Large deviations in the Langevin dynamics of a random field Ising model

Gérard Ben Arous, Michel Sortais

Research output: Contribution to journalArticle

Abstract

We consider a Langevin dynamics scheme for a d-dimensional Ising model with a disordered external magnetic field and establish that the averaged law of the empirical process obeys a large deviation principle (LDP), according to a good rate functional Ja having a unique minimiser Q. The asymptotic dynamics Q may be viewed as the unique weak solution associated with an infinite-dimensional system of interacting diffusions, as well as the unique Gibbs measure corresponding to an interaction Ψ on infinite dimensional path space. We then show that the quenched law of the empirical process also obeys a LDP, according to a deterministic good rate functional Jq satisfying: Jq≥Ja, so that (for a typical realisation of the disordered external magnetic field) the quenched law of the empirical process converges exponentially fast to a Dirac mass concentrated at Q.

Original languageEnglish (US)
Pages (from-to)211-255
Number of pages45
JournalStochastic Processes and their Applications
Volume105
Issue number2
DOIs
StatePublished - Jun 1 2003

Fingerprint

Langevin Dynamics
Ising model
Empirical Process
Large Deviations
Random Field
Ising Model
Large Deviation Principle
Magnetic fields
External Field
Magnetic Field
Interacting Diffusions
Path Space
Infinite-dimensional Systems
Gibbs Measure
Infinite-dimensional Spaces
Paul Adrien Maurice Dirac
Weak Solution
Converge
Interaction
Large deviations

Keywords

  • Disordered systems
  • Interacting diffusion processes
  • Large deviations
  • Statistical mechanics

ASJC Scopus subject areas

  • Statistics, Probability and Uncertainty
  • Mathematics(all)
  • Statistics and Probability

Cite this

Large deviations in the Langevin dynamics of a random field Ising model. / Arous, Gérard Ben; Sortais, Michel.

In: Stochastic Processes and their Applications, Vol. 105, No. 2, 01.06.2003, p. 211-255.

Research output: Contribution to journalArticle

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