATO (Università La Sapienza)

Title: Fluctuations of random walks on quasi 1D lattices and applications to biophysical systems

Abstract: A broad class of kinetic models for molecular motors is given by random walks on quasi 1D lattices with random holding times, not necessarly exponential. We derive information on the asymptotic velocity (law of large numbers), gaussian fluctuations (invariance principle) and large fluctuations (large deviation principle). As applications, we consider some special models and give explicit formulas. We also discuss some theoretical results on Gallavotti-Cohen type symmetries for molecular motors. Joint work with D. Di Pietro, V. Silvestri.

Speaker: DIMITRI KOROLIOUK (Accademia delle Scienze Ucraina)

Title: Binary statistical experiments with persistent regression

Abstract: We deal with a mathematical model of binary statistical experiments, based on statistical data, for the validation of the elementary hypothesis about the presence or absence of a predefined attribute A in the set of elements that make up a complex system. It is assumed that: 1) All the elements that make up the system can gain or lose the given attribute A over time, that is the frequency of the attribute A is a dynamic variable; 2) The basic object of the study are the statistical experiments, which are characterized by relative frequencies of the presence or absence of the attribute A in the sample of fixed volume at each time instant. 3) It is assumed the dependency of (average) results of the next experiment on the average result of the present experiment. This relationship is called the property of persistent regression and used as fundamental condition for the subsequent analysis of the model.

Speaker: MASSIMILIANO GUBINELLI (Università Dauphine, Parigi)

Title: Stochastic quantization, paraproducts and all that

Abstract: The stochastic quantization equation is a simple model for the kind of problems linked to locality and non-triviality of quantum field theories. In this talk we review recent advances in undestanding of the functional analytic structure of solutions to non-linear SPDEs and their application to the study of the stochastic quantization of a scalar field in 3 dimensions. These advances have been possible thanks to a generalization of the theory of (controlled) rough paths which allows a pathwise formulation to stochastic differential equations driven by irregular signal. In particular we discuss the role of multiscale decomposition of distributions and of the notion of paraproduct in the analysis of this problem.

Speaker: GIACOMO DI GESU' (Università La Sapienza)

Title: Spectral asymptotics for discrete metastable diffusions via Witten complex techniques

Abstract: I will consider a metastable diffusion moving in a multiwell potential on the rescaled n-dimensional integer lattice. From a purely spectral point of view metastability effects correspond to the presence of nearly degenerate small eigenvalues of the generator, each one linked to a well of the potential. I will present a result providing complete asymptotic expansions of these small eigenvalues. The proof, inspired by previous work of B. Helffer, M. Klein and F. Nier in continuous setting, is based on tools of semiclassical analysis (Harmonic approximation, WKB expansions) and on a supersymmetric extension à la Witten of the generator on the level of discrete 1-forms.