Edwards-Wilkinson fluctuations for the Anisotropic KPZ in the weak coupling regime

Abstract: In this talk, we present recent results on an anisotropic variant of the Kardar-Parisi-Zhang equation, the Anisotropic KPZ equation (AKPZ), in the critical spatial dimension d=2. This is a singular SPDE which is conjectured to capture the behaviour of the fluctuations of a large family of random surface growth phenomena but whose analysis falls outside of the scope not only of classical stochastic calculus but also of the theory of Regularity Structures and paracontrolled calculus. We first prove the conjecture made in [Cannizzaro, Erhard, Toninelli, "The AKPZ equation at stationarity: logarithmic superdiffusivity"], i.e. we show that the nonlinearity causes a logarithmically superdiffusive behaviour at large scales and more precisely that correlation length of the solution grows like t1/2 (log t)1/4 up to lower order correction. Motivated by the previous, we consider the AKPZ equation in the so-called weak coupling regime, i.e. the equation regularised at scale N and the coefficient of the nonlinearity tuned down by a factor (log N)-1/2, and prove that, for N going to infinity, its solution converges to a linear stochastic heat equation with renormalised coefficients. The talk is based on (ongoing) joint work with D. Erhard and F. Toninelli.

mathematical physicsprobability

Audience: researchers in the topic

Probability and the City Seminar

Series comments: The Probability and the City Seminar is organized jointly by the probability groups of Columbia University and New York University.

Video recordings of talks are posted online at www.youtube.com/channel/UC0CXjG-ZSIZHy0S40Px2FEQ .