Symmetric Matrix Ensemble and Integrable Hydrodynamic Chains

TL;DR

This paper links the symmetric matrix ensemble's partition function to integrable hydrodynamic chains, revealing new solvable structures and reductions in the thermodynamic limit.

Contribution

It establishes a connection between the symmetric matrix ensemble's tau-function and integrable hydrodynamic chains, providing new insights into their solvability and reductions.

Findings

Partition function identified with tau-function of Pfaff Lattice

Hydrodynamic chain is diagonalisable and admits Riemann invariant reductions

Results applicable in the thermodynamic limit for even power interactions

Abstract

The partition function of the Symmetric Matrix Ensemble is identified with the $\tau-$function of a particular solution of the Pfaff Lattice. We show that in the case of even power interactions, in the thermodynamic limit, the $\tau-$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.