2023/12/21 by David Vernotte, Vernotte, David
Mathematics · #Stochastic processes and statistical mechanics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2312.14205
Our study concerns the large scale geometry of the excursion set of planar random fields: E ℓ = x ∈ R 2 |f (x) ≥-ℓ, where ℓ ∈ R is a real parameter and f is a continuous, stationary, centered, planar Gaussian field satisfying some regularity assumptions (in particular, this study applies to the planar Bargmann-Fock field). It is already known that under those hypotheses there is a phase transition at ℓc = 0. When ℓ > 0, we are in a supercritical regime and almost surely E ℓ has a unique unbounded connected component. We prove that in this supercritical regime, whenever two points are in the same connected components of E ℓ then, with high probability, the chemical distance (the length of the shortest path in E ℓ between these points) is close to the Euclidean distance between those two points Contents