Singular perturbation near mode-coupling transition

TL;DR

This paper develops a systematic perturbation theory near the mode-coupling transition in the spherical p-spin glass model, revealing how late-stage invariance determines relaxation dynamics and alpha-relaxation time.

Contribution

It introduces a novel perturbation framework with multiple time scales to analyze the mode-coupling transition in spin glasses.

Findings

Identifies invariance under time dilatation in late stages.

Derives an exact expression for alpha-relaxation time.

Provides insights into the structure of solutions near the transition.

Abstract

We study the simplest mode-coupling equation which describes the time correlation function of the spherical p-spin glass model. We formulate a systematic perturbation theory near the mode-coupling transition point by introducing multiple time scales. In this formulation, the invariance with respect to the dilatation of time in a late stage yields an arbitrary constant in a leading order expression of the solution. The value of this constant is determined by a solvability condition associated with a linear singular equation for perturbative corrections in the late stage. The solution thus constructed provides exactly the alpha-relaxation time.