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Register a new userAsymptotics of the Airy-kernel determinant
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The authors use Riemann-Hilbert methods to compute the constant that arises in the asymptotic behavior of the Airy-kernel determinant of random matrix theory.
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We provide an alternative proof of the classical single-term asymptotics for Toeplitz determinants whose symbols possess Fisher-Hartwig singularities. We also relax the smoothness conditions on the regular part of the symbols and obtain an estimate for the error term in the asymptotics. Our proof is based on the Riemann-Hilbert analysis of the related systems of orthogonal polynomials and on differential identities for Toeplitz determinants. The result discussed in this paper is crucial for the proof of the asymptotics in the general case of Fisher-Hartwig singularities and extensions to Hankel and Toeplitz+Hankel determinants in [15].
We obtain asymptotics in n for the n-dimensional Hankel determinant whose symbol is the Gaussian multiplied by a step-like function. We use Riemann-Hilbert analysis of the related system of orthogonal polynomials to obtain our results.
The Bessel process models the local eigenvalue statistics near $0$ of certain large positive definite matrices. In this work, we consider the probability begin{align*} mathbb{P}Big( mbox{there are no points in the Bessel process on } (0,x_{1})cup(x_{2},x_{3})cupcdotscup(x_{2g},x_{2g+1}) Big), end{align*} where $0<x_{1}<cdots<x_{2g+1}$ and $g geq 0$ is any non-negative integer. We obtain asymptotics for this probability as the size of the intervals becomes large, up to and including the oscillations of order $1$. In these asymptotics, the most intricate term is a one-dimensional integral along a linear flow on a $g$-dimensional torus, whose integrand involves ratios of Riemann $theta$-functions associated to a genus $g$ Riemann surface. We simplify this integral in two generic cases: (a) If the flow is ergodic, we compute the leading term in the asymptotics of this integral explicitly using Birkhoffs ergodic theorem. (b) If the linear flow has certain good Diophantine properties, we obtain improved estimates on the error term in the asymptotics of this integral. In the case when the flow is both ergodic and has good Diophantine properties (which is always the case for $g=1$, and almost always the case for $g geq 2$), these results can be combined, yielding particularly precise and explicit large gap asymptotics.
We obtain large gap asymptotics for a Fredholm determinant with a confluent hypergeometric kernel. We also obtain asymptotics for determinants with two types of Bessel kernels which appeared in random matrix theory.
We study the multipoint distribution of stationary half-space last passage percolation with exponentially weighted times. We derive both finite-size and asymptotic results for this distribution. In the latter case we observe a new one-parameter process we call half-space Airy stat. It is a one-parameter generalization of the Airy stat process of Baik-Ferrari-Peche, which is recovered far away from the diagonal. All these results extend the one-point results previously proven by the authors.
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