Probability and Stochastic Analysis Seminar 

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 less than or equal to one half, 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.

The loop-erased random walk (LERW) is a fundamental model for random self-avoiding curves. Since its introduction by Lawler in the 1980s, the scaling limits of LERW have been thoroughly studied. While these limits are well-understood in dimensions two, four, and higher, the three-dimensional case continues to present unique challenges.

In this talk, I will present some recent results for general non-local branching particle process or general non-local superprocess, in both cases, with and without immigration. Under the assumption that the mean semigroup has a Perron-Frobenious type behaviour, for the immigrated mass, as well as the existence of second moments, we consider necessary and sufficient conditions that ensure limiting distributional stability. More precisely, our first main contribution pertains to proving the asymptotic Kolmogorov survival probability and Yaglom limit for critical non-local branching particle systems and superprocesses under a second moment assumption on the offspring distribution. Our results improve on existing literature by removing the requirement of bounded offspring in the particle setting and to include non-local branching mechanisms. Our second main contribution pertains to the stability of both critical and sub-critical non-local branching particle systems and superprocesses with immigration. At criticality, we show that the scaled process converges to a Gamma distribution under a necessary and sufficient integral test. At subcriticality we show stability of the process, also subject to an integral test. In these cases, our results complement classical results for (continuous-time) Galton-Watson processes with immigration and continuous-state branching processes with immigration.

We are going to present recent results and ongoing research concerning the Ising chain (i.e., the Ising model in dimension 1) with homogeneous (large) spin-spin interaction, but subjected to a random external field, the latter being sampled from an i.i.d sequence. The framework is that of disordered systems. We will first present the pure model (homogeneous external field) which is exactly solvable. Then we will turn to the disordered model, showing that for this model the disorder is strongly relevant, through estimates on the free energy and a description of the typical configurations. There are two distinct cases with qualitatively different behaviours: the cases of centered or uncentered disorder. We will also present a continous analogue of the model, obtained by a weak disorder limit, which has the advantage of allowing some explicit computations. Our approach confirms and deepens claims made in the physics literature by D. Fisher and collaborators, based on a study of the Glauber dynamics of the model using the renormalisation group method.

The Totally Asymmetric Simple Exclusion Process (TASEP) on a ring of size $ N $ with $ p $ particles is a key model in non-equilibrium statistical physics. While its stationary state is well understood, the relaxation dynamics and current fluctuations in finite systems are less explored. We introduce a deformation parameter $ \gamma $ defining a tilted operator controlling the time-integrated current statistics. Using the coordinate Bethe ansatz, we obtain implicit formulas for the scaled cumulant generating function (SCGF) and spectral gap, expressed via Bethe roots and characterized asymptotically by the Cassini oval geometry.

In the thermodynamic limit with fixed particle density, a dynamical phase transition emerges between fluctuation regimes: for $ \gamma > 0 $, the SCGF scales ballistically with system size, $ \lambda_1 \sim N $, and the spectral gap closes polynomially, $ \Delta \sim N^{-1} $, indicating rapid relaxation. For $ \gamma < 0 $, the SCGF converges to $-1$, with an exponentially closing gap, $ \Delta \sim \exp(-cN) $, signaling metastability. These non-perturbative results provide new insights into large deviations and relaxation dynamics in driven particle systems.