dorsal/arxiv
View SchemaOn the relation between orthogonal, symplectic and unitary matrix ensembles
| Authors | Harold Widom |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9804005 |
| URL | https://arxiv.org/abs/solv-int/9804005 |
| DOI | 10.1023/A:1004536018336 |
| Journal | J.Statist.Phys. 94 (1999) 347-364 |
Abstract
For the unitary ensembles of $N\times N$ Hermitian matrices associated with a weight function $w$ there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For the orthogonal and symplectic ensembles of Hermitian matrices there are $2\times2$ matrix kernels, usually constructed using skew-orthogonal polynomials, which play an analogous role. These matrix kernels are determined by their upper left-hand entries. We derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles. We also show that whenever $w'/w$ is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of $w'/w$. General formulas are obtained for these extra terms. We do not use skew-orthogonal polynomials in the derivations.
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"abstract": "For the unitary ensembles of $N\\times N$ Hermitian matrices associated with a\nweight function $w$ there is a kernel, expressible in terms of the polynomials\northogonal with respect to the weight function, which plays an important role.\nFor the orthogonal and symplectic ensembles of Hermitian matrices there are\n$2\\times2$ matrix kernels, usually constructed using skew-orthogonal\npolynomials, which play an analogous role. These matrix kernels are determined\nby their upper left-hand entries. We derive formulas expressing these entries\nin terms of the scalar kernel for the corresponding unitary ensembles. We also\nshow that whenever $w\u0027/w$ is a rational function the entries are equal to the\nscalar kernel plus some extra terms whose number equals the order of $w\u0027/w$.\nGeneral formulas are obtained for these extra terms. We do not use\nskew-orthogonal polynomials in the derivations.",
"arxiv_id": "solv-int/9804005",
"authors": [
"Harold Widom"
],
"categories": [
"solv-int",
"hep-th",
"math.SP",
"nlin.SI"
],
"doi": "10.1023/A:1004536018336",
"journal_ref": "J.Statist.Phys. 94 (1999) 347-364",
"title": "On the relation between orthogonal, symplectic and unitary matrix ensembles",
"url": "https://arxiv.org/abs/solv-int/9804005"
},
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