CS42: Recent Advances in Random Walks and Random Polymers in Random Environments

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

Central Limit Theorem for 2d directed polymers

Anna Donadini

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.