平均场自旋玻璃动力学REM的普适性
Universality of the REM for dynamics of mean-field spin glasses
摘要：
我们考虑一个版本的Glauber动力学的p-自旋谢灵顿-柯克帕特里克模型的自旋玻璃，可以看作是一个时变的简单随机游动的N维超立方体。我们证明了，对于任意p>2和任意逆温度β>0，存在常数g>0，使得对于所有指数时间尺度，$\exp(γN)$，其中$γ< g$，适当重新标度的时钟过程（时变过程），收敛于α稳定从属子，其中α=γ/β^2<1。此外，在这些时间尺度上，动力学表现出老化，时间-时间相关函数收敛到该α稳定从属子的反正弦律。换句话说，直到重新标度，在这些时间尺度上（比系统的平衡时间更短），p自旋模型的动力学以与REM相同的方式老化，并通过扩展Bouchaud的REM类陷阱模型，证实后者是广泛系统的普遍老化机制。SK模型（p=2的情况）似乎属于不同的普适类。
英文摘要：
We consider a version of a Glauber dynamics for a p-spin Sherrington--Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for any p>2 and any inverse temperature β>0, there exist constants g>0, such that for all exponential time scales, $\exp(γN)$, with $γ< g$, the properly rescaled clock process (time-change process), converges to an α-stable subordinator where α=γ/β^2<1. Moreover, the dynamics exhibits aging at these time scales with time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system), the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud's REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p=2) seems to belong to a different universality class.
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