# Real Space Migdal-Kadanoff Renormalisation of Glassy Systems: Recent Results and a Critical Assessment

## TL;DR

This paper reviews recent advances in applying the Migdal-Kadanoff renormalisation group to finite dimensional glassy systems, critically assesses its limitations, and discusses its predictions for glass and spin-glass transitions, especially in relation to infinite dimensions.

## Contribution

It provides a critical assessment of MK-RG's effectiveness in describing glassy systems and explores its predictions for phase transitions and fixed points in various dimensions.

## Key findings

- MK-RG predicts zero-temperature fixed points govern glass and spin-glass transitions.
- Fixed points exist only in dimensions greater than 3 but influence lower-dimensional flows.
- The method suggests large energy and time scales near transitions, consistent with experiments.

## Abstract

In this manuscript, in honour of L. Kadanoff, we present recent progress obtained in the description of finite dimensional glassy systems thanks to the Migdal-Kadanoff renormalisation group (MK-RG). We provide a critical assessment of the method, in particular discuss its limitation in describing situations in which an infinite number of pure states might be present, and analyse the MK-RG flow in the limit of infinite dimensions. MK-RG predicts that the spin-glass transition in a field and the glass transition are governed by zero-temperature fixed points of the renormalization group flow. This implies a typical energy scale that grows, approaching the transition, as a power of the correlation length, thus leading to enormously large time-scales as expected from experiments and simulations. These fixed points exist only in dimensions larger than $d_L>3$ but they nevertheless influence the RG flow below it, in particular in three dimensions. MK-RG thus predicts a similar behavior for spin-glasses in a field and models of glasses and relates it to the presence of avoided critical points.