Symmetric random matrices and the Pfaff lattice

Details Symmetric random matrices and the Pfaff lattice Symmetric random matrices and the Pfaff lattice

1999-03-08

arxiv.org/abs/solv-int/9903009v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

44 pages

Scientific paper

Consider a symmetric (finite) matrix ensemble, with a certain probability distribution. What is the probability that the spectrum belongs to a certain interval or union of intervals on the real line? In this paper, we show that, upon introducing an appropriate time parameter, this probability is intimately related to Pfaffians, which as a vector satisfy the so-called Pfaff lattice. The latter is a particular reduction of the 2d-Toda lattice. In particular, they satisfy a KP-like equation, but with a right hand side, depending on nearest neighbors. They also satisfy Virasoro constraints, which combined with the KP-like equation lead to inductive equations for the probabilities.

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