dorsal/arxiv
View SchemaOn the bilinear equations for Fredholm determinants appearing in random matrices
| Authors | J. Harnad |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9906004 |
| URL | https://arxiv.org/abs/solv-int/9906004 |
| Journal | J. Nonlinear Math. Phys., volume 9, no. 4 (2002) 530-550 |
Abstract
It is shown how the bilinear differential equations satisfied by Fredholm determinants of integral operators appearing as spectral distribution functions for random matrices may be deduced from the associated systems of nonautonomous Hamiltonian equations satisfied by auxiliary canonical phase space variables introduced by Tracy and Widom. The essential step is to recast the latter as isomonodromic deformation equations for families of rational covariant derivative operators on the Riemann sphere and interpret the Fredholm determinants as isomonodromic $\tau$-functions.
{
"annotation_id": "269650b2-b2d6-4c04-be6b-969400595620",
"date_created": "2026-03-02T18:02:51.603000Z",
"date_modified": "2026-03-02T18:02:51.603000Z",
"file_hash": "96fbf61cf10b94308fe86d987f53d0ecc296c85118872bef197b9fc964e63745",
"private": false,
"record": {
"abstract": "It is shown how the bilinear differential equations satisfied by Fredholm\ndeterminants of integral operators appearing as spectral distribution functions\nfor random matrices may be deduced from the associated systems of nonautonomous\nHamiltonian equations satisfied by auxiliary canonical phase space variables\nintroduced by Tracy and Widom. The essential step is to recast the latter as\nisomonodromic deformation equations for families of rational covariant\nderivative operators on the Riemann sphere and interpret the Fredholm\ndeterminants as isomonodromic $\\tau$-functions.",
"arxiv_id": "solv-int/9906004",
"authors": [
"J. Harnad"
],
"categories": [
"solv-int",
"hep-th",
"math-ph",
"math.MP",
"nlin.SI"
],
"journal_ref": "J. Nonlinear Math. Phys., volume 9, no. 4 (2002) 530-550",
"title": "On the bilinear equations for Fredholm determinants appearing in random matrices",
"url": "https://arxiv.org/abs/solv-int/9906004"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "2a733387-e640-4c65-9f53-92f5a1fa2aa4",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}