### Abstract

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 ∫^{t}_{0} 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1281-1298 |

Number of pages | 18 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 49 |

Issue number | 12 |

DOIs | |

State | Published - Dec 1996 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics