Dynamical freezing in a spin glass system with logarithmic correlations

Aser Cortines; Julian Gold; Oren Louidor

doi:10.1214/18-EJP181

Dynamical freezing in a spin glass system with logarithmic correlations

Aser Cortines, Julian Gold, Oren Louidor

Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

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.

Original language	English (US)
Article number	59
Journal	Electronic Journal of Probability
Volume	23
DOIs	https://doi.org/10.1214/18-EJP181
State	Published - 2018

Funding

The work of A.C. and O.L. was supported in part by the European Union’s-Seventh Framework Program (FP7/2007-2013) under grant agreement no. 276923 – M-MOTIPROX. The work of A.C. was also supported by the Israeli Science Foundation grant no. 1723/14 – Low Temperature Interfaces: Interactions, Fluctuations and Scaling; and by the Swiss National Science Foundation 200021_163170. The work of O.L. was also supported by the Israeli Science Foundation grant no. 1328/17 and by grant I-2494-304.6/2017 from the German-Israeli Foundation for Scientific Research and Development. The work of J.G. was supported in part by NSF grant DMS-1407558, by NSF grant DMS-1502632, and by a UCLA Dissertation Year Fellowship.

Keywords

Aging
Dynamical freezing
Gaussian free field
K-process
Random walk in a random potential
Spin-glasses
Trap models

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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10.1214/18-EJP181

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title = "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.",

keywords = "Aging, Dynamical freezing, Gaussian free field, K-process, Random walk in a random potential, Spin-glasses, Trap models",

author = "Aser Cortines and Julian Gold and Oren Louidor",

note = "Funding Information: The work of A.C. and O.L. was supported in part by the European Union{\textquoteright}s-Seventh Framework Program (FP7/2007-2013) under grant agreement no. 276923 – M-MOTIPROX. The work of A.C. was also supported by the Israeli Science Foundation grant no. 1723/14 – Low Temperature Interfaces: Interactions, Fluctuations and Scaling; and by the Swiss National Science Foundation 200021_163170. The work of O.L. was also supported by the Israeli Science Foundation grant no. 1328/17 and by grant I-2494-304.6/2017 from the German-Israeli Foundation for Scientific Research and Development. The work of J.G. was supported in part by NSF grant DMS-1407558, by NSF grant DMS-1502632, and by a UCLA Dissertation Year Fellowship. Publisher Copyright: {\textcopyright} 2018, University of Washington. All rights reserved.",

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doi = "10.1214/18-EJP181",

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volume = "23",

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AU - Gold, Julian

AU - Louidor, Oren

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PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

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Dynamical freezing in a spin glass system with logarithmic correlations

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