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

TL;DR

This paper investigates the Liouville heat kernel for specific Gaussian fields on the 2D torus, showing deviations from prior predictions and demonstrating nonuniversality in the behavior of these kernels.

Contribution

It constructs particular Gaussian fields with covariance perturbations and derives short-time estimates for their Liouville heat kernels, challenging existing theoretical predictions.

Findings

Liouville heat kernel exhibits different short-time behavior than Watabiki's predictions.

Existence of Gaussian fields with covariance perturbations affecting heat kernel estimates.

Demonstrates nonuniversality in Liouville heat kernel behavior for certain Gaussian fields.

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 $\varepsilon>0$ 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, $$ \exp \left( - t^{ - \frac 1 { 1 + \frac 1 2 \gamma^2 } - \varepsilon } \right) \le p_t^\gamma (x, y) \le \exp \left( - t^{- \frac 1 { 1 + \frac 1 2 \gamma^2 } + \varepsilon } \right) , $$ 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.