Analytical approaches to time and length scales in models of glasses

TL;DR

This paper reviews analytical results on the growth of static correlation lengths in glassy systems and their relation to dynamic correlation times, including specific models where these lengths are computed.

Contribution

It provides a comprehensive overview of recent analytical methods for understanding length scales in glasses and their connection to dynamics, with explicit calculations in certain models.

Findings

Correlation length growth is linked to equilibrium correlation time.

Explicit calculations of correlation lengths in mean-field and Kac models.

Rigorous bounds connect static length scales to dynamics.

Abstract

The goal of this chapter is to review recent analytical results about the growth of a (static) correlation length in glassy systems, and the connection that can be made between this length scale and the equilibrium correlation time of its dynamics. The definition of such a length scale is first given in a generic setting, including finite-dimensional models, along with rigorous bounds linking it to the correlation time. We then present some particular cases (finite connectivity mean-field models, and Kac limit of finite dimensional systems) where this length can be actually computed.