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Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum. (English) Zbl 1033.82005

For Hermitian, symmetric and symplectic ensembles it is shown that the probability of belonging the spectrum to one or several intervals satisfies a nonlinear partial differential equation. In the case of Hermitian ensemble this equation is connected with the Toda lattice and KP equation, for the symmetric and symplectic ones with Pfaff lattice and Pfaff-KP equation. The used method is connected with time variables inserting in integrals with consequent verification of integrable lattice equations and Virasoro constraints.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
37K60 Lattice dynamics; integrable lattice equations
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
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