arXiv subjects
Publications and source records attributed to Luming Yao.
By showing that the symmetrically transformed Bessel kernel admits a full asymptotic expansion for large parameter, we establish a hard-to-soft edge transition expansion. This resolves a conjecture recently proposed by Bornemann.
In this paper, we are concerned with the following noncommutative Painlevé II equation \begin{equation*} \mathbf{D}^2 β_1 = 4\mathbf{s} β_1 +4 β_1 \mathbf{s} +8 β_1^3, \end{equation*} where $β_1=β_1(\vec{s})$ is an $n \times n$ matrix-valued function of $\vec{s}=(s_1,\ldots,s_n)$, $\mathbf{s}=\diag(s_1,\ldots,s_n)$ and $\mathbf{D}=\sum_{j=1}^n\frac{\partial}{\partial s_j}$. If $n=1$, it reduces to the classical Painlevé II equation up to a scaling. Given an arbitrary $n \times n$ constant matrix $C=\left(c_{j k}\right)_{j, k=1}^n$, a remarkable result due to Bertola and Cafasso asserts that there exists a unique solution $β_1(\vec{s})=β_1(\vec{s};C)$ of the noncommutative PII equation such that its $(k,l)$-th entry behaves like $-c_{kl} \Ai (s_k+s_l)$ as $S= \frac{1}{n}\sum_{i=1}^n s_j\to+\infty$, where $\Ai$ stands for the standard Airy function. For a class of structured matrices $C$, we establish asymptotics of the associated solutions as $S \to -\infty$, which particularly include the so-called connection formulas. In the present setting, it comes out that the solution exhibits a hybrid behavior in the sense that each entry corresponds to either an extension of the Hastings-McLeod solution or an extension of the Ablowitz-Segur solution for the PII equation. It is worthwhile to emphasize the asymptotics of the $(k,l)$-th entry as $S \to -\infty$ cannot be deduced solely from its behavior as $S \to +\infty$ in general, which actually also depends on the positive infinity asymptotics of the $(l,k)$-th entry. This new and intriguing phenomenon disappears in the scalar case.
A special type of geometric situation in ensembles of non-intersecting paths occurs when the non-intersecting trajectories are required to be nonnegative so that the limit shape becomes tangential to the hard-edge $0$. The local fluctuation is governed by the universal hard edge tacnode process, which also arises from some tiling problems. It is the aim of this work to explore the integrable structure and asymptotics for the gap probability of the hard edge thinned/unthinned tacnode process over $(0,s)$. We establish an integral representation of the gap probability in terms of the Hamiltonian associated with a system of differential equations. With the aids of some remarkable differential identities for the Hamiltonian, we are able to derive the associated large gap asymptotics, up to and including the constant term in the thinned case. Some applications of our results are discussed as well.
In this paper, we investigate a determinantal point process on the interval $(-s,s)$, associated with the confluent hypergeometric kernel. Let $\mathcal{K}^{(α,β)}_s$ denote the trace class integral operator acting on $L^2(-s, s)$ with the confluent hypergeometric kernel. Our focus is on deriving the asymptotics of the Fredholm determinant $\det(I-γ\mathcal{K}^{(α,β)}_s)$ as $s \to +\infty$, while simultaneously $γ\to 1^-$ in a super-exponential region. In this regime of double scaling limit, our asymptotic result also gives us asymptotics of the eigenvalues $λ^{(α, β)}_k(s)$ of the integral operator $\mathcal{K}^{(α,β)}_s$ as $s \to +\infty$. Based on the integrable structure of the confluent hypergeometric kernel, we derive our asymptotic results by applying the Deift-Zhou nonlinear steepest descent method to analyze the related Riemann-Hilbert problem.
The tacnode process is a universal determinantal point process arising from non-intersecting particle systems and tiling problems. It is the aim of this work to explore the integrable structure and large gap asymptotics for the gap probability of the thinned/unthinned tacnode process over $(-s,s)$. We establish an integral representation of the gap probability in terms of the Hamiltonian associated with a system of differential equations. With the aids of some remarkable differential identities for the Hamiltonian, we also compute large gap asymptotics, up to and including the constant term in the thinned case. As direct applications, we obtain expectation, variance and a central limit theorem for the associated counting function