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