Olivier Daviaud | Searcharxiv

arXiv2006

Extremes of the discrete two-dimensional Gaussian free field

We consider the lattice version of the free field in two dimensions and study the fractal structure of the sets where the field is unusually high (or low). We then extend some of our computations to the case of the free field conditioned on being everywhere nonnegative. For example, we compute the width of the largest downward spike of a given length. Through the prism of these results, we find that the extrema of the free field under entropic repulsion (minus its mean) and those of the unconditioned free field are identical. Finally, when compared to previous results these findings reveal a suggestive analogy between the square of the free field and the two-dimensional simple random walk on the discrete torus.

math.PR↗

arXiv2003

Thick points for the Cauchy process

Let $\mathcal{T}(x,\eps)$ denote the occupation measure of an interval of length $2\eps$ centered at $x$ by the Cauchy process run until it hits $(-\infty,-1]\cup [1,\infty)$. We prove that $\sup_{|x|\leq 1}\mathcal{T}(x,\eps)/(\eps(\ln\eps)^2)\to 2/π$ a.s. as $\eps\to 0$. We also obtain the multifractal spectrum for thick points, i.e. the Hausdorff dimension of the set of $α$-thick points $x$ for which $\lim_{\eps \to 0} \mathcal{T}(x,\eps)/(\eps(\ln\eps)^2) = α> 0$.

math.PR↗