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.
References
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