Random matrix ensembles and integrable differential identities

Integrable differential identities, together with ensemble-specific initial conditions, provide an effective approach for the characterisation of relevant observables and state functions in random matrix theory. We develop this approach for the unitary and orthogonal ensembles. In particular, we focus on a reduction where the probability measure is induced by a Hamiltonian expressed as a formal series of even interaction terms. We show that the order parameters for the unitary ensemble, that is associated with the Volterra lattice, provide a solution of the modified KP equation. The analogous reduction for the orthogonal ensemble, associated with the Pfaff lattice, leads to a new integrable chain. A key step for the calculation of order parameters for the orthogonal ensemble is the evaluation of the initial condition by using a map from orthogonal to skew-orthogonal polynomials. The thermodynamic limit leads to an integrable system (a chain for the orthogonal ensemble) of hydrodynamic type. Intriguingly, we find that the solution to the initial value problem for both the discrete system and its continuum limit are given by the very same semi-discrete dynamical chain.