Rigorous Inequalities between Length and Time Scales in Glassy Systems

TL;DR

This paper establishes rigorous bounds linking length and time scales in glassy systems, demonstrating that increased relaxation times imply growing correlation lengths, applicable to various models including p-spin glasses.

Contribution

It provides the first rigorous proof of a purely dynamical phase transition in p-spin glass models without thermodynamic singularities.

Findings

Growth of length scales with relaxation time in glassy systems

Existence of a dynamical phase transition in p-spin models

Applicability to finite-dimensional and mean field systems

Abstract

Glassy systems are characterized by an extremely sluggish dynamics without any simple sign of long range order. It is a debated question whether a correct description of such phenomenon requires the emergence of a large correlation length. We prove rigorous bounds between length and time scales implying the growth of a properly defined length when the relaxation time increases. Our results are valid in a rather general setting, which covers finite-dimensional and mean field systems. As an illustration, we discuss the Glauber (heat bath) dynamics of p-spin glass models on random regular graphs. We present the first proof that a model of this type undergoes a purely dynamical phase transition not accompanied by any thermodynamic singularity.