Third-order phase transition: random matrices and screened Coulomb gas with hard walls

TL;DR

This paper proves the existence of a third-order phase transition in constrained random matrix eigenvalues and screened Coulomb gases, providing exact formulas for free energy and identifying electrostatic pressure as the transition's order parameter.

Contribution

It establishes the third-order phase transition for both one-dimensional random matrices with hard walls and higher-dimensional Yukawa gases under confinement, with explicit free energy formulas.

Findings

Third-order phase transition identified in eigenvalues and Yukawa gases.

Exact free energy formulas with a third-derivative jump.

Electrostatic pressure as the transition order parameter.

Abstract

Consider the free energy of a $d$-dimensional gas in canonical equilibrium under pairwise repulsive interaction and global confinement, in presence of a volume constraint. When the volume of the gas is forced away from its typical value, the system undergoes a phase transition of the third order separating two phases (pulled and pushed). We prove this result i) for the eigenvalues of one-cut, off-critical random matrices (log-gas in dimension $d=1$) with hard walls; ii) in arbitrary dimension $d\geq1$ for a gas with Yukawa interaction (aka screened Coulomb gas) in a generic confining potential. The latter class includes systems with Coulomb (long range) and delta (zero range) repulsion as limiting cases. In both cases, we obtain an exact formula for the free energy of the constrained gas which explicitly exhibits a jump in the third derivative, and we identify the 'electrostatic pressure' as the order parameter of the transition. Part of these results were announced in [F. D. Cunden, P. Facchi, M. Ligab\`o and P. Vivo, J. Phys. A: Math. Theor. 51, 35LT01 (2018)].