Seminário de Probabilidade e Análise Estocástica 

Exclusion processes with non-reversible boundary dynamics are one-dimensional interacting particle systems evolving on a finite lattice. These systems arise from the superposition of two different types of dynamics. In the bulk, particles evolve according to the usual symmetric simple exclusion dynamics, that is, at most one particle is allowed per site and jumps occur through nearest-neighbor interactions. Near the boundary, within a window of fixed size $l\geq 1$, particles may be created or annihilated according to rates depending on the local configuration in a finite neighborhood of the boundary. This choice of boundary dynamics is non-conservative and is considered under very general rates, allowing, in particular, the simultaneous creation of more than one particle. This flexibility creates a crucial distinction between our model and the simplified version first introduced in 2011 to describe boundary current fluctuations, where a very specific choice of boundary rates was considered. In this sense, our model generalizes earlier works by capturing not only non-linear evolutions of the particle density, but also, and more importantly, the surprising emergence of multiple stationary solutions to the hydrodynamic equation describing the density profile. In this talk, we define the model, discuss its hydrodynamic limit, and analyze several properties of the stationary solutions of the associated hydrodynamic equation. This presentation is based on joint works with Claudio Landim and João Pedro Mangi.

Take some array, waiting for some numbers to be written. For each column c, toss X(c) some Bernoulli random variable with parameter p. Besides, for each column c, select randomly, in any way, a cell S(c) in this column. The only independence assumption we make is that the random variables X(c) form an independent family. For each column c, when X(c) = 1, write "1" in the cell S(c). Is it always possible to fill the remaining cells of the array so that all cells are independent Bernoulli(p) variables? Studying this question permits to revisit some results from percolation theory. Based on a joint word with Rémy Poudevigne–Auboiron and Paul Rax.

We derive a scaling limit in law as N tends to infinity for the cover time by a simple random walk of the subgraph of the square lattice obtained by discretizing an N-scale blow up of a planar domain and adding a wired boundary. The limiting distribution is that of a Gumbel random variable shifted by an independent (random) quantity which is equal to the full mass of a variant of the critical Liouville Quantum Gravity Measure on the same domain. We also derive a limit in law for the rescaled location of the last visited vertex by the walk. Here the limit turns out to be precisely the (expected) critical Liouville Measure, normalized by its total mass. Both limits hold jointly with the limiting joint law explicitly described. These results resolve well-known open problems in the field, in the case of wired boundary conditions. The proof is based on comparison with the extremal landscape of the discrete Gaussian Free Field and, in particular, with that of the discrete Gaussian Free Field conditioned to have zero average. Joint work with Oren Louidor (Technion).

When studying systems of particles, the very first step before any qualitative analysis is to establish the well-posedness of the dynamics of the system. In the case of hard spheres, whose trajectories are piecewise affine, the singularities arising at collision events prevent the direct use of Cauchy-Lipschitz-type of arguments. This issue was addressed by Alexander (1975) in the elastic case, where the kinetic energy is conserved during the collisions. For dissipative systems, the question remains largely open, due to the possibility that infinitely many collisions take place in finite time, a phenomenon known as inelastic collapse. We will discuss the case of a particular class of inelastic hard sphere systems, in which a fixed amount of kinetic energy is lost in each sufficiently energetic collision. The results were obtained in collaboration with Juan J. L. Velázquez (Universität Bonn), and are published in arXiv:2403.02162v2 (to appear in Communications in Mathematical Physics).

We investigate the Stochastic Heat Equation (SHE) in the critical space dimension two, where it is ill-defined. A non-trivial solution, known as the critical 2D Stochastic Heat Flow (SHF), can be constructed through regularisation and renormalisation. We investigate the SHF in the strong-disorder regime, showing that it vanishes locally and identifying the spatial scale governing the transition from extinction to an averaged behavior. Corresponding results are established for the partition functions of 2D directed polymers, which shed light into the SHE regularised via space-time discretisation: when the disorder strength is kept fixed, the solution exhibits fluctuations on a superdiffusive scale. Based on joint works with Quentin Berger and Nicola Turchi.

Mixing times are a classical tool to describe the speed of convergence to the stationary distribution for Markov chains. In this talk, we investigate mixing times for exclusion processes. More precisely, we discuss recent progress on determining the exact transition from unmixed to mixed, called the limit profile, as well as open questions.
This talk is based on joint work with Jimmy He, and with Sam Olesker-Taylor.

The aim of this talk is to construct a measure-valued Markov process, representing an evolving population, such that its genealogies are given, in some convenient sense, by the Lambda-coalescents (Pitman 1999). Our starting point for this work is the celebrated duality between Fleming-Viot processes and coalescents (Perkins 1992. Bertoin and Le Gall 2003). Our motivation comes from several recent works establishing such connections when the evolving process has the branching property. We are able to extend this duality between two Markov additive processes, a forward one related to the Fleming-Viot process and a backward one related to the coalescent. After some transformation and time change of the forward process, we obtain a wide new class of measure-valued self-similar Markov processes, characterized by the self-similarity index and a Lévy triplet, that fulfills the desired conditions. This provides a change of paradigm in population genetics where the branching property is let aside. This is a joint work with Alejandro H. Wences.

