Chemical Distances for Percolation of Planar Gaussian Free Fields and Critical Random Walk Loop Soups

Abstract

We initiate the study on chemical distances of percolation clusters for level sets of twodimensional discrete Gaussian free fields as well as loop clusters generated by two-dimensional random walk loop soups. One of our results states that the chemical distance between two macroscopic annuli away from the boundary for the random walk loop soup at the critical intensity is of dimension 1 with positive probability. Our proof method is based on an interesting combination of a theorem of Makarov, isomorphism theory and an entropic repulsion estimate for Gaussian free fields in the presence of a hard wall.where G VN (u, v) is the Green's function for simple random walk, i.e., the expected number of visits to v before reaching ∂V N for a simple random walk started at u. The first goal of the present paper is to study chemical distances (i.e., graph distances) on percolation clusters for level sets of GFFs. Precisely, for any λ ∈ R, we let H N,λ = {v ∈ V N : η N,v λ} be the λ-level set, i.e., the collection of all vertices

Liouville first passage percolation: geodesic length exponent is strictly larger than 1 at high temperatures

“…Assuming (13), by (6) and (12), we have Eη j (x)η j (y) ≥ −2(3C 1 + 2C 2 + 1)C 3 , completing the proof. It remains to prove (13). Without loss of generality, suppose x = (0, 0) and…”

“…In [13], the chemical distance (i.e., the graph distance in the induced open cluster) for the percolation of level sets of the two-dimensional DGFF was studied. In particular, [13,Theorem 1.1] implies that there exists a path of length exponent 1 and Liouville FPP weight O N 1+o(1) joining the two boundaries of an annulus. This can be regarded as a complement of our Theorem 1.2.…”

Euclidean and chemical distances in ellipses percolation

“…To understand geometric properties of infinite clusters is a problem of major interest in percolation theory. Models for which the chemical distance was studied include Bernoulli percolation and first-passage percolation [1,11,12]; random interlacements [6]; random walk loop soup [7]; and Gaussian free field [8,9]. General conditions for a percolation model on Z d to have a unique infinite cluster in which Euclidean and chemical distances are comparable are provided in [9].…”

Percolation for level-sets of Gaussian free fields on metric graphs

“…For d = 2 we take a sequence φN of fields defined on the metric graphs GN associated to V N (with Dirichlet boundary conditions). For h ∈ R we let Ẽ≥h N = {v ∈ GN : φN,v ≥ h} be the level set, or excursion set, of φN above hnote that our choice of level set is different from that of [6] by a flipping symmetry, in order to be consistent with the majority of the literature. Further, for u, v ∈ Ẽ≥h N , we let the chemical distance D N,h (u, v) be the graph distance between u and v in Ẽ≥h N , with…”