Heat kernel for Liouville Brownian motion and Liouville graph distance

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 $$\underset{t\to 0}{\lim }\frac{\log \mathrm{\mid }\log {\mathsf{p}}_t^{\gamma }(u,v)\mathrm{\mid }}{\mathrm{\mid }\log t\mathrm{\mid }}=\frac{\chi }{2-\chi },\mathrm{a}\mathrm{.}\mathrm{s}\mathrm{.}$$