Probability and Stochastic Analysis Seminar 

The stochastic Curie-Weiss model is probably the simplest example of a non-trivial Glauber dynamics. This model has been extensively studied, and in the high-temperature regime, Levin-Luczak-Peres computed the mixing time up to an optimal constant of order O(n). In the classical definition of mixing time, one takes the least-favorable case as initial condition of the dynamics. A natural question to tackle is the dependence of the mixing time on the initial condition of the dynamics. In order to solve this question, we develop a new framework, which we call sharp convergence. We show sharp convergence of the Curie-Weiss model in the whole high-temperature regime, including non-zero magnetic field, and as an application we compute the mixing time of the Curie-Weiss model up to order o(n), and we also show that the mixing time improves if we take initial conditions with the ‘right’ density. Joint work with Freddy Hernández (Universidad Nacional de Colombia)

We consider the classical 2-opinion dynamics known as the voter model on finite graphs. It is well known that this interacting particle system is dual to a system of coalescing random walkers and that under so-called mean-field geometrical assumptions, as the graph size increases, the characterization of the time to reach consensus can be reduced to the study of the first meeting time of two independent random walks starting from equilibrium.

As a consequence, several recent contributions in the literature have been devoted to making this picture precise in certain graph ensembles for which the above mentioned meeting time can be explicitly studied. I will first review this type of results and then focus on the specific geometrical setting of random regular graphs, both static and dynamic (i.e. edges of the graphs are rewired at random over time), where in recent works we study precise first order behaviour of the involved observables. We will in particular show a quasi-stationary-like evolution for the discordant edges (i.e. with different opinions at their end vertices) which clarify what happens before the consensus time scale both in the static and in the dynamic graph setting. Further, in the dynamic geometrical setting we can see how consensus is affected as a function of the graph dynamics.

Based on recent and ongoing joint works with Rangel Baldasso, Rajat Hazra, Frank den Hollander and Matteo Quattropani.

The Averaging process (a.k.a. repeated averages) is a 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. At each discrete time an edge is sampled uniformly at random, and 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. Despite the simplicity of its formulation, even in the case of seemingly straightforward geometries—such as the complete graph or the 1-d torus— the analysis of the mixing behavior of this process can be handled only by means of non trivial probabilistic and functional analytic techniques. In the talk I will focus on the (mean-field) hypergraph case, in which edges are replaced by a more general notion of “blocks”. I will discuss the speed of convergence to equilibrium as a function of the blocks size, providing sharp conditions for the emergence of the cutoff phenomenon, i.e., a dynamical phase transition in which equilibrium is reached abruptly on a given time scale.

The talk is based on joint works with Pietro Caputo (Roma) and Federico Sau (Trieste).

Dense-packing problems on two and three dimensional lattices will be discussed, and their role will be shown in the analysis of the phase diagram of the hard-core model. Namely, high-density pure phases/extreme Gibbs distributions of the model are generated by certain ground states - in accordance with the Pirogov-Sinai theory - which are dense-packings for this model in the high-density regime.

In 1987, Harris and Vickers [HV87] proposed a model of a race in which two firms invest resources, each trying to be the first to secure a patent. They called the model tug of war. A counter moves randomly left or right on an integer interval, with the odds of a rightward move at each turn determined by the resources expended by each of the firms at the turn. The game ends when the counter reaches one or another end of the interval, the patent thus accorded to one or another firm. In 2009, Peres, Schramm, Sheffield and Wilson [PSSW09] independently introduced a similar game, which they also named tug of war. Two players also push a counter on a board, with a trivial rule for turn victor selection, via the toss of a fair coin; but also with a much richer geometric setting: by considering the game played in the limit of small step size in a domain in Euclidean space, the value of the game was found to be infinity harmonic, namely to satisfy an infinity version of the usual Laplace equation. These two works, HV87 and PSSW09, have unleashed two big but thus far non-interacting waves of attention, from economists and mathematicians respectively. In this talk, we will discuss the Trail of Lost Pennies, which is a tug-of-war game played on the integers in which the resources that each player expends during the game are deducted from her terminal receipt. The solution of this game is remarkably sensitive to the relative incentives of the two players, with a change in incentive of order $10^{-4}$ being sufficient to fundamentally alter outcomes. Overall we will indicate how stake-governed tug-of-war may weave the two strands of research in economics and mathematics, with the geometrically richer mathematical setting being allied to the resource-based rules for turn victor that is characteristic of the economics research strand.

