Electronic Journal of Probability

On the Liouville heat kernel for $k$-coarse MBRW

Jian Ding, Ofer Zeitouni, and Fuxi Zhang

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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.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 62, 20 pp.

Received: 11 January 2017
Accepted: 12 June 2018
First available in Project Euclid: 21 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G60: Random fields 60G15: Gaussian processes

Liouville quantum gravity Liouville Brownian motion Liouville heat kernel

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Ding, Jian; Zeitouni, Ofer; Zhang, Fuxi. On the Liouville heat kernel for $k$-coarse MBRW. Electron. J. Probab. 23 (2018), paper no. 62, 20 pp. doi:10.1214/18-EJP189. https://projecteuclid.org/euclid.ejp/1529546430

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