Upper Bounds on Liouville First‐Passage Percolation and Watabiki's Prediction

Abstract

AbstractGiven a planar continuum Gaussian free field h𝒰 in a domain 𝒰 with Dirichlet boundary condition and any δ > 0, we let be a real‐valued smooth Gaussian process where is the average of h𝒰 along a circle of radius δ with center v. For γ > 0, we study the Liouville first‐passage percolation (in scale δ), i.e., the shortest path distance in 𝒰 where the weight of each path P is given by . We show that the distance between two typical points is for all sufficiently small but fixed γ > 0 and some constant c* > 0. In addition, we obtain similar upper bounds on the Liouville first‐passage percolation for discrete Gaussian free fields, as well as the Liouville graph distance, which roughly speaking is the minimal number of euclidean balls with comparable Liouville quantum gravity measure whose union contains a continuous path between two endpoints. Our results contradict some reasonable interpretations of Watabiki's prediction (1993) on the random distance of Liouville quantum gravity at high temperatures.© 2019 Wiley Periodicals, Inc.