dorsal/arxiv
View SchemaSymmetric random matrices and the Pfaff lattice
| Authors | M. Adler, P. van Moerbeke |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9903009 |
| URL | https://arxiv.org/abs/solv-int/9903009 |
Abstract
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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"abstract": "Consider a symmetric (finite) matrix ensemble, with a certain probability\ndistribution. What is the probability that the spectrum belongs to a certain\ninterval or union of intervals on the real line? In this paper, we show that,\nupon introducing an appropriate time parameter, this probability is intimately\nrelated to Pfaffians, which as a vector satisfy the so-called Pfaff lattice.\nThe latter is a particular reduction of the 2d-Toda lattice. In particular,\nthey satisfy a KP-like equation, but with a right hand side, depending on\nnearest neighbors. They also satisfy Virasoro constraints, which combined with\nthe KP-like equation lead to inductive equations for the probabilities.",
"arxiv_id": "solv-int/9903009",
"authors": [
"M. Adler",
"P. van Moerbeke"
],
"categories": [
"solv-int",
"nlin.SI"
],
"title": "Symmetric random matrices and the Pfaff lattice",
"url": "https://arxiv.org/abs/solv-int/9903009"
},
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