Electronic Journal of Probability

Dynamical freezing in a spin glass system with logarithmic correlations

Aser Cortines, Julian Gold, and Oren Louidor

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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

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

Received: 26 November 2017
Accepted: 23 May 2018
First available in Project Euclid: 11 June 2018

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Primary: 60K37: Processes in random environments 82C44: Dynamics of disordered systems (random Ising systems, etc.) 60G57: Random measures 60G70: Extreme value theory; extremal processes 60G15: Gaussian processes 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

K-process trap models aging Gaussian free field random walk in a random potential dynamical freezing spin-glasses

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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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