CS42: Some recent advances in Random walks and Directed Polymers in random environment

date: 7/14/2025, time: 16:00-17:30, room: ICS 13

Organizer: Loukianova Dasha (Evry-Paris-Saclay University)

Chair: Loukianova Dasha (Evry-Paris-Saclay University)

Central Limit Theorem for 2d directed polymers

Anna Donadini (Università degli Studi di Milano-Bicocca)

Directed polymers in random environment describe the behaviour of a long directed chain of monomers in presence of random impurities. In the most common setting, the trajectory of the polymer is given by a nearest-neighbour random walk path on the \(d\)-dimensional lattice, while the impurities (also called the environment) are given by a collection of i.i.d. centered random variables.

In the recent years, there has been much focus on studying the scaling properties of the model around the critical point in spatial dimension \(d=2\). Remarkably, the work of F. Caravenna, R. Sun and N. Zygouras (Ann. Appl. Prob. 2017) showed that the log-partition function of the polymer converges in law to a normal distribution when considering disorder in the subcritical regime. In this talk, we present an alternative, yet more elementary, proof of this result which relies on an \(L^2\) decomposition of the partition function into the product of smaller-scale independent partition functions and central limit theorem arguments. This presentation is based on a joint work with Clément Cosco.

Limit Laws and LAMN Property for Recurrent RWRE

Oleg Loukianov (LaMME (Evry University) and Paris-Est University)

We present some recent results for nearest-neighbour, recurrent random walks \((X_k)\) in i.i.d. random environment \(\omega(\cdot)\) on \(\mathbb{Z}\).

First, as shown in \([1]\), we recall that the empirical measure of the “shifted” environment \(\omega(X_k+\cdot)\) converges in law to some explicitly described random measure, thereby extending to the recurrent case the method of the “environment viewed from the particle”, originally introduced for transient ballistic RWRE. This allows, in particular, to describe the limit distribution of \[\tag{1} \frac 1n \sum_{k=1}^n f(\omega(X_k), \Delta X_k),\] where \(\Delta X_k=X_{k+1}-X_k\) and \(f\) is a bounded function.

Next, assuming that the environment has a finite support, which is treated as a parameter, we establish in \([2]\) a Local Asymptotic Mixed Normality property for this parameter. We also show that the Maximum Likelihood Estimator of the support parameter converges in law at the rate \(\sqrt n\) to a mixture of normal distributions and is asymptotically efficient. The proofs rely on the convergence result for expressions \((1)\), which appear quite naturally in this framework.

Bibliography

\([1]\) Francis Comets, Oleg Loukianov and Dasha Loukianova, “Limit of the environment viewed from Sinaï’s walk”, Stochastic Processes and their Applications, vol. 168, Year 2024

\([2]\) Oleg Loukianov, Dasha Loukianova, Thi Phuong Thuy Vo, “LAMN property for recurrent Random Walk in Random Environment”, Preprint

On the moments of the mass of shrinking balls under the 2d critical stochastic heat flow

Ziyang Liu (University of Warwick)

The Critical 2d Stochastic Heat Flow (SHF) is a measure valued stochastic process on R^2 that defines a non-trivial solution to the two-dimensional stochastic heat equation with multiplicative space-time noise. Its one-time marginals are a.s. singular with respect to the Lebesgue measure, meaning that the mass they assign to shrinking balls decays to zero faster than their Lebesgue volume. In this work we explore the intermittency properties of the Critical 2d SHF by studying the asymptotics of the h-th moment of the mass that it assigns to shrinking balls of radius ε and we determine that its ratio to the Lebesgue volume is of order ((1/ε))^p up to possible lower order corrections. This also exhibits the breakdown of gradual independence of pair collisions of critical attractive random walk.