Spin depolarization decay rates in α-symmetric stable fields on cubic lattices
✍ Scribed by Sunder Sethuraman; Lin Xu
- Book ID
- 102651427
- Publisher
- John Wiley and Sons
- Year
- 1996
- Tongue
- English
- Weight
- 632 KB
- Volume
- 49
- Category
- Article
- ISSN
- 0010-3640
No coin nor oath required. For personal study only.
✦ Synopsis
We study the asymptotic, long-time behavior of the energy function where {X, : 0 5 s < XI} is the standard random walk on the &dimensional lattice Zd, 1 < a 5 2, and f : R+ + R+ is any nondecreasing concave function. In the special case f(z) = z, our setting represents a lattice model for the study of transverse magnetization of spins diffusing in a homogeneous, a-stable, i.i.d., random, longitudinal field {XV(z) : z E Z d } with common marginal distribution; the standard a-symmetric stable distribution; the parameter X describes the intensity of the field.
Using large-deviation techniques, we show that S,(X, a, f) = limt,, E(t; A; f) exists.
Moreover, we obtain a variational formula for this decay rate S,. Finally, we analyze the behavior Sc(X, a, f) as X + 0 when f(z) = so for all 1 2 p > 0.
Consequently, several physical conjectures with respect to lattice models of transverse magnetization are resolved by setting , CJ = 1 in our results. We show that Sc(X,a, 1) N A" for d 2 3, Xa(ln t)n-l in d = 2, and X. +l in d = 1. @I% John Wiley & Sons, Inc.
2a 1 The Model and Main Results
In [ll, lo], an analysis of the tranverse magnetization of spins diffusing in a random longitudinal field is proposed. These papers give convincing physical arguments when the randomness is Gaussian or derived from an a-stable symmetric distribution. In [2], these arguments are substantiated using largedeviation techniques. Mitra and Doussal in [ 1 I] also address the question of universality by developing a discrete lattice model in which similar results for the continuum case are derived.
We now describe this model: Let {V(z) : 2 E E d } be an i.i.d. random field with common distribution, the a-stable symmetric distribution, having characteristic function