Franco Severo

ETH Zürich ETH Zürich

On the off-critical level sets of smooth Gaussian fields

Probability Seminar

18th May 2022, 4:00 pm – 5:00 pm
Online, Zoom

We consider the level sets of smooth Gaussian fields on $\mathbb{R}^d$ below a parameter $\ell\in \mathbb{R}$. As $\ell$ varies this defines a percolation model, whose critical point is denoted by $\ell_c$. In this talk we will discuss the behaviour of these level sets on the off-critical regime, i.e. for $\ell \neq \ell_c$. Our main result states that, for fields with positive and sufficiently fast decaying correlations, the connection probabilities decay exponentially for $\ell<\ell_c$ and percolation occurs in sufficiently thick 2D slabs for $\ell>\ell_c$. This result, often referred to as (subcritical and supercritical, respectively) sharpness of phase transition, is typically the starting point for the study of finer properties of the off-critical phases. The result follows from a global comparison with a truncated and discretised version of the model, which may be of independent interest. The proof of this comparison relies on an interpolation scheme that integrates out the long-range and infinitesimal correlations of the model while compensating them with a small change in the parameter $\ell$.

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