Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the spectral gap of spherical spin glass dynamics

Reza Gheissari and Aukosh Jagannath

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We consider the time to equilibrium for the Langevin dynamics of the spherical $p$-spin glass model of system size $N$. We show that the log-Sobolev constant and spectral gap are order $1$ at sufficiently high temperatures whereas the spectral gap decays exponentially in $N$ at sufficiently low temperatures. These verify the existence of a dynamical high temperature phase and a dynamical glass phase at the level of the spectral gap. Key to these results are the understanding of the extremal process and restricted free energy of Subag–Zeitouni and Subag.


Nous considérons le temps d’atteinte de l’équilibre pour la dynamique de Langevin du modèle de verre de $p$-spin sphérique de taille $N$. Nous montrons que la constante de log-Sobolev et le trou spectral sont d’ordre $1$ à température suffisamment grande, alors que le trou spectral décroit exponentiellement en $N$ à température suffisamment basse. Ceci confirme l’existence d’une phase dynamique de haute température et d’une phase dynamique verre concernant le trou spectral. Les arguments clés de ces résultats sont la compréhension du processus extrémal et de l’énergie libre restreinte de Subag–Zeitouni et Subag.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 756-776.

Received: 21 June 2017
Revised: 27 February 2018
Accepted: 14 March 2018
First available in Project Euclid: 14 May 2019

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Digital Object Identifier

Primary: 82C44: Dynamics of disordered systems (random Ising systems, etc.)
Secondary: 82D30: Random media, disordered materials (including liquid crystals and spin glasses) 82C26: Dynamic and nonequilibrium phase transitions (general) 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Spin glass Langevin dynamics Glassy dynamics Spectral gap Log-Sobolev


Gheissari, Reza; Jagannath, Aukosh. On the spectral gap of spherical spin glass dynamics. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 756--776. doi:10.1214/18-AIHP897.

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