The Vertex Reinforced Jump Process and a random Schrödinger operator on finite graphs

Sabot, Christophe; Tarrès, Pierre; Zeng, Xiaolin

doi:10.1214/16-aop1155

Cited by 32 publications

(74 citation statements)

References 18 publications

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“…There are at least three different proofs of this fact, given in [DSZ10], [ST15], and [STZ17]. Precise references and some more comments on this are given in appendix C, which reviews known results used here.…”

Section: Extension Of the Susy Hyperbolic Nonlinear Sigma Modelmentioning

confidence: 96%

Convergence of vertex-reinforced jump processes to an extension of the supersymmetric hyperbolic nonlinear sigma model

Merkl

Rolles

Tarrès

2018

Probab. Theory Relat. Fields

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In this paper, we define an extension of the supersymmetric hyperbolic nonlinear sigma model introduced by Zirnbauer. We show that it arises as a weak joint limit of a time-changed version introduced by Sabot and Tarrès of the vertex-reinforced jump process. It describes the asymptotics of rescaled crossing numbers, rescaled fluctuations of local times, asymptotic local times on a logarithmic scale, endpoints of paths, and last exit trees. 1

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Section: Extension Of the Susy Hyperbolic Nonlinear Sigma Modelmentioning

confidence: 96%

Convergence of vertex-reinforced jump processes to an extension of the supersymmetric hyperbolic nonlinear sigma model

Merkl

Rolles

Tarrès

2018

Probab. Theory Relat. Fields

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“…More recently Sabot, Tarrès, and Zeng developed further the random Schrödinger operator interpretation [STZ17] [SZ15]. In particular they derived the explicit law for the random potential, and constructed two families of martingales in discrete time.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…These variables were introduced in [STZ17]. We drop the dependence on V , W , or both if there is no risk of confusion.…”

Section: Results In Finite Volumementioning

confidence: 99%

Martingales and some generalizations arising from the supersymmetric hyperbolic sigma model

Disertori¹,

Merkl²,

Rolles³

2019

ALEA

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We introduce a family of real random variables (β, θ) arising from the supersymmetric nonlinear sigma model and containing the family β introduced by Sabot, Tarrès, and Zeng [STZ17] in the context of the vertex-reinforced jump process. Using this family we construct an exponential martingale generalizing the one considered in [DMR17]. Moreover, using the full supersymmetric nonlinear sigma model we also construct a generalization of the exponential martingale involving Grassmann variables. 2010 MSC: 60G60 (primary), 60G42, 82B44 (secondary).

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“…Turning to the vertex reinforced jump process (abbreviate to VRJP in the sequel and is defined in 4.2), a breakthrough in the study of this self interacting processes with linearly reinforcement is the discovery of its mixing measure, with explicit density function, in a series of papers [12,33,34]. This mixing measure also plays a role in the supersymmetric hyperbolic sigma model introduced by Zirnbauer [39], and studied in e.g.…”

Section: Introductionmentioning

confidence: 99%

Interacting particle systems

et al. 2017

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Abstract. We present in this paper a number of recent and various works of statistical physics, which involve an interacting particle system. In kinetically constrained models, the particles are placed on Z d , with local constraints on the dynamics, that can slow down the evolution and reproduce the behaviour of glassy systems. The contact process (or Suspected-Infected-Susceptible) is a simple model for the spread of an infection on a (general) graph. In the Kuramoto model, the motion of the particles is determined by a diffusion system with mean-field interaction. Finally, in the modeling of the vertex reinforced jump process (a particle that interacts with itself via its path), we show an unexpected link with an interacting particle system. Résumé. Nous présentons dans ce papier plusieurs travaux récents et divers de physique statistique,qui mettent en jeu un système de particules en interaction. Dans les modèles cinétiquement contraints, les particules sont placées sur Z d avec des contraintes locales sur la dynamique qui peuvent ralentir celle-ci et reproduire le comportement des systèmes vitreux. Le processus de contact (ou SusceptibleInfected-Susceptible) modélise quant à lui la propagation d'une infection sur un graphe quelconque. Dans le modèle de Kuramoto, le mouvement des particules est déterminé par un système de diffusions, avec interaction de type champ moyen. Enfin, pour le processus de sauts renforcé par site, la particule interagit avec elle-même (avec sa trajectoire), la modélisation présentant un lien inattendu avec un système de particules en interaction.

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The Vertex Reinforced Jump Process and a random Schrödinger operator on finite graphs

Abstract: International audienc

Cited by 32 publications

References 18 publications

Convergence of vertex-reinforced jump processes to an extension of the supersymmetric hyperbolic nonlinear sigma model

Convergence of vertex-reinforced jump processes to an extension of the supersymmetric hyperbolic nonlinear sigma model

Martingales and some generalizations arising from the supersymmetric hyperbolic sigma model

Interacting particle systems

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