Large Gap Asymptotics at the Hard Edge for Product Random Matrices and Muttalib–Borodin Ensembles

Abstract

We study the distribution of the smallest eigenvalue for certain classes of positive-definite Hermitian random matrices, in the limit where the size of the matrices becomes large. Their limit distributions can be expressed as Fredholm determinants of integral operators associated to kernels built out of Meijer G-functions or Wright's generalized Bessel functions. They generalize in a natural way the hard edge Bessel kernel Fredholm determinant. We express the logarithmic derivatives of the Fredholm determinants identically in terms of a 2×2 Riemann-Hilbert problem, and use this representation to obtain the so-called large gap asymptotics.

Fredholm Determinant Solutions of the Painlevé II Hierarchy and Gap Probabilities of Determinantal Point Processes

“…An important ingredient in this analysis is the construction of a suitable g-function, for which we need to exploit results related to the theory of topological minimal models of type A n . The general strategy to prove Theorem 1.4 and Theorem 1.6 shows similarities with the one from [13], where large gap asymptotics were obtained for Fredholm determinants arising from product random matrices. Finally, in Appendix A, we prove that the kernels (1.1) define point processes.…”

Gap Probability at the Hard Edge for Random Matrix Ensembles with Pole Singularities in the Potential

“…It would then be natural to derive the large s asymptotics of det(I − K PIII ) and to ask for its Painlevé type formula, which will be the main results of the present work stated in what follows. We note that similar problems have been addressed for other generalizations of Bessel kernel recently in [16,26,35,45].…”

“…When E(s) admits a representation in terms of a Fredholm determinant associated with certain kernel, this conjecture is supported by all the classical kernels encountered in random matrix theory, which include the sine kernel [23] (β = 0), the Airy kernel [39] (β = 1/2) and the Bessel kernel (1.29) (β = −1/2). Other evidences beyond these classical cases include higher Painlevé I kernels [17] (β = 2l + 1/2, l ∈ N), the Painlevé II kernel [7] (β = 2), the Meijer G-kernels (β = − l l+1 , l ∈ N) and Wright's generalized Bessel kernels (β = − 1 1+θ , θ > 0) investigated in [16].…”

Higher Order Large Gap Asymptotics at the Hard Edge for Muttalib–Borodin Ensembles

“…There are several expressions available in the literature for the kernel K(x, y) in (1.2); in [6] it is written as a series, and also in terms of Wright's generalized Bessel functions. For us, the following double contour integral expression (from [12]) will be important:…”