Heat kernel for Liouville Brownian motion and Liouville graph distance

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 $$ \lim_{ t \to 0} \frac{\log |\log {\mathsf p}_t^γ(u,v)| }{|\log t|}=\fracχ{2-χ}, \ \mbox{\rm a.s.} $$

References in corpus (4)

• The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds
• Subsequential scaling limits for Liouville graph distance
• On the Liouville heat kernel for k-coarse MBRW and nonuniversality
• Liouville metric of star-scale invariant fields: tails and Weyl scaling

Cited by in corpus (11)

• The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds
• Mating of trees for random planar maps and Liouville quantum gravity: a survey
• Subsequential scaling limits for Liouville graph distance
• Geodesic networks in Liouville quantum gravity surfaces
• Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity
• Tightness of supercritical Liouville first passage percolation
• Liouville metric of star-scale invariant fields: tails and Weyl scaling
• Comparison of discrete and continuum Liouville first passage percolation
• Introduction to the Liouville quantum gravity metric
• Volume of metric balls in Liouville quantum gravity
• On the chemical distance exponent for the two-sided level-set of the 2D Gaussian free field