Electronic Journal of Probability

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

Jian Ding, Ofer Zeitouni, and Fuxi Zhang

Full-text: Open access

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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1529546430

Digital Object Identifier
doi:10.1214/18-EJP189

Mathematical Reviews number (MathSciNet)
MR3827969

Zentralblatt MATH identifier
06924674

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

Keywords
Liouville quantum gravity Liouville Brownian motion Liouville heat kernel

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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