In this talk, I am going to present a generalization of the Porous Media Model (PMM) analogue of the Bernstein polynomial basis, in the context of gradient models. The PMM is a symmetric nearest-neighbour process associated with the Porous Media Equation, and the corresponding dynamics are kinetically constrained, in the sense that particles diffuse in the lattice under a set of conditions on local configurations. While in the PMM, the occupation values of two neighbouring sites are exchanged only if there are "enough" groups of particles around them, our generalized model describes a system where, at very low densities, there is no interaction, while at high densities, there is no space for movement. I am going to present the construction of the model and its main properties, and, if time permits, discuss its extension in a long-range interaction context.

In this talk, I will overview works on random walks in dynamical random environments. I will recall a result obtained in collaboration with Hilario and Teixeira and then I will focus on a work with Conchon--Kerjan and Rodriguez. Our main interest is to investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, with density in [0,1]. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole. We prove that the speed of the walk, seen as a function of the density, exists for all density but at most one, and that it is strictly monotonic. We will explain how this can be seen as a sharpness result and provide an outline of the proof, whose general strategy is inspired by techniques developed for studying the sharpness of strongly-correlated percolation models.

Bootstrap percolation is a classical statistical physics model displaying metastable behaviour. Let each site of the square lattice be infected independently with a fixed probability. At each round, infect each site with at least two infected neighbours and do not remove any infections. How long does it take before the origin is infected? We start by reviewing the rich history of this problem and some of the classical arguments used to tackle it. We then give a very precise answer to the above question in the relevant regime of sparse infection. The key to the proof is a new locality approach to bootstrap percolation, which also resolves the bootstrap percolation paradox concerning the failure of numerical predictions in the field. The talk is based on joint work with Augusto Teixeira available at https://arxiv.org/abs/2404.07903.

We consider a large class of supercritical spatially embedded random graphs, including among others long-range percolation and geometric inhomogeneous random graphs, and identify a single exponent zeta depending on the model parameters that describes the asymptotics of

1. the probability that the largest connected component is much smaller than expected;
2. the size of the second-largest component;
3. the distribution of the size of the component containing a distinguished vertex.

In the talk, I will explain the relation between the three quantities and give some intuition for the values of zeta in different regimes.

Joint work with Joost Jorritsma and Júlia Komjáthy.

The Directed Polymer in a Random Environment is a statistical mechanics model, which has been introduced (in dimension 1) as a toy model to study the interfaces of the planar Ising model with random coupling constants. The model was shortly afterwards generalized to higher dimensions. In this latter case, rather than an effective interface model, the directed polymer in a random environment can be thought of as modeling the behavior of a stretched polymer in a solution with impurities. The interest in the model model, triggered by its rich phenomenology, has since then generated a plentiful literature in theoretical physics and mathematics. An important topic for the directed polymer is the so-called localization transition. This transition can be defined in terms of the asymptotic behavior of the renormalized partition function of the model. If the finite volume partition function converges to an almost surely positive limit we say that weak disorder holds. On the other hand, if it converges to zero almost surely, we say that strong disorder holds. It has been proved that weak disorder implies that the distribution of the rescaled polymer converges to standard Brownian motion while some localization results have been proved under the strong disorder assumptions. Much stronger characterizations of disorder-induced localization have been obtained under the stronger assumption that the partition function converges to zero.

(joint with Stefan Junk, Gakushuin University)

We consider large systems of jump processes that interact locally with respect to an underlying (possibly random) graph. Such processes model diverse phenomena including the spread of diseases, opinion dynamics and gas dynamics. Under a broad set of assumptions, we show that the empirical measure satisfies a large deviation principle in the sparse regime, that is, when the sequence of graphs converges locally to a limit graph. As a corollary we establish (quenched) hydrodynamic limits for the sequence of interacting jump processes. In addition, for a sub-class of processes that include the SIR process, we obtain a fairly explicit characterization of this limit and provide numerical evidence to show that it serves as a good approximation for finite systems of moderate size.

This is based on various joint works with I-Hsun Chen, Juniper Cocomello and Sarath Yasodharan.

Thanks to the theories of Regularity Structures, Paracontrolled Distributions and Renormalisation we now have a robust framework for singular SPDEs, which are “sub-critical” in the sense of renormalisation. Recently, there have been efforts to approach the situation of “critical” SPDEs and statistical mechanics models. A first such treatment has been through the study of the two-dimensional stochastic heat equation, which has revealed a certain phase transition and has led to the construction of the novel object called the Critical 2d Stochastic Heat Flow. In this talk we will present some aspects of this model and its construction. We will also present developments relating to other critical SPDEs.
Parts of this talk are based on joint works with Caravenna and Sun and others with Rosati and Gabriel.

We derive the heat equation for the thermal energy under diffusive space-time scaling from a purely deterministic microscopic dynamics satisfying Newton equations perturbed by an external chaotic force acting like a magnetic field.

Joint work with Giovanni Canestrari and Carlangelo Liverani.

