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

We study the asymptotic, long-time behavior of the energy function E(t; λ; f) = 1/t ln E exp { -tf [1/t ∑cursive Greek chi∈Z^d (λ ∫^t₀ δ{cursive Greek chi} (X_s) ds)^α]} where {X_s : 0 ≤ s < ∞} is the standard random walk on the d-dimensional lattice ℤ^d, 1 < α ≤ 2, and f : ℝ⁺ → ℝ⁺ is any nondecreasing concave function. In the special case f(cursive Greek chi) = cursive Greek chi, 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(cursive Greek chi) : cursive Greek chi ∈ ℤ^d} 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 S_c(λ, α, f) = lim_t→∞ E(t; λ; f) exists. Moreover, we obtain a variational formula for this decay rate S_c. Finally, we analyze the behavior S_c(λ, α, f) as λ → 0 when f(cursive Greek chi) = cursive Greek chi^β 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 S_c(λ, α, 1) ∼ λ^α for d ≥ 3, λα(ln 1/λ)^α-1 in d = 2, and λ^2α/α+1 in d = 1.