Heat kernel for Liouville Brownian motion and Liouville graph distance

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