## Electronic Journal of Probability

### Dynamical freezing in a spin glass system with logarithmic correlations

#### Abstract

We consider a continuous time random walk on the two-dimensional discrete torus, whose motion is governed by the discrete Gaussian free field on the corresponding box acting as a potential. More precisely, at any vertex the walk waits an exponentially distributed time with mean given by the exponential of the field and then jumps to one of its neighbors, chosen uniformly at random. We prove that throughout the low-temperature regime and at in-equilibrium timescales, the process admits a scaling limit as a spatial K-process driven by a random trapping landscape, which is explicitly related to the limiting extremal process of the field. Alternatively, the limiting process is a supercritical Liouville Brownian motion with respect to the continuum Gaussian free field on the box. This demonstrates rigorously and for the first time, as far as we know, a dynamical freezing in a spin glass system with logarithmically correlated energy levels.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 59, 31 pp.

Dates
Accepted: 23 May 2018
First available in Project Euclid: 11 June 2018

https://projecteuclid.org/euclid.ejp/1528704077

Digital Object Identifier
doi:10.1214/18-EJP181

Mathematical Reviews number (MathSciNet)
MR3814253

Zentralblatt MATH identifier
06924671

#### Citation

Cortines, Aser; Gold, Julian; Louidor, Oren. Dynamical freezing in a spin glass system with logarithmic correlations. Electron. J. Probab. 23 (2018), paper no. 59, 31 pp. doi:10.1214/18-EJP181. https://projecteuclid.org/euclid.ejp/1528704077

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