Anderson's tight binding model predicts that a random Schrodinger operator on Z^d exhibits a transition from delocalized to localized behavior as the disorder strength increases. This conjecture has motivated the study of non-mean-field random matrix models that exhibit similar properties while being more accessible to analysis. In this talk, I will present recent results establishing delocalization and bulk universality for several such models in dimensions d\ge 3: random band matrices, block Anderson model, and Wegner orbital model. Based on joint work with Fan Yang, Horng-Tzer Yau, and Jun Yin.

In this talk, we will present some recent results concerning the vector-valued mean-field spherical model in a random external field, which is a mean-field generalization of the Berlin-Kac model subject to an additional external random field term in the Hamiltonian. Through the random field term, the Gibbs measures become random Gibbs measures, and their convergence in the infinite volume limit can be studied in different probabilistic modes. For this particular model, we are able to characterize exactly the limit points, convergence in distribution, and convergence in empirical distributions. The properties of the model are closely connected to corresponding limit theorems for random walks, and they exhibit similar dimensional dependence for phenomena like transience and recurrence. Time permitting, we will also discuss the spin glass features of the model. This talk is based on the joint work with Professor Christof Külske.

The seminal Edwards-Sokal coupling allows to express correlations in the Potts model via connection probabilities in the random-cluster model. In this talk we discuss how similar ideas can be applied in a number of other settings. Specifically:

• the planar Potts model can be related to the Ashkin-Teller model, and this brings new results for both models, including convergence of interfaces to Brownian bridges and the wetting phenomenon (j.w. Moritz Dober and Sébastien Ott);
• the loop $O(n)$ model can be resampled via a divide-and-color procedure, and this can be used to establish a big part of its phase diagram.

Another important ingredient is our proof of a conjecture of Benjamini and Schramm, in particular:

• we show that $p_c$ is at least 1/2 on any unimodular invariantly amenable planar graph (j.w. Matan Harel and Nathan Zelesko);
• we also show delocalisation of two-dimensional height functions related to these models (j.w. Piet Lammers).

How do competing pathogen strains evolve within and across a population of hosts? We propose a simple stochastic model in which the type composition within each host evolves according to a family of Markov kernels. When hosts evolve independently, the model reveals a moment duality with genealogies related to the Ancestral Selection Graph and, under suitable scaling, converges to a Wright–Fisher diffusion with drift. When hosts interact through the population distribution, the system becomes weakly interacting. We prove propagation of chaos and show that the dynamics of a typical host converge to a McKean–Vlasov diffusion. As an illustration, we consider mutation rates depending on the current population state and study ergodicity of the resulting mean-field dynamics. This talk is based on join work with Leonardo Videla (Universidad de Santiago) and Héctor Olivero (Universidad de Valparaiso).

The Stochastic Burgers Equation (SBE) is a singular, non-linear Stochastic Partial Differential Equation (SPDE) which was introduced in the eighties by van Beijren, Kutner and Spohn to describe, on mesoscopic scales, the fluctuations of stochastic driven diffusive systems with one conserved scalar quantity. Following a Physics heuristics, the non-linearity is usually chosen to be quadratic as this is the first term that cannot be removed via simple transformations and should thus provide the first non-trivial contribution to the dynamics. As shown by Hairer and Quastel in dimension 1 in the so-called weakly asymmetric scaling, such derivation is not fully correct and, when considering a generic non-linearity, higher order terms do contribute to the limit. In the present talk, we consider the critical dimension 2 and prove that, under a logarithmically superdiffusive scaling (no weak asymmetry is required), the same is true, meaning that the limiting contribution of the non-linearity comes in via the second order coefficient of its Hermite expansion. We conclude with some remarks in higher dimension, for which instead every term in the expansion does contribute to the limit. This is a joint work with Q. Moulard and T. Klose.

