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Large deviations and exponential decay for the magnetization in a Gaussian random field
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  • Published: 01 October 2002

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

  • René A. Carmona1 &
  • Lin Xu1 

Probability Theory and Related Fields volume 106, pages 233–247 (1996)Cite this article

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Summary.

We consider a continuous model for transverse magnetization of spins diffusing in a homogeneous Gaussian random longitudinal field \(\{\lambda V(x);\, x \in {\Bbb R}^{d} \} \), where \(\lambda \) is the coupling constant giving the intensity of the random field. In this setting, the transverse magnetization is given by the formula \(M(t)={\Bbb E} \exp \{ -{\lambda}^{2} \int_{0}^{t} \int_{0}^{t} K(B_r-B_s) \; ds dr \} \), where \(\{B_t;\,t\ge 0\}\) 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 \(S(\lambda)=\lim_{t \rightarrow \infty} \frac{1}{t} \ln M(t)\) exists. We also determine the small \(\lambda\) behavior of the rate \(S(\lambda)\) and show that it is indeed decaying as conjectured in the physics literature.

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Authors and Affiliations

  1. Department of Mathematics, University of California at Irvine, Irvine, CA 92717, USA (e-mail: rcarmona@princeton.edu), 92717, Irvine, CA, US

    René A. Carmona & Lin Xu

Authors
  1. René A. Carmona
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  2. Lin Xu
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Received: 30 June 1995 / In revised form: 26 January 1996

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Carmona, R., Xu, L. Large deviations and exponential decay for the magnetization in a Gaussian random field. Probab Theory Relat Fields 106, 233–247 (1996). https://doi.org/10.1007/s004400050063

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  • Published: 01 October 2002

  • Issue Date: October 1996

  • DOI: https://doi.org/10.1007/s004400050063

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  • Mathematics Subject Classification (1991):60H25
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