Probability Days

Abstract:

Geometric network models formalize the natural idea that similar vertices are likely to connect. Therefore, geometric models capture many common structural properties of real-world networks. However, if one observes only the network connections, the presence of geometry is not always evident. Currently, triangle counts and clustering coefficients are the standard statistics to signal the presence of geometry. We show that triangle counts or clustering coefficients are insufficient because they fail to detect geometry induced by hyperbolic spaces, or in networks with power law degrees. We therefore introduce different statistics, based on weighted subgraph counts that can even detect geometry in the 'weak geometry' regime, where the geometric effects converge to zero.

Abstract:

Introductory lecture: The story of mean-field spin glasses

Mean-field spin glasses are paradigmatic examples of complex systems. They involve a large (N->inf) number of simple “spin” variables, which are subject to disordered random interactions that cause highly complex and fascinating collective behavior. They were introduced in statistical physics to model disordered magnetic materials – subsequently models with very different origins but fundamentally similar behavior were discovered in theoretical computer science and theoretical statistics. All together these form a kind of universality class of mean-field spin glass models, that spans several areas of science.

This talk will consist of a high-level narrative retelling of the history of mean-field spin glasses, including original motivations, the remarkable description of their complex behavior discovered by Giorgio Parisi (Nobel Prize in Physics 2021), and the subsequent effort to put it on a mathematically rigorous footing.

Advanced lecture: The Thouless-Andersson-Palmer (TAP) approach to mean-field spin glasses

The Thouless-Andersson-Palmer (TAP) approach to mean-field spin glasses provides a particularly attractive lens on the phenomena of the models, thanks to its geometric and potentially constructive nature. In the story of spin glasses the TAP approach has played an important, but ultimately supporting, role – in both physics and mathematics the main tools come from other very different approaches. This talk will be about ongoing efforts to turn the TAP approach into a self-contained mathematically rigorous theory of mean-field spin glasses, which is my main research interest in the area.

Abstract:

I will first review the generalized strong invariance principle and law of iterated logarithm for i.i.d. random variables with infinite variance, but still in the domain of a normal law (superdiffusive).

In the second part I will present recent results which naturally generalize the almost sure invariance principle and the law of iterated logarithm to superdiffusive processes with heavy dependencies, focusing on the set-up of the Lorentz gas with infinite horizon. This is joint work with Peter Balint.

Abstract:

In this talk we will discuss the contact process on graphs. We will introduce the model, which can be used to describe for example the spread of a disease, and discuss the classical results on critical infection rates on infinite graphs such as the square lattice. Then we shift to the contact process on finite graphs, and discuss the concept of exponential extinction time. We discuss some recent results on this, and also our own contribution. Finally, we discuss the matter of metastability and the corresponding metastable distribution. This is a largely unexplored part in the literature and as a teaser, we will discuss some conjectures based on relevant local considerations.

Abstract:

In the introductory lecture, I will discuss a bit of statistical physics background/motivations about "driven diffusive systems", and explain how they can be mathematically modeled, either via discrete Markov chains (interacting particle systems, like the "Asymmetric Simple Exclusion Process") or via stochastic PDEs (like the Stochastic Burgers equation). I will also give a panorama of expected and proven results about "large-scale Gaussian fluctuations in dimension d\ge2". This part will include no proofs or technical details.

In the more technical lecture, I will focus on the stochastic Burgers equation (and similar stochastic PDEs) in the critical dimension d=2. I will discuss what is the "weak coupling limit" and I will formulate a  theorem (based on 2 joint recent works ( arXiv:2304.05730, arXiv:2108.09046) with G. Cannizzaro, D. Erhard, M. Gubinelli in different combinations) that says that these equations, in the weak coupling regime, have a Gaussian scaling limit at large scales. I plan to explain a bit the ideas behind the proof. 

Abstract:

We consider a large class of spatially-embedded random graphs that includes among others long-range percolation, continuum scale-free percolation/geometric inhomogeneous random graphs and the age-dependent random connection model. Assume that the parameters are such that there is an infinite component. We identify the stretch-exponent zeta in (0,1) of the subexponential decay of the cluster-size distribution. That is, with C(0) denoting the number of vertices in the component of the vertex at the origin,

  P ( C is larger than k but finite ) = exp ( -k^zeta) as k tends to infinity

The value of zeta undergoes several phase transitions with respect to three main model parameters: the Euclidean dimension d, the power-law tail exponent tau of the degree distribution and a long-range parameter alpha governing the presence of long edges in Euclidean space.

In this seminar I will describe the connection to the second largest component, and present the key ideas of the proof. Based on joint work with Joost Jorritsma (Tu/e) and Dieter Mitsche (Lyon).

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.

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).

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.

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.

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).

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.

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).

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.

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.

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.