On the Bilinear Equations for Fredholm Determinants Appearing in Random Matrices

@article{Harnad1999OnTB,
title={On the Bilinear Equations for Fredholm Determinants Appearing in Random Matrices},
journal={Journal of Nonlinear Mathematical Physics},
year={1999},
volume={9},
pages={530 - 550}
}
• Published 10 June 1999
• Mathematics
• Journal of Nonlinear Mathematical Physics
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…
• Mathematics
• 2022
The Hamiltonian approach to isomonodromic deformation systems is extended to include generic rational covariant derivative operators on the Riemann sphere with irregular singularities of arbitrary
It was proved by Akemann et al (2013 Phys. Rev. E 88 052118) that squared singular values of products of M complex Ginibre random matrices form a determinantal point process whose correlation kernel
• Mathematics
• 2003
For one-matrix models with polynomial potentials, the explicit relationship between the partition function and the isomonodromic tau function for the 2 × 2 polynomial differential systems satisfied
Two approaches to the derivation of integrable differential equations for random matrix probabilities are compared and Orthogonal function systems and Toda lattice are seen as the core structure of both approaches and their relationship.
• Physics
• 1999
if their associatedBoltzmann weights satisfy the factorization or star-triangle equations of Mc-Guire [1], Yang [2] and Baxter [3]. For such models the free energy per siteand the one-point
An attempt is made to describe random matrix ensembles with unitary invariance of measure (UE) in a unified way, using a combination of Tracy-Widom (TW) and Adler-Shiota-Van Moerbeke (ASvM)
During the last 15 years or so, and since the pioneering work of E. Wigner, F. Dyson and M.L. Mehta, random matrix theory, combinatorial and perco- lation questions have merged into a very lively

References

SHOWING 1-10 OF 33 REFERENCES

The Hamiltonian structure of the monodromy preserving deformation equations of Jimboet al [JMMS] is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to
• Mathematics
• 1993
The level spacing distributions in the Gaussian Unitary Ensemble, both in the “bulk of the spectrum,” given by the Fredholm determinant of the operator with the sine kernel \(\frac{{\sin \;\pi (x -
• Mathematics
• 1990
AbstractThe approach to isospectral Hamiltonian flow introduced in part I is further developed to include integration of flows with singular spectral curves. The flow on finite dimensional