On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo-Miwa-Ueno differential

Abstract

Several distribution functions in the classical unitarily invariant matrix ensembles are prime examples of isomonodromic tau functions as introduced by Jimbo, Miwa and Ueno (JMU) in the early 1980s \cite{JMU}. Recent advances in the theory of tau functions \cite{ILP}, based on earlier works of B. Malgrange and M. Bertola, have allowed to extend the original Jimbo-Miwa-Ueno differential form to a 1-form closed on the full space of extended monodromy data of the underlying Lax pairs. This in turn has yielded a novel approach for the asymptotic evaluation of isomonodromic tau functions, including the exact computation of all relevant constant factors. We use this method to efficiently compute the tail asymptotics of soft-edge, hard-edge and bulk scaled distribution and gap functions in the complex Wishart ensemble, provided each eigenvalue particle has been removed independently with probability $1-\gamma \in (0,1]$.

44 pages, 14 figures

References in corpus (8)

• Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function
• Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions
• Randomly incomplete spectra and intermediate statistics
• Large deformations of the Tracy-Widom distribution I. Non-oscillatory asymptotics
• Monodromy dependence and connection formulae for isomonodromic tau functions
• Deformations of the Tracy-Widom distribution
• On the connection problem for Painlevé I
• The sine process under the influence of a varying potential

Cited by in corpus (15)

• Fredholm determinants, full counting statistics and Loschmidt echo for domain wall profiles in one-dimensional free fermionic chains
• Irregular conformal blocks and connection formulae for Painlevé V functions
• On some Hamiltonian properties of the isomonodromic tau functions
• Gap probability of the circular unitary ensemble with a Fisher-Hartwig singularity and the coupled Painlevé V system
• Large gap asymptotics for the generating function of the sine point process
• The sine process under the influence of a varying potential
• Gap probability for the hard edge Pearcey process
• The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem
• On the deformed Pearcey determinant
• Short distance asymptotics for a generalized two-point scaling function in the two-dimensional Ising model
• Global rigidity and exponential moments for soft and hard edge point processes
• Entanglement entropies of an interval in the free Schrödinger field theory on the half line
• Asymptotics of the deformed higher order Airy-kernel determinants and applications
• On integrals of the tronquée solutions and the associated Hamiltonians for the Painlevé II equation
• Geometry of the Ising persistence problem and the universal Bonnet-Manin Painlevé VI distribution