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

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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∈Zd (λ ∫t0 δ{cursive Greek chi} (Xs) ds)α]} 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(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 ∈ 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(cursive Greek chi) = cursive Greek chiβ 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, λα(ln 1/λ)α-1 in d = 2, and λ2α/α+1 in d = 1.

Original languageEnglish (US)
Pages (from-to)1281-1298
Number of pages18
JournalCommunications on Pure and Applied Mathematics
Issue number12
StatePublished - Dec 1996

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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