# Thermodynamic Limit and Dispersive Regularisation in Matrix Models

## TL;DR

This paper investigates phase transitions in Hermitian matrix models, revealing a dispersive regularisation mechanism that resolves singularities and leads to complex, chaotic behaviors, with implications for understanding nonlinear matrix dynamics.

## Contribution

It introduces a novel dispersive regularisation framework for phase transitions in matrix models, linking them to Toda hierarchy reductions and dispersive shocks.

## Key findings

- Dispersive regularisation resolves singularities at critical points.
- Chaotic behaviors emerge from dispersive shocks in matrix models.
- The mechanism applies to nonlinearities of arbitrary order.

## Abstract

We show that Hermitian matrix models support the occurrence of a new type of phase transition characterised by dispersive regularisation of the order parameter near the critical point. Using the identification of the partition function with a solution of a reduction of the Toda hierarchy, known as Volterra system, we argue that the singularity is resolved by the onset of a multi-dimensional dispersive shock of the order parameter in the space of coupling constants. This analysis explains the origin and mechanism leading to the emergence of chaotic behaviours observed in M^6 matrix models and extends its validity to even nonlinearity of arbitrary order.