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

We consider a continuous model for transverse magnetization of spins diffusing in a homogeneous Gaussian random longitudinal field {XV (x); x ∈ ℝ^d}, 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{-λ²∫^t₀ ∫^t₀ K(B_r - B_s) dsdr}, where {B_t; 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) = lim_t→∞ 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.