Let $L = (L(t))_{t\geq 0}$ be a multivariate Lévy process with Lévy measure $\nu(dy) = \exp(-f(|y|)) dy$ for a smoothly regularly varying function $f$ of index $\alpha>1$. The process $L$ is renormalized as $X^\epsilon(t) = \epsilon L(r_\epsilon t)$, $t\in [0, T]$, for a scaling parameter $r_\epsilon = o(\epsilon^{-1})$, as $\epsilon \to 0$. We study the behavior of the bridge $Y^{\epsilon, x}$ of the renormalized process $X^\epsilon$ conditioned on the event $X^\epsilon(T) = x$ for a given end point $x\neq 0$ and end time $T>0$ in the regime of small $\epsilon$. Our main result is a sample path large deviations principle (LDP) for $Y^{x, \epsilon}$ with a specific speed function $S(\epsilon)$ and an entropy-type rate function $I_{x}$ on the Skorokhod space in the limit $\epsilon \to 0^+$. We show that the asymptotic energy minimizing path of $Y^{\epsilon, x}$ is the linear parametrization of the straight line between $0$ and $x$, while all paths leaving this set are exponentially negligible. Since on these short time scales ($r_\epsilon = o(\epsilon^{-1})$) direct LDP methods cannot be adapted we use an alternative direct approach based on convolution density estimates of the marginals $X^{\epsilon}(t)$, $t\in [0, T]$, for which we solve a specific nonlinear functional equation.

I will consider a Markov chain on a finite state space and having exponentially small transition rates on a diverging parameter $N$. I will discuss the asymptotic behavior in the parameter $N$ of the invariant measure. In particular I will discuss the large deviations behavior with an associated discrete Hamilton-Jacobi equation and a recursive construction of the limiting measure. The main ingredient used is the Markov chain tree theorem.

In a classical game two players, Alice and Bob, take turns to play $n$ moves each. Alice starts. For each move each player has two options, 1 and 2. The outcome is determined by the exact sequences of moves played by each player. Prior to the game, a winner is assigned to each of the $2^{2n}$ possible outcomes in an i.i.d. fashion, where $p$ is the probability that Bob is the winner for a given outcome. Then it is known that there exists a value $p_c\in (0,1)$ such that the probability that Bob has a winning strategy for large $n$ tends to one if $p>p_c$ and to zero if $p< p_c$. We study a modification of this game for which the outcome is determined by the exact sequence of moves played by Alice as before, but in the case of Bob all that matters is how often he has played move 1. We show that also in this case, there exists a sharp threshold $p'_c$ that determines which player has with large probability a winning strategy in the limit as $n$ tends to infinity. Joint work with Anja Sturm (Göttingen) and Natalia Cardona-Tobón (Bogotá).

Consider a tail trivial, positively associated site percolation process such that the set of open vertices is stochastically dominated by the set of closed ones. We show that, for any planar graph $G$, such a process must contain zero or infinitely many infinite connected components. The assumptions cover Bernoulli site percolation at parameter $p\leq 1/2$, resolving a conjecture of Benjamini and Schramm. As a corollary, we prove that $p_c$ is greater than or equal to $1/2$ for any unimodular, invariantly amenable planar graphs. We will then apply this percolation statement to the loop $O(n)$ model on the hexagonal lattice, and show that, whenever $n$ is between $1$ and $2$ and $x$ is between $1/\sqrt{2}$ and $1$, the model exhibits infinitely many loops surrounding every face of the lattice, giving strong evidence for conformally invariant behavior in the scaling limit (as conjectured by Nienhuis).

We provide a probabilistic characterization of the class of probability measures that can be represented by the Matrix Product Ansatz (MPA). We describe a constructive procedure, based on a suitable enlargement of the state space, showing that a probability measure can be expressed in terms of non-negative matrices via the MPA if and only if it can be written as a mixture of inhomogeneous product measures, where the mixing law is given by a Markov bridge. We illustrate this construction by applying it to examples of interacting particle systems. Finally, we discuss how the resulting probabilistic structure can be exploited to obtain large deviation principles for this class of measures. Joint work with Davide Gabrielli.

Which graphs $G$ admit a percolating phase (i.e. $p_c(G)\lt 1$)? This seemingly simple question is one of the most fundamental ones in percolation theory. A famous argument of Peierls implies that if the number of minimal cutsets of size $n$ from a vertex to infinity in the graph grows at most exponentially in $n$, then $p_c(G)\lt 1$. Our first theorem establishes the converse of this statement. This implies, for instance, that if a (uniformly) percolating phase exists, then a strongly percolating one also does. In a second theorem, we show that if the simple random walk on the graph is uniformly transient, then the number of minimal cutsets is bounded exponentially (and in particular $p_c\lt 1$). Both proofs rely on a probabilistic method that uses a random set to generate a random minimal cutset whose probability of taking any given value is lower bounded exponentially on its size. Joint work with Philip Easo and Vincent Tassion.

We study the fluctuations of the size (that is, the number of vertices) of the giant component in the Erdős–Rényi random graph process. The functional CLT in the supercritical case was recently obtained by Enriquez, Faraud and Lemaire. Our approach is based on an exploration algorithm called the simultaneous breadth-first walk, introduced by Limic in 2019, which encodes the dynamics of the evolution of the sizes of the connected components of random graph processes. We will also discuss how our method can be adapted to establish a similar functional CLT in the barely supercritical regime.

This is joint work with Vlada Limic and Sophie Lemaire.