Large deviations and exponential decay for the magnetization in a Gaussian random field

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Abstract

We consider a continuous model for transverse magnetization of spins diffusing in a homogeneous Gaussian random longitudinal field {XV (x); x ∈ ℝd}, where λ is the coupling constant giving the intensity of the random field. In this setting, the transverse magnetization is given by the formula M(t) = double-struck E signexp{-λ2t0t0 K(Br - Bs) dsdr}, where {Bt; t ≥ 0} is the standard process of Brownian motion and K(x) is the covariance function of the original random field V(x). We use large deviation techniques to show that the limit S(λ) = limt→∞ 1/t In M (t) exists. We also determine the small λ behavior of the rate S(λ) and show that it is indeed decaying as conjectured in the physics literature.

Original languageEnglish (US)
Pages (from-to)233-247
Number of pages15
JournalProbability Theory and Related Fields
Volume106
Issue number2
DOIs
StatePublished - Oct 1996

All Science Journal Classification (ASJC) codes

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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