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.
Citation
Jian Ding. Ofer Zeitouni. Fuxi Zhang. "On the Liouville heat kernel for $k$-coarse MBRW." Electron. J. Probab. 23 1 - 20, 2018. https://doi.org/10.1214/18-EJP189
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