Plenary session 2

Tuesday, 7/15, 9:00-12:00, Congress Centre

Superdiffusive Central Limit Theorem for the critical Stochastic Burgers Equation

Giuseppe Cannizzaro (University of Warwick)

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. The classical example of such systems is the Asymmetric (Simple) Exclusion process (ASEP), an interacting particle system whose large-scale behaviour has fascinated probabilists and mathematicians for more than half a century. In the subcritical spatial dimension d=1, the SBE coincides with the derivative of the celebrated Kardar-Parisi-Zhang equation, which is polynomially superdiffusive and whose fluctuations are described by the KPZ Fixed Point, while in the super-critical dimensions d>2, it was recently shown to be diffusive and rescale to a biased Stochastic Heat equation. The present talk focuses on the critical dimension d=2, which falls outside of the ground-breaking theory of Regularity Structures and whose large-scale behaviour had long been open. In their seminal work, van Beijren, Kutner and Spohn conjecture that the SBE should be logarithmically superdiffusive with a precise exponent but this has only been shown up to lower order corrections (both for ASEP in a landmark paper of H.-T. Yau, and, more recently for the SPDE). We pin down the logarithmic superdiffusivity exactly by identifying the asymptotic behaviour of the so-called diffusion matrix and show that, once the logarithmic corrections to the scaling are taken into account, the solution of the SBE satisfies a central limit theorem. This is joint work with Q. Moulard and F. Toninelli.

Energy propagation in stochastically perturbed harmonic chains

Tomasz Komorowski (IM PAN)

Nature has a hierarchical structure with macroscopic behavior arising from the dynamics of atoms and molecules. The connection between different levels of the hierarchy is however not always straightforward, as seen in the emergent phenomena, such as phase transition and heat convection. Establishing in a mathematical precise way the connection between the different levels is the central problem of rigorous statistical mechanics. One of the methods leading to such results is to introduce some stochasticity inside the system.

A classical microscopic model of the thermal energy transport is provided by a chain of coupled oscillators on a integer lattice, that describes atoms (or molecules) in a crystal. We summarise some of the results obtained recently concerning the derivation of the macroscopic heat equation from the microscopic behaviour of a harmonic chain with a stochastic perturbation. We focus our attention on the emergence of macroscopic boundary conditions. The results have been obtained in collaboration with Joel Lebowitz, Stefano Olla, Marielle Simon.

Critical long-range percolation

Tom Hutchcroft (Caltech)

Many statistical mechanics models on the lattice (including percolation, self-avoiding walk, the Ising model and so on) have natural "long-range" versions in which vertices interact not only with their neighbours, but with all other vertices in a way that decays with the distance. When this decay is described by a power-law, it can lead to new kinds of critical phenomena that are not present in the short-range models. I will describe a new approach to the study of these models that allows us to go well beyond what is known for short-range models, particularly in intermediate dimensions.