2025/03/28 by David Vernotte, Vernotte, David
Mathematics · #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2503.22434
Gaussian percolation can be seen as the generalization of standard Bernoulli percolation on ℤd. Instead of a random discrete configuration on a lattice, we consider a continuous Gaussian field f and we study the topological and geometric properties of the random excursion set E_ℓ(f) := \x∈ ℝd | f(x)≥ -ℓ\ where ℓ∈ ℝ is called a level. It is known that for a wide variety of fields f, there exists a phase transition at some critical level ℓc. When ℓ> ℓc, the excursion set E_ℓ(f) presents a unique unbounded component while if ℓ<ℓc there are only bounded components in E_ℓ(f). In the supercritical regime, ℓ>ℓc, we study the geometry of the unbounded cluster. Inspired by the work of Peter Antal and Agoston Pisztora for the Bernoulli model \citeAntal, we introduce the chemical distance between two points x and y as the Euclidean length of the shortest path connecting these points and staying in E_ℓ(f). In this paper, we show that when ℓ>-ℓc then with high probability, the chemical distance between two points has a behavior close to the Euclidean distance between those two points.