On the Liouville heat kernel for k-coarse MBRW and nonuniversality

Abstract

We study the Liouville heat kernel (in the $L^2$ phase) associated with a class of logarithmically correlated Gaussian fields on the two dimensional torus. We show that for each $\epsilon >0$ there exists such a field, whose covariance is a bounded perturbation of that of the two dimensional Gaussian free field, and such that the associated Liouville heat kernel satisfies the short time estimates, $$\exp \left(-t^{-\frac{1}{1+\frac{1}{2}{\gamma }^2}-\epsilon }\right)\le p_t^{\gamma }(x,y)\le \exp \left(-t^{-\frac{1}{1+\frac{1}{2}{\gamma }^2}+\epsilon }\right),$$ for $\gamma <1/2$. In particular, these are different from predictions, due to Watabiki, concerning the Liouville heat kernel for the two dimensional Gaussian free field.

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