Spin depolarization decay rates in α‐symmetric stable fields on cubic lattices

Abstract

We study the asymptotic, long-time behavior of the energy function where {Xs : 0 ≤ s < ∞} is the standard random walk on the d-dimensional lattice Zd, 1 < α ≤ 2, and f:R+ → R+ is any nondecreasing concave function. In the special case f(x) = x, our setting represents a lattice model for the study of transverse magnetization of spins diffusing in a homogeneous, α-stable, i.i.d., random, longitudinal field {λV(x) : x ∈ Zd} with common marginal distribution, the standard α-symmetric stable distribution; the parameter λ describes the intensity of the field. Using large-deviation techniques, we show that Sc(λ α f) = limt→∞ E(t; λ f) exists. Moreover, we obtain a variational formula for this decay rate Sc. Finally, we analyze the behavior Sc(λ α f) as λ → 0 when f(x) = xβ for all 1 ≥ β > 0. Consequently, several physical conjectures with respect to lattice models of transverse magnetization are resolved by setting β = 1 in our results. We show that Sc(λ, α, 1) ≈ λα for d ≥ 3, λagr;(ln 1/λ)α−1 in d = 2, and in d = 1. © 1996 John Wiley & Sons, Inc.