We study the $\Phi_2^3$-measure in the infinite volume limit. This Gibbs-type measure inspired by similar objects appearing in QFT also correspondis to the invariant measure for the nonlinear SPDE known as the stochastic quantization equations. We give sharp estimates for the partition function in the large torus limit which imply strong concentration of the field around zero.

Joint work with K. Seong.

Physicists and mathematicians have for decades been interested in the Abelian sandpile model (ASM), a simple dynamical system that exhibits self-organized criticality. In the present work we asked us the question: what function of a Gaussian field produces the same scaling limit of its joint moments as the height-one field of the ASM? The squared norm of the gradient of the discrete Gaussian free field (GFF) almost does the job, with a subtle yet important difference. It turns out that, if one replaces the variables by Grassmann (or fermionic) objects, we obtain the exact same moments, which begs the follow-up question: why? What is the relationship between these two models? We will show that this relationship lies on the uniform spanning tree (UST) model. This observation will allow us to extend the calculations to square lattices in any dimensions, as well as the triangular and hexagonal lattices in d=2, pointing towards a potential universality property. We also extend our results by giving exact closed-form expressions for the scaling limit of the PMF of the degree field of the UST in these lattices. Joint work with L. Chiarini (U Durham), A. Cipriani (UCL), R.S. Hazra (U Leiden), W.M. Ruszel (U Utrecht).

A cellular automaton describes a process in which cells evolve according to a set of rules. Which rule is applied to a specific cell only depends on the states of the neighboring and the cell itself. Considering a one-dimensional cellular automaton with finite range, the positive rates conjecture states that and under the presence of noise the associated stationary measure must be unique. We restrict ourselves to the case of nearest-neighbor interaction where simulations suggest that the positive rates conjecture is true. After discussing a simple criterion to deduce decay of correlations, we show that the positive rates conjecture is true for almost all nearest-neighbor cellular automatons. The main tool is comparing a one-dimensional cellular automaton to a properly chosen two-dimensional Ising-model. We outline a pathway to resolve the remaining open cases. This presentation is based on collaborative work with Maciej Gluchowski from the University of Warsaw and Jacob Manaker from UCLA

In this talk, we will consider two models for diffusing particles in time-dependent random environments: the discrete random walk in random environment (RWRE) and a continuum scaling limit of the RWRE called sticky Brownian motion. We will present some recent results on the weak convergence of both models to the KPZ equation in the moderate deviation regime. We will also discuss an application to the fluctuations of the maximal particle in these models. This is joint work with Sayan Das and Shalin Parekh.

Birkhoff's ergodic theorem is a cornerstone in Mathematics, with a huge range of applications. We present here an extended form of the multidimensional ergodic theorem with weights, which allows to derive large scale averaging of stationary ergodic random measures on $\mathbb{R}^d$, also when the testing observables are not compactly supported. This control at infinity of random measures plays a crucial role when analysing the large scale behaviour of RWs and IPSs on weighted random graphs on $\mathbb{R}^d$ built on simple point processes. In particular, this allows us to obtain homogenization and hydrodynamics under weaker conditions.

We will then discuss several examples in order to emphasize the universality of our results.

The model of directed polymers describe the behavior of a long, directed chain that spreads among an inhomogeneous environment which may attract or repulse the polymer. When the spacial dimension is larger than three, a phase transition occurs between diffusivity (high temperature) and localization (low temperature). On the other hand, in dimensions one and two the polymer is always localized. Dimension two is however critical, as one can recover a phase transition by letting the temperature tend to infinity under a specific parametrization (Caravenna-Sun-Zygouras 17’). In this talk, I will present some of the main results that are known about this scaling regime, and discuss the recent advances that have occurred in the past few years. In particular, I will describe some results that I have obtained with my coauthors (Anna Donadini, Shuta Nakajima and Ofer Zeitouni) on the diffusive phase and its relation to Gaussian logarithmically correlated fields. I will also discuss connexions of the model with the Kardar-Parisi-Zhang (KPZ) equation and the stochastic heat equation.

We introduce a novel variant of the exclusion process where particles make asymmetric nearest neighbor jumps across a bond (k,k+1) only when the site k-1 to the left of the bond is empty. This next-nearest-neighbor interaction significantly enriches the model's behavior. We show that for a system with periodic boundary conditions, ergodicity is ensured only for systems that are strictly less than half-filled. For half-filling the system segregates into two distinct ergodic components, and we provide the invariant measure for each component and prove that it is reversible. The combinatorial properties of this invariant measure are intimately related to the q-Catalan numbers, where q represents the asymmetry of the two elementary hopping events. Exploiting this relation allows us to extract the asymptotic behavior both in the strongly and weakly asymmetric regimes. We report a phase separation characterized by different critical exponents for which we provide an intuitive geometrical interpretation. This is joint work with Gunter Schütz.