Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials

TL;DR

This paper establishes Russo-Seymour-Welsh (RSW) estimates for the nodal structures of Gaussian random fields, including the Kostlan ensemble, demonstrating uniform crossing probability bounds across polynomial degrees and scales.

Contribution

It provides the first RSW-type estimates for the Kostlan ensemble, extending prior local scaling results to a global, degree-uniform setting.

Findings

RSW estimates hold uniformly for Kostlan polynomials

Crossing probabilities are controlled across all relevant scales

Results apply to quads with controlled geometry

Abstract

We study the percolation properties of the nodal structures of random fields. Lower bounds on crossing probabilities (RSW-type estimates) of quads by nodal domains or nodal sets of Gaussian ensembles of smooth random functions are established under the following assumptions: (i) sufficient symmetry; (ii) smoothness and non-degeneracy; (iii) local convergence of the covariance kernels; (iv) asymptotically non-negative correlations; and (v) uniform rapid decay of correlations. The Kostlan ensemble is an important model of Gaussian homogeneous random polynomials. An application of our theory to the Kostlan ensemble yields RSW-type estimates that are uniform with respect to the degree of the polynomials and quads of controlled geometry, valid on all relevant scales. This extends the recent results on the local scaling limit of the Kostlan ensemble, due to Beffara and Gayet.