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

Given a planar continuum Gaussian free field h(& x1d4b0;) in a domain & x1d4b0; with Dirichlet boundary condition and any delta > 0, we let {h delta U(v):v is an element of U} be a real-valued smooth Gaussian process where h delta U(v) is the average of h(& x1d4b0;) along a circle of radius delta with center v. For gamma > 0, we study the Liouville first-passage percolation (in scale delta), i.e., the shortest path distance in & x1d4b0; where the weight of each path P is given by integral Pe gamma h delta U(z)|dz|. We show that the distance between two typical points is O(delta c*gamma 4/3/log gamma-1) for all sufficiently small but fixed gamma > 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.(c) 2019 Wiley Periodicals, Inc.