Symmetric matrix ensemble and integrable hydrodynamic chains

Abstract

The partition function of the Symmetric Matrix Ensemble is identified with the $$\tau $$ τ -function of a particular solution of the Pfaff Lattice. We show that, in the case of even power interactions, in the thermodynamic limit, the $$\tau $$ τ -function corresponds to the solution of an integrable chain of hydrodynamic type. We prove that the hydrodynamic chain so obtained is diagonalisable and admits hydrodynamic reductions in Riemann invariants in an arbitrary number of components.

Symmetries and criticality of generalised van der Waals models

“…The problem of constructing suitable partial differential equations for state functions of thermodynamic systems and the study of critical properties in terms of critical asympotics of the solutions to these equations is an active field of research which brought further insights on a variety of classical systems, see e.g. [33][34][35][36][37][38], and appears to be promising for the study of complex systems [39,40]. Studies similar to the present work can be put forward therefore for other systems of physical interest.…”

On solutions to a novel non-evolutionary integrable 1 + 1 PDE

“…Extension of our approach to integrable hydrodynamic chains may also prove to be insightful, e.g. in the context of random matrix models [49]. Proof.…”

Symmetries and criticality of generalised van der Waals models

“…The problem of constructing suitable partial differential equations for state functions of thermodynamic systems and the study of critical properties in terms of critical asymptotics of the solutions to these equations is an active field of research which brought further insights on a variety of classical systems, see e.g. [34][35][36][37][38][39], and appears to be promising for the study of complex systems [40,41]. Studies exploiting the Lie symmetry analysis can be therefore carried out for other systems of physical interest.…”