Probability and Stochastic Analysis Seminar 

It has been known for a long time that the Brownian motion is an invariant measure for the Kardar-Parisi-Zhang equation on the real line. For KPZ growth on bounded domains, however, the situation is more complicated. Stationary measures are typically not invariant by translation, non Gaussian, and they depend non trivially on boundary conditions. I will review recent progress on the characterization of invariant measures for the KPZ equation on a segment $[0,L]$ and on the half-line $\mathbb{R}_+$. Based on joint works with Pierre Le Doussal and Ivan Corwin.

Groeneboom in 1989 discovered an explicit formula for the law of the entropy solution to Burgers' equation when the initial condition is a white noise. The method of his proof relied extensively on probabilistic methods and in particular on the sophisticated excursion theory for diffusions. Recently, by verifying a conjecture of Menon and Srinivasan, Kaspar and Rezakhanlou managed to prove a closure theorem for Markovian solutions to scalar conservation laws which bridged the probabilistic problem to kinetic theory. In this talk, I present a new and significantly shorter proof of Groeneboom's results. This approach builds on these recent developments, and a central limit theorem for certain Markovian jump processes. I also discuss how a kinetic theory can be developed when we add an external force to the Burgers' equation.

Let X be a simple random walk in $\mathbb{Z}_n^d$ with $d\geq 3$ and let $t_{\rm{cov}}$ be the expected time it takes for $X$ to visit all vertices of the torus. In joint work with Prévost and Rodriguez we study the set $\mathcal{L}_\alpha$ of points that have not been visited by time $\alpha t_{\rm{cov}}$ and prove that it exhibits a phase transition: there exists $\alpha_*$ so that for all $\alpha>\alpha_*$ and all $\epsilon>0$ there exists a coupling between $\mathcal{L}_\alpha$ and two i.i.d. Bernoulli sets $\mathcal{B}^{\pm}$ on the torus with parameters $n^{-(a\pm\epsilon)d}$ with the property that $\mathcal{B}^-\subseteq \mathcal{L}_\alpha\subseteq \mathcal{B}^+$ with probability tending to $1$ as $n\to\infty$. When $\alpha\leq \alpha_*$, we prove that there is no such coupling.

I will start by presenting a number of known published rigorous upper and lower bounds for the number of collisions of a finite number of billiard balls in free space (i.e., a billiard table with no walls). The known bounds are not sharp. Those questions inspired a model with "pinned billiard balls" that have "velocities" and collide with each other but do not move. The evolution of pseudo-velocities seems to be well represented by modulated white noise. The parameters of white noise, the mean and standard deviation, are deterministic functions of space and time. They satisfy a simple system of PDEs with complicated boundary conditions.
Joint work with J. Athreya, M. Duarte, J. Hoskins, S. Steinerberger and J. Sylvester.

Percolation on the Euclidean d-dimensional lattice has been studied for over sixty years and is still a fascinating source of interesting mathematical problems. The fact that this model undergoes a non-trivial phase transition is well-understood since the early studies in the Bernoulli setting, where the lattice is regular and there are no inhomogeneities on the parameters. One way to introduce random disorder is, for example, either passing to a dilute lattice where lower dimensional affine hyperplanes are removed or, alternately, introducing inhomogeneities on the parameter along such hyperplanes. In these situations, even to establish that non-trivial phase transition takes place may be a hard task. In this talk we review some recent results on this topic and discuss some open problems.

We consider the hydrodynamic mass scaling limit of a zero-range particle system on a $1D$ discrete torus with a defect at one site. In such a model, a particle at $x$ jumps equally likely to a neighbor with rate depending only on a function of $k$, the number of particles at $x$, say $g(k)=k^{-\alpha}$. A defect, however, may be present at specific sites in that the jump rate is slowed down there to $N^{-\beta}g(k)$. Here, in diffusion scale, the grid spacing is seen as $1/N$ and time is speeded up by $N^2$. In three regimes, when $\beta <\alpha$, $\beta=\alpha$, and $\beta>\alpha$, the scaling pde limit is different, with boundary conditions reflecting interaction with the slow site and condensation on it. This is work with Jianfei Xue.

Kinetically constrained models are interacting particle systems on $\mathbb{Z}^d$, in which particles can appear/disappear only if a given local constraint is satisfied. This condition complexifies significantly the dynamics. In particular, it deprives the system of monotonicity properties, which leaves us with few tools to study the dynamics when it is initially not at equilibrium. I will review the results and techniques we have in this direction.

We consider Bernoulli percolation on transitive graphs of polynomial growth. In the subcritical regime ($p < p_c$), it is well known that the connection probabilities decay exponentially fast. In this talk, we discuss the supercritical phase $p > p_c$, where we prove the exponential decay of the truncated connection probabilities (probabilities that two points are connected by an open path, but not to infinity). This sharpness result was established by Chayes, Chayes and Newman on $\mathbb{Z}^d$ and uses the difficult slab result of Grimmett and Marstrand. However, the techniques used there are very specific to hypercubic lattices and do not extend to more general geometries. Our approach involves new robust techniques based on the recent progress in the theory of sharp thresholds and the sprinkling method of Benjamini and Tassion. In this talk, we will mainly discuss the methods on $ \mathbb{Z}^d $, and give a completely new proof of the slab result of Grimmett and Marstrand.

Based on joint work with Sébastien Martineau and Vincent Tassion.

We discuss deviations for the number of intersections of two independent infinite-time ranges in dimensions five and more. This settles a conjecture of van den Berg, Bolthausen and den Hollander. Moreover, we obtain the scenario leading to this deviation. (joint work with B. Schapira).

We consider diffusive particle systems evolving on a lattice of mesh $1/N$ and study the large deviations for the current on a time window $[0,T]$. We consider two different asymptotics. In the first case we send before $N\to +\infty$ with a diffusive hydrodynamic rescaling keeping $T$ fixed, and then send $T\to +\infty$. In the second case we send before $T\to +\infty$ keeping $N$ fixed and then send $N\to +\infty$.

We rigorously formulate and prove for a relatively general class of interactions the characterization of shift-invariant closed $L^2$-forms for a large scale interacting system. Such characterization of closed forms has played an essential role in proving the hydrodynamic limit of nongradient systems. The universal expression in terms of conserved quantities was sought from observations for specific models, but a precise formulation or rigorous proof up until now had been elusive. To obtain this, we first prove the universal characterization of shift-invariant closed “local” forms in a completely geometric way, that is, in a way that has nothing to do with probability measures. Then, we apply the result to characterize the $L^2$-forms. Our result is applicable for many important models including generalized exclusion processes, multi-species exclusion processes, exclusion processes on crystal lattices and so on. This talk is based on a joint work with Kenichi Bannai and Yukio Kametani, and a joint work with Kenichi Bannai.

Dynamic large deviations for additive path functionals of stochastic processes have attracted recent research interest, in particular in the context of stochastic particle systems and statistical physics. Efficient numerical 'cloning' algorithms have been developed to estimate the scaled cumulant generating function, based on importance sampling via cloning of rare event trajectories. Adapting previous results from the literature of particle filters and sequential Monte Carlo methods, we use Feynman-Kac models to establish fully rigorous bounds on systematic and random errors of cloning algorithms in continuous time. To this end we develop a method to compare different algorithms for particular classes of observables, based on the martingale characterization and related to the propagation of chaos for mean-field models. Our results apply to a large class of jump processes on locally compact state space, and provide a framework that can also be used to evaluate and improve the efficiency of algorithms. This is joint work with Letizia Angeli, Adam Johansen and Andrea Pizzoferrato.

The Airy_2 process is a universal distribution which describes fluctuations in models in the Kardar--Parisi--Zhang (KPZ) universality class, such as the asymmetric simple exclusion process (ASEP) and the Gaussian Unitary Ensemble (GUE). Despite its ubiquity, there are no proven results for analogous fluctuations of multi--species models. Here, we will discuss one model in the KPZ universality class, the q--Boson. We will show that the joint multi--point fluctuations of the single--species q--Boson match the single--point fluctuations of the multi--species q--Boson. Therefore the single--point fluctuations of multi--species models in the KPZ class ought to be the Airy_2 process. The proof utilizes the underlying algebraic structure of the multi--species q--Boson, namely the quantum group symmetry and Coxeter group actions.

The stochastic six-vertex model is an example of a discrete random surface, which can be viewed as an interacting particle system that is discrete in both space and time. In this talk we describe several asymptotic properties for this model, including its relation to asymmetric simple exclusion processes, its limit shapes, local statistics, and translation-invariant Gibbs measures

Liouville conformal field theory was introduced by Polyakov in 1981 as an essential ingredient in his path integral construction of string theory. Since then Liouville theory has appeared in a wide variety of contexts ranging from random conformal geometry to 4d Yang-Mills theory with supersymmetry. Recently, a probabilistic (or constructive) construction of Liouville theory was provided using the 2d Gaussian Free Field. This construction can be seen as a rigrous construction of the 2d path integral introduced in Polyakov's 1981 work. In contrast to this construction, modern conformal field theory is based on representation theory and the so-called bootstrap procedure (based on recursive techniques) introduced in 1984 by Belavin-Polyakov-Zamolodchikov. In particular, a bootstrap contruction for Liouville theory has been proposed in the mid 90's by Dorn-Otto-Zamolodchikov-Zamolodchikov (DOZZ). The aim of this talk is to review ongoing work which aims at showing the equivalence between the probabilistic (or path integral) construction and the bootstrap construction of Liouville theory. Based on numerous joint works with F. David, C. Guillarmou, A. Kupiainen, R. Rhodes.

We investigate mixing properties of RWs on random Cayley graphs of a finite group G with k ≫ 1 independent, uniformly random generators. Denote this Gₖ. Assume that 1 ≪ log k ≪ log |G|. Aldous and Diaconis (1985) conjectured that the RW exhibits cutoff for any group G whenever k ≫ log |G| and further that the time depends only on k and |G|. This was verified for Abelian groups by Dou and Hildebrand (1994, 1996). Their upper bound holds for all groups. We establish cutoff for the RW on Gₖ for all Abelian groups when 1 ≪ k ≲ log |G|, subject to some 'almost necessary' conditions. We also exhibit a non-Abelian matrix group which contradicts the second part of the AD conjecture. Lastly, we upper bound the mixing time of a RW on a nilpotent group by that of the RW on a corresponding Abelian group.

Inhomogeneous random graphs are a natural generalization of the wellknown Erdös-Renyi random graph, where vertices are characterized by a type and edges are present independently according to the type of the vertices that they are connecting. In the sparse regime, these graphs undergo a phase transition in terms of the emergence of a giant component exactly as the classical Erd}os{R enyi model. In this talk we will present an alternative approach, via large deviations, to prove this phase transition. This allows a comparison with the gelation phase transition that characterizes some coagulation process and with phase transitions of condensation type emerging in several systems of interacting components. This is an ongoing joint work with Wolfgang Koenig (WIAS and TU Berlin), Tejas Iyer (WIAS), Heide Langhammer (WIAS), Robert Patterson (WIAS).

We consider a particle moving in continuous time as a Markov jump process; its discrete chain given by an ordinary random walk on Z^d (with finite second moments), and its jump rate at (x,t) given by a fixed function f of the state of a simple birth-and-death (BD) process at x on time t. BD processes at different sites are independent and identically distributed, and f is assumed non increasing and vanishing at infinity. We present an argument to obtain a CLT for the particle position when the environment is ergodic. In the absence of a viable uniform lower bound for the jump rate, we resort instead to stochastic domination, as well as to a subadditive argument to control the time spent by the particle to give n jumps (both ingredients rely on the monotonicity of f); and we also impose conditions on the initial (product) environmental initial distribution. We also discuss the asymptotic form of the environment seen by the particle. Joint work with Maicon Pinheiro and Pablo Gomes.

In this talk, we consider the Ising and Potts model defined on large lattices of dimension two or three at very low temperature regime. Under this regime, each monochromatic spin configuration is metastable in that exit from the energetic valley around that configuration is exponentially difficult. It is well-known that, under the presence of external magnetic fields, the metastable transition from a monochromatic configuration to another one is characterized solely by the appearance of a critical droplet. On the other hand, for the model without external field, the saddle structure is no longer characterized by a sharp droplet but has a huge and complex plateau structure. In this talk, we explain our recent research on the analysis of this energy landscape and its application to the demonstration of Eyring-Kramers formula for models on fixed two or three dimensional lattice (cf. https://arxiv.org/abs/2102.05565) or models on growing two-dimensional lattice (cf. https://arxiv.org/abs/2109.13583).

This talk is based on joint works with Seonwoo Kim.

I will discuss a class of interacting particle systems in continuous space. Such models are known to "homogenize", in the sense that the behavior of the cloud of particles is approximately described by a partial differential equation over large scales. In the talk, I will describe a first step towards making this result quantitative. The approach is inspired by recent developments in the homogenization of elliptic equations with random coefficients. Joint work with Arianna Giunti and Chenlin Gu.