  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.
