Heat kernel for Liouville Brownian motion and Liouville graph distance

TL;DR

This paper establishes the existence of a scaling exponent for Liouville graph distance and provides short-time estimates for the Liouville heat kernel, advancing understanding of Liouville quantum gravity in two dimensions.

Contribution

It introduces the first rigorous proof of the scaling exponent for Liouville graph distance and characterizes the short-time behavior of the Liouville heat kernel.

Findings

Existence of the scaling exponent  in a specific range.

Asymptotic behavior of the Liouville heat kernel as time approaches zero.

Quantitative relationship between the heat kernel decay and the graph distance exponent.

Abstract

We show the existence of the scaling exponent $\chi\in (0,4[(1+\gamma^2/4)- \sqrt{1+\gamma^4/16}]/\gamma^2]$ of the graph distance associated with subcritical two-dimensional Liouville quantum gravity of paramater $\gamma<2$ on $\mathbb V =[0,1]^2 $. We also show that the Liouville heat kernel satisfies, for any fixed $u,v\in \mathbb V^o$, the short time estimates $$ \lim_{ t \to 0} \frac{\log |\log {\mathsf p}_t^\gamma(u,v)| }{|\log t|}=\frac{\chi}{2-\chi}, \ \mbox{\rm a.s.} $$