Dynamics for Spherical Spin Glasses: Disorder Dependent Initial Conditions
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Dynamics for Spherical Spin Glasses: Disorder Dependent Initial Conditions Amir Dembo1,2 · Eliran Subag3 Received: 1 September 2019 / Accepted: 9 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We derive the thermodynamic limit of the empirical correlation and response functions in the Langevin dynamics for spherical mixed p-spin disordered mean-field models, starting uniformly within one of the spherical bands on which the Gibbs measure concentrates at low temperature for the pure p-spin models and mixed perturbations of them. We further relate the large time asymptotics of the resulting coupled non-linear integro-differential equations, to the geometric structure of the Gibbs measures (at low temperature), and derive their FDT solution (at high temperature). Keywords Interacting random processes · Disordered systems · Statistical mechanics · Langevin dynamics · Aging · Spin glass models Mathematics Subject Classification Primary: 82C44 · Secondary: 82C31 · 60H10 · 60F15 · 60K35
1 Introduction The thermodynamic limits of a wide class of Markovian dynamics with random interactions, exhibit complex long time behavior, which is of much interest in out of equilibrium statistical physics (c.f. the surveys [14,15,22] and the references therein). This work is about the ther-
Communicated by Ivan Corwin. Research partially supported by BSF Grant 2014019 (A.D. & E.S.), NSF Grants #DMS-1613091, #DMS-1954337 (A.D), and the Simons Foundation (E.S.).
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Eliran Subag [email protected] Amir Dembo [email protected]
1
Department of Statistics, Stanford University, Stanford, CA 94305, USA
2
Department of Mathematics, Stanford University, Stanford, CA 94305, USA
3
Courant Institute, New York University, New York, NY 10012, USA
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A. Dembo, E. Subag
modynamic (N → ∞), long time (t → ∞), behavior of a certain class of systems composed of N Langevin particles xt = (xti )1≤i≤N ∈ R N , interacting with each other through a random potential. More precisely, one considers a diffusion of the form dxt = − f (||xt ||2 /N )xt dt − β∇ HJ (xt )dt + dBt ,
(1.1)
where Bt is an N -dimensional Brownian motion, ||x|| denotes the Euclidean norm of x ∈ R N and differentiable fast growing functions f = f L such that e− f L (r ) approximates √ as L → ∞ N −1 ( N ) of radius the indicator on r = 1, effectively restricting x to the sphere S := S t N √ N . In particular, the spherical, mixed p-spin model (with p ≤ m), has a centered Gaussian potential HJ : R N −→ R of non-negative definite covariance structure Cov HJ (x), HJ (y) = N ν N −1 x, y ,
ν(r ) :=
m
b2p r p
(1.2)
p=2
(see Remark 1.8 on a possible extension to m = ∞). Hereafter we shall realize this potential as HJ (x) =
m
bp
Ji1 ···i p x i1 · · · x i p , bm = 0
(1.3)
1≤i 1 ≤···≤i p ≤N
p=2
for independent centered Gaussian coupling constants J = {Ji1 ···i p }, such that p! Var(Ji1 ...i p ) = N − p+1 , k lk !
(1.4)
where (l1 , l2 , . . .) are the multiplicities of the different elements of the set {i 1 ,
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