[Paper Review] Aging and spin-glass dynamics

Gérard Ben-Arous|ArXiv.org|Apr 24, 2003

Theoretical and Computational Physics15 references20 citations

TL;DR

This paper investigates aging phenomena in disordered spin-glass systems using mathematical models, focusing on the Bouchaud trap model and the spherical Sherrington-Kirkpatrick (SSK) model. It demonstrates that aging emerges via slow relaxation in deep traps or weakly curved directions in high-dimensional energy landscapes, with distinct aging behaviors in different time regimes: power-law decay in the REM-like trap model and a non-trivial limit function $ f(\theta) $ in the SSK model at high inverse temperature $ \beta > \beta_c $, confirming aging through non-Markovian correlation decay.

ABSTRACT

We survey the recent mathematical results about aging in certain simple disordered models. We start by the Bouchaud trap model. We then survey the results obtained for simple models of spin-glass dynamics, like the REM (the Random Energy Model, which is well approximated by the Bouchaud model on the complete graph), then the spherical Sherrington-Kirkpatrick model. We will insist on the differences in phenomenology for different types of aging in different time scales and different models. This talk is based on joint works with A.Bovier, J.Cerny, A.Dembo, V.Gayrard, A.Guionnet, as well as works by C.Newman, R.Fontes, M.Isopi, D.Stein.

Motivation & Objective

• To understand the mathematical foundations of aging in disordered systems, particularly in spin-glass dynamics.
• To analyze aging mechanisms in the Bouchaud trap model on various graphs, especially in the low-temperature phase with heavy-tailed trap times.
• To study the Langevin dynamics of the spherical Sherrington-Kirkpatrick (SSK) model and establish the existence of aging in the low-temperature phase.
• To compare aging phenomenology across different models and time scales, distinguishing between trap-based aging (e.g., REM) and curvature-induced aging (e.g., SSK).
• To derive rigorous results on the limiting two-time correlation function $ C(t_w, t_w + t) $ in the thermodynamic limit $ N \to \infty $.

Proposed method

• Modeling aging via the Bouchaud trap model with i.i.d. exponential energies $ E_x $, leading to heavy-tailed waiting times $ \tau(x) = e^{\beta E_x} $.
• Using the Random Hopping Times (RHT) dynamics ($ a = 0 $) where jump rates depend only on local trap depth $ \tau(x) $, simplifying analysis.
• Analyzing the quenched two-time correlation function $ R^\omega(t_w, t_w + t) = P^\omega(X(t_w) = X(t_w + t)) $ to detect aging.
• Studying the spherical SK model via a stochastic differential equation with a Lagrange multiplier enforcing the spherical constraint $ \sum_i (u^i_t)^2 = N $.
• Reducing the high-dimensional SSK dynamics to an effective system of Ornstein-Uhlenbeck processes driven by eigenvalues of a GOE matrix, enabling explicit computation of the limiting correlation function.
• Proving that the limiting two-time correlation $ C(t_w, t_w + t) $ converges to a non-trivial function $ f(\theta) $ when $ \beta > \beta_c $, with $ \theta = t/t_w $.

Experimental results

Research questions

• RQ1 Does aging occur in the Bouchaud trap model on general graphs, and what is the nature of the aging behavior in the low-temperature phase?
• RQ2 How does the aging behavior in the Random Energy Model (REM), approximated by the Bouchaud model on the complete graph, differ from that in the spherical SK model?
• RQ3 What is the limiting form of the two-time correlation function $ C(t_w, t_w + t) $ in the spherical SK model at high $ \beta $, and does it exhibit aging?
• RQ4 How do the time-scale dependencies of aging differ between trap-based mechanisms (e.g., REM) and curvature-based mechanisms (e.g., SSK)?
• RQ5 Can the non-Markovian behavior of the limiting empirical measure in the SSK model be characterized through the correlation function $ C(t_w, t_w + t) $?

Key findings

• For $ \beta < \beta_c $, the two-time correlation function decays exponentially: $ C(t_w, t_w + t) \leq C_\beta \exp(-\delta_\beta |t - s|) $, indicating no aging.
• At $ \beta = \beta_c $, the correlation decays polynomially as $ t^{-1/2} $ when $ t_w/t $ is bounded, and as $ t_w^{1/2}/(t_w + t) $ otherwise, indicating weak aging.
• For $ \beta > \beta_c $, the limit $ \lim_{t_w \to \infty} C(t_w, t_w + \theta t_w) = f(\theta) $ exists and is non-trivial, confirming aging in the long-time regime.
• When $ t \gg t_w \gg 1 $, the correlation satisfies $ C(t_w, t_w + t) \cdot (t / t_w)^{3/4} \in (c, C) $ for some positive constants $ c, C $, showing slow decay consistent with aging.
• The aging in the spherical SK model arises from slow equilibration in directions corresponding to the top eigenvectors of the random matrix $ J $, where curvature is weak.
• The limiting correlation function in the SSK model is computable via an autonomous renewal equation due to rotational symmetry, enabling rigorous analysis.