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

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.