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 (15)

• Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric
• The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds
• Quantum Loewner Evolution
• Liouville Brownian motion
• KPZ formula derived from Liouville heat kernel
• Liouville heat kernel: regularity and bounds
• Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity
• Liouville first-passage percolation: subsequential scaling limits at high temperature
• Subsequential scaling limits for Liouville graph distance
• Spectral dimension of Liouville quantum gravity
• Non-universality for first passage percolation on the exponential of log-correlated Gaussian fields
• A distance exponent for Liouville quantum gravity
• Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding
• 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