**March 6 ^{th}, 2020**

~~Location: (different location!) Aula Magna 327 Viale Morgagni 40, Firenze. Google maps link~~

**The conference has been postponed as a safety precaution for the coronavirus outbreak. A new date will be set when the circumstances allow it.**

**Morning lecturer: Massimiliano Gubinelli (Hausdorff Center for Mathematics)**

**Title: Universality in slow growth phenomena and singular martingale problems.
**

Abstract: In the last years there have been steady progress in understanding the large scale properties of one dimensional growing interfaces. In the regime where the growth is comparable to thermal fluctuations the interface are described via the Kardar-Parisi-Zhang equation (KPZ). This is a stochastic partial differential equation which contains a non-linearity whose precise meaning is not apriori clear. There are various analytic approaches to make sense of such an equation like regularity structures, paracontrolled distributions, rough paths. In this talk we will describe a further approach to the KPZ equation based on the probabilistic notion of martingale problem. This approach can be used to prove the scaling limit of interface fluctuations in a wide class of models. Due to the singular nature of the equation, the martingale problem has to be formulated in a non-standard way and several new ideas are needed to obtain a mathematically satisfactory theory. The aim of the first part of talk will be to give a wide perspective on the phenomenon of universality of the KPZ and related equations and in the issues involved in their definition. In the second part we will discuss more in details the well-posedness of the martingale problem, in particular the uniqueness problem.

**Afternoon lecturer: Julia Komjathy (Eindhoven University of Technology)**

**Title: How to stop explosion by penalising transmission to hubs.
**

Abstract:

In this talk we study the spread of information on infinite inhomogeneous spatial random graphs.

We take a scale-free spatial random graph, where the degree of a vertex follows a power law with exponent tau >1.

Examples of such graphs include: Scale free percolation, Geometric Inhomogeneous Random Graphs, and Hyperbolic Random Graphs.

Then we equip each edge with a random and iid transmission delay L, and study the ball-growth of the first-passage infected cluster around the source vertex as a function of time. For a second, more realistic spreading model, the iid random transmission delay L through an edge with expected degrees W and Z is multiplied by a factor that is a polynomial of W,Z, (the penalty factor).

We call the model outwards (inwards) explosive if it is possible to reach infinitely many vertices within finite time (if infinitely many vertices can reach a target vertex within finite time).

We will discuss the criterion for explosion in the original model (no penalty factor) and in the penalised model. In particular, we will discuss that asymmetric penalty functions can lead to `outwards’ explosion but no `inwards’ explosion or the other way round.

Joint work with John Lapinskas and Johannes Lengler.

**November 22 ^{th}, 2019**

**Location: Aula Tricerri**

**Morning lecturer: Sabine Jansen (LMU Munich)**

**Title: Large deviations and metastability for the Widom-Rowlinson model.
**

Abstract: The Widom-Rowlinson model is one of the few models in statistial mechanics for which a phase transition is rigorously proven. It is also popular in stochastic geometry and spatial statistics where it is called area interaction model and belongs to the broader class of quermass interaction models. After reviewing some relevant background about Gibbs measures for continuum interacting particle systems, I will discuss large deviations for the Widom-Rowlinson model in a joint high-density / low-temperature limit. I will also discuss metastability for a spatial birth and death process, a.k.a. continuum Glauber or grand-canonical Monte-Carlo, for which the Gibbs measure is reversible. Based on joint work with Frank den Hollander, Roman Kotecký and Elena Pulvirenti.

**Afternoon lecturer: Luca Avena (University of Leiden)**

**Title: Explorations of networks through random spanning forests: theory and applications.
**

Abstract:

David Wilson in the 1990s described a simple and efficient algorithm based on loop-erased random walks to sample uniform spanning trees and, more generally, weighted rooted trees or forests spanning a given graph. The goal of this lecture is to describe the resulting probability measure when Wilson’s algorithm is used to sample rooted spanning forests. This forest-measure has a rich, flexible and explicit mathematical structure which makes it a powerful tool to design different algorithms to explore a given network.

In the first part of the lecture, I will focus on fundamental aspects of this measure and how it relates to other objects of interest in statistical physics such as the well known Random-cluster model. I will in particular describe the main properties of related observables (e.g. set of roots, induced partition) which turn out to be determinantal processes with simple kernels and then discuss some progress in understanding related scaling limits.

The second part of the lecture will be devoted to applications. In particular, depending on time, I plan to discuss four different algorithms aiming at: (1) sampling well-distributed points in a graph, (2) coarse-graning a given network, (3) processing signals on graphs (a novel gaph wavelet transform), (4) estimating the spectrum of the graph Laplacian. The core of this lecture is based on different joint collaborations with the following colleagues and students: Castell, Gaudillere, Melot, Milanesi (Marselle), Quattropani (Rome), Driessen, Koperberg, Magrini (Leiden) , Amblard, Barthelme, Tremblay (Grenoble).

**September 27 ^{th}, 2019**

**Location: Aula Tricerri**

**Morning lecturer: Pierre Picco (Marseille)**

**Title: One-dimentional Ising model with long range interactions. A review of results.**

Abstract: In the first talk I will make an quick historical survey of the rigorous results on the one-dimensional Ising model with long-range interactions. A first part will be dedicated to uniqueness of the Gibbs states (Ruelle (1968); Dobrushin (1968); Bricmont, Lebowitz & Pfister (1986)) and the regularity of the free energy when the decay of the potential is fast +enough (Dobrushin (1973) Cassandro & Olivieri (1981) and its extensions in particular Capocaccia, Campanino & Olivieri (1983).

A second part will be dedicated to the existence of phase transition starting from the Kac-Thompson conjecture (1968) the Dyson results (1969), the Frohlich \& Spencer result (1982), the Imbrie result (1982) the Aizenmann, Chayes, Chayes & Newman result on the Thouless effect (1988), Imbrie & Newman result on the Berezinsky, Kosterlitz & Thouless transition (1988).

A third part will be dedicated to present results in the phase transition regime that started with Frohlich & Spencer (1981), Cassandro, Ferrari, Merola & Presutti (2001) and its extensions in particular by Cassandro, Merola, Picco & Rosikov (2014) on the definition of an interface and its fluctuations, and on a Minlos & Sinai theorem on the phase separation problem by Cassandro, Merola & Picco (2017).

In the second talk I will review heuristic arguments that were invoked to conjecture the existence of a phase transition at low temperature in particular the Landau argument. I will present toy models where the fluctuation of interfaces and localisation of the droplet in the Minlos & Sinai theory will be explained. I will give an algorithmic definition of one-dimensional contours of Cassandro, Ferrari, Merola & Presutti.

**Afternoon lecturer: Rui Pires da Silva Castro (Eindhoven)**

**Title: Testing for the presence of communities in inhomogeneous random graphs**

Abstract: Many complex systems can be viewed as a network/graph consisting of vertices (e.g., individuals) connected by edges (e.g., a friendship relation). Often one believes there is some sort of community structure, where some vertices are naturally grouped together (e.g., more densely connected between themselves than to the rest of the network). Much of the community detection literature is concentrated around methods that extract communities from a given network. Our goal in this work is different, and we attempt to understand how difficult is it to determine if a network has real communities. Furthermore, we are primarily interested in the case of small or very small communities, for which many existing results and methods are not applicable.

We cast this problem as a binary hypothesis test, where the null model corresponds to a graph without community structure, and the alternative model almost the same, but it also includes a planted community – that is, a small subset of the vertices has higher connection probability than under the null. The main question is to determine the minimal size and â€œstrengthâ€ of the planted community that will allow detection. The seminal work of Arias-Castro and Verzelen tackled this problem when the null model is a homogeneous random graph. In our work, however, we consider the case where the null model is inhomogeneous, as this is somewhat closer to realistic scenarios. In particular, we present a scan test and provide conditions under which it is able to detect the presence of a small community. These results are valid for a wide variety of parameter choices. Furthermore, we show that for some parameters choices the scan test is optimal, and no other test can perform better (e.g, detect smaller or weaker planted communities). Finally, we extend this scan test to adapt to many parameters of the model when the null is a rank-1 generalized random graph.

In the first part of the talk I will describe the above formulation and ensuing results, with illustrative examples and briefly touching upon the analytical methodology. In addition, I will discuss the related problem of characterizing cliques in rank-1 random graphs, which provides some insights on the role of inhomogeneity. The second part of the talk will go deeper into more technical aspects and ensuing insights. This presentation is based on joint work with Kay Bogerd and Remco van der Hofstad (https://arxiv.org/abs/1805.01688 and ongoing work).

**March 22nd, 2019**

**Location: Aula Tricerri**

**Morning lecturer: Giovanni Gallavotti (Roma)**

**Title: Statistical ensembles, entropy and probability in statistical mechanics, and extension to chaotic motions** (slides part one and slides part two)

Abstract: a historical view on the theoretical developments generated by Boltzmann’s attempt to find a mechanical interpretation of the second principle, from the action principle to the Boltzmann’s equation to phase transitions and their universality to the modern developments in the non-equilibrium thermodynamics. In the second part the case of fluid mechanics and an interpretation of viscosity and irreversibility will be analyzed and related to an extension of the statistical ensembles to non-equilibrium phenomena.

**Afternoon lecturer: Silke Rolles (Technical University of Munich)**

**Title: Processes with reinforcement **(slides)

Abstract: In 1986, Persi Diaconis introduced edge-reinforced random walk as a simple model for a tourist exploring an unknown city. Already then, he raised the question of recurrence and transience of this process on the d-dimensional integer lattice. Since edge-reinforced random walk is more likely to traverse edges that have been traversed often before and simple random walk is recurrent in dimension 2, recurrence of edge-reinforced random walk on the two-dimensional integer lattice may seem intuitively clear. However, a proof of this result was only found in 2015 by Sabot and Zeng. For dimensions larger or equal to 3 a phase transition between recurrence and transience was shown by Disertori, Sabot and Tarres in 2011 and 2014. In the talk I will give an overview of the subject and present some basic techniques. In particular, the edge-reinforced random walk is a mixture of reversible Markov chains with an explicitly known mixing measure. In a special case, this can be illustrated with an analogous result for the Polya urn.

**November 23rd, 2018 **(announcement)

** Location: Aula Tricerri**

** Morning lecturer: Giovanni Jona Lasinio (Roma)**

** Title: Singular stochastic partial differential equations** (slides)

Abstract: Singular stochastic partial differential equations (SSPDE) first appeared in rather special contexts like the stochastic quantization of field theories or in the problem of crystal growth, the well known KPZ equation. In the last decade these equations have been intensely studied giving rise to an important branch of mathematics possibly relevant for physics. This talk will review some aspects and open problems in the subject.

**Afternoon lecturer: Giambattista Giacomin (Paris)**

** Title: Infinite disorder renormalization fixed point: the big picture and one specific result** (slides)

Abstract: the natural question of the effect of a random environment («disorder») on phase transitions and critical phenomena has attracted a lot of attention. I will give an introduction to this domain of research via an overview of some of the physical predictions and of the mathematical approaches and challenges. I will in particular develop the notion of disorder relevance and irrelevance. The focus will be on a very basic class of statistical mechanics model – called pinning models – for which in the last years the mathematical work matched the physical counterpart and, in some cases, went beyond. Nevertheless, also for pinning models the results in the regime in which disorder is relevant are rather weak and many of the physical predictions do not appear to be solid or coherent. But the situation has evolved very recently and a certain consensus has grown in favor of a very strong smoothing effect of the disorder for this class of models when disorder is relevant. This is part of a very intriguing and challenging general physical picture. The aim of the second part of the seminar is to present a very specific pinning model in which we have been able to pinpoint this strong smoothing effect (work in collaboration with Quentin Berger and Hubert Lacoin, arXiv:1712.02261). I hope I will be able to explain why we could tackle this case (and not other ones) and to develop (or sketch) at least one of the main technical ideas that are at the center of our approach.

**September 14th, 2018 **(**announcement)**

** Location: Aula Magna via San Gallo**

** Morning lecturer: Stefano Olla (Université Paris Dauphine)**

** Hyperbolic Hydrodynamic Limits** (slides)

Abstract: I will present a review of old and new results (and open problems) concerning scaling limits for conservation laws in the hyperbolic space-time scale, for a system of anharmonic oscillators with external boundary tension. The macroscopic equation is given by the compressible Euler system, with corresponding boundary conditions. The problem is particularly challenging when shockwaves are present. Some results exists when the microscopic dynamics is perturbed by a conservative stochastic viscosity. Works in Collaboration with Stefano Marchesani (GSSI) and Lu Xu (CEREMADE).

** Afternoon lecturer: Raú Rechtman (Universidad Nacional Autónoma de México)**

** Title: Chaos and damage spreading in a probabilistic cellular automaton**

Abstract: Deterministic Boolean cellular automata (CA) are discrete maps F:B^N -> B^N, B={0,1}, x(t+1)=F(x(t)) with x in B^N, N large and t= 0,1,… . The vector x is the state of the cellular automaton with components x[i], i=0,…,N-1 the state of cell I. Each cell is connected to others, generally in a uniform and local way, and one can define an adjacency matrix a[ij]=1 is cell j is connected to cell i and zero otherwise. The global map F is determined by the parallel application of a local function f, such that x[i](t+1) = f(v[i](t)), where v[i] denotes the state of cells connected to cell i. Deterministic CA are thus the discrete equivalent of dynamical systems, and many concepts like trajectory (the sequence of configurations x (t)), fixed points and limit cycles can be used. There are cellular automata for which a small modification in an initial configuration propagates to the whole system, a situation similar to chaos in continuous systems, and indeed one can extend the concept of the largest Lyapunov exponent to deterministic CA using Boolean derivatives. One of the main inconvenient is that these systems do not have continuous parameters to be tuned, in order to study bifurcations.

In probabilistic cellular automata, the function f (and thus F) is defined in terms of transition probabilities so that deterministic CA can be seen as the extreme cases of probabilistic ones, when the transition probabilities are either zero or one. Probabilistic CA can be seen also as Markov chains, and one can observe interesting phase transitions after changing the transition probabilities that are therefore continuous control parameters.

A realization of a specific trajectory is determined by the extraction of one or more of random numbers for each cell. By extracting these numbers at the beginning of the simulation, for all cells and all times, one converts a probabilistic CA into a deterministic one, running over a quenched random field. One can therefore use the concepts of deterministic CA, like damage spreading and maximum Lyapunov exponent also for probabilistic CA, with the advantage of having the possibility of fine-tuning the control parameters.

In particular, we investigate a probabilistic cellular automaton which can be considered an extension of a model in the universality class of directed percolation models, but with two absorbing states. In the first part of the talk all the concepts mentioned above are defined and in the second part, the probabilistic cellular automaton is studied numerically. We show that the phase transitions when the order parameter is the average damage do not coincide with those found for the Lyapunov exponent and the reason of this is the presence of absorbing states.

**May 25th, 2018 **(announcement)

** Location: Aula Magna via San Gallo**

**Morning lecturer: Francis Comets (Université Paris-Diderot Paris 7)**

** Title: Cover time, cover process, random interlacements for random walk on the torus**

Abstract:

**Afternoon lecturer: Remco van der Hofstad (TU/e, Eindhoven Technical University)**

** Title: Ising models on random graphs**

Abstract: The Ising model is one of the simplest statistical mechanics models that displays a phase transition. While invented by Ising and Lenz to model magnetism, for which the Ising model lives on regular lattices, it is now widely used for other real-world applications as a model for cooperative behavior and consensus between people. As such, it is natural to consider the Ising model on complex networks. Since complex networks are modelled using random graphs, this leads us to study the Ising model on random graphs. In this talk, we discuss some recent results on the stationary distribution of the Ising model on locally tree-like random graphs. We start by giving an extensive introduction to random graph models for complex networks, to set the stage of the graphs on which our Ising models live. Real-world networks tend to be highly inhomogeneous, a fact that is most prominently reflected in their degree distributions having heavy tails as described by power laws. Due to the randomness of the graphs on which the Ising model lives, there are different settings for the Ising model on it. The quenched setting describes the Ising model on the random graph as it is, while the averaged quenched setting takes the expectation w.r.t. the randomness of the graph. As such, it takes the expectation of the Boltzman distribution, which is a ratio of an exponential involving the Hamiltonian, and the partition function. In the annealed setting, the expectation is taken on both sides of the ratio. These different settings each describe different physical realities. We discuss the thermodynamic limit of the Ising model, which can be used to define the phase transition in the Ising model on locally tree-like random graphs, by describing when spontaneous magnetization exists and when not, extending work by Dembo and Montanari. We give an explicit expression for the critical value and the critical exponents for the magnetization close to it. These critical exponents depend on the power-law exponent of the degree distribution in the random graph. We also discuss central limit theorems for the total spin in the uniqueness regime, as well as a non-classical limit theorem for the total spin at the critical point in the special setting of the annealed generalized random graph. This talk is based on several joint works with Sander Dommers, Cristian Giardina, Claudio Giberti and Maria Luisa Prioriello.

**March 16th, 2018 **(announcement)

** Location: Aula Magna via San Gallo**

** Morning lecturer: Frank den Hollander (Leiden University)**

** Title: Large deviations for the Wiener sausage **(slides)

Abstract: The Wiener sausage is the 1-environment of Brownian motion. It is an important mathematical object because it is one of the simplest non-Markovian functionals of Brownian motion. The Wiener sausage has been studied intensively since the 1970’s. It plays a key role in the study of various stochastic phenomena, including heat conduction, trapping in random media, spectral properties of random Schrödinger operators, and Bose-Einstein condensation. In these lectures we look at two specific quantities: the volume and the capacity. After an introduction to the Wiener sausage, we show that both the volume and the capacity satisfy a downward large deviation principle. We identify the rate and the rate function, and analyse the properties of the rate function. We also explain how the large deviation principles are proved with the help of the skeleton approach. Joint work with Michiel van den Berg (Bristol) and Erwin Bolthausen (Zurich).

**Afternoon lecturer: Giovanni Peccati (University of Luxembourg)**

** Title: Stein’s method and stochastic geometry** (slides)

Abstract: The so-called ‘Stein’s method’ for probabilistic approximations is a collection of powerful analytical techniques, allowing one to explicitly assess the distance between the distributions of two random objects, by using caracterizing differential operators. Originally developed by Ch. Stein at the end of the sixties for dealing with one-dimensional normal approximations under weak dependence assumptions, Stein’s method has rapidly become a crucial tool in many areas of modern stochastic analysis, ranging from random matrix theory and random graphs, to mathematical physics, geometry, combinatorics and statistics. In the first part of my talk, I will provide a self-contained introduction to Stein’s method for normal approximations, by focussing on some connection with generalised integration by parts formulae, both in a continuous and discrete setting. In the second part of my talk, I will present some recent applications of Stein’s method in stochastic geometry, with specific emphasis on the geometry of random fields, and on random geometric graphs.

**December 15th, 2017 **(announcement)

** Location: Aula Magna via San Gallo**

** Morning lecturer: Milton Jara (IMPA, Rio de Janeiro)**

** Title: Weak universality of the stationary KPZ equation**

Abstract: A basic question about Markov chains is the asymptotic behavior of integrals of some function of the chain along its trajectory. In the literature, those integrals are sometimes called ‘Birkhoff averages’ or ‘additive functionals’. In the first talk, I will introduce a general strategy to estimate moment generating functions of these additive functionals in terms of the relative entropy of the chain with respect to carefully constructed reference measures. In the second talk, I will explain how to use this strategy to prove that for a large class of weakly asymmetric stochastic systems, the density of particles is well approximated by the stationary KPZ equation. This proof does not require explicit knowledge of the stationary measures of the stochastic systems, which was a major drawback of previous results.

**Afternoon lecturer: Nikos Zygouras (University of Warwick)**

** Title: Combinatorial structures in KPZ stochastic models** (slides)

Abstract: It was proposed by Kardar, Parisi and Zhang in the 1980s that a large class of randomly growing interfaces exhibit universal fluctuations described mathematically by a nonlinear stochastic partial differential equation, which is now known as the Kardar-Parisi-Zhang or KPZ equation. Examples of physical systems which fall in this class are percolation of liquid in porous media, growth of bacteria colonies, currents in one dimensional traffic or liquid systems, liquid crystals etc. Surprisingly the fluctuations of such random interfaces are governed by exponents and distributions that differ from the predictions given by the classical central limit theorem and in dimension one are linked to laws emerging from random matrix theory. The link between random growth and random matrices (which still demands deeper investigation) is certain combinatorial structures. In these talks I review the current status of the field and describe some of the combinatorial structures and the advances (both older and more recent) that these have led to.

**May 26th, 2017 **(announcement)

**Location: Aula Magna via San Gallo**

** Morning lecturer: R. Fernandez (Mathematics Department, Utrecht)**

**Title: Signal description: process or Gibbs?** (slides)

Abstract: The distribution of signals such as spike trains is naturally modeled through stochastic processes where the probability of future states depend on the pattern of past spikes. Mathematically, this corresponds to distributions *conditioned on the past*. From a signal-theoretic point of view, however, one could wonder whether a more efficient description could be obtained through the simultaneous conditioning of past *and* future. Furthermore, such a formalism could be appropriate when discussing string without a particular “time” order, such as the distribution of DNA nucleotides, or even issues related to anticipation and prediction in neuroscience. On the mathematical level this double conditioning would correspond to a Gibbsian description analogous to the one adopted in statistical mechanics. In this talk I will introduce and contrast both approaches -process and Gibbsian based- reviewing existing results on scope and limitations of them.

**Afternoon lecturer: Robert Morris (IMPA, Rio de Janeiro)**

** Title: Monotone cellular automata** (slides)

Abstract: Cellular automata are interacting particle systems whose update rules are local and homogeneous. Since their introduction by von Neumann almost 50 years ago, many particular such systems have been investigated, but no general theory has been developed for their study, and for many simple examples surprisingly little is known. Understanding their (typical) global behaviour is an important and challenging problem in statistical physics, probability theory and combinatorics. In this talk I will outline some recent progress in understanding the behaviour of a particular (large) family of monotone cellular automata – those which can naturally be embedded in d-dimensional space – with random initial conditions. For example, in the case where a site updates (from inactive to active) if at least r of its neighbours are already active, these models are known as bootstrap percolation, and have been extensively studied for various specific underlying graphs. Apart from their inherent mathematical interest, the study of these processes is motivated by their close connection to models in statistical physics, and I will discuss some applications to a family of models of the liquid-glass transition known as kinetically constrained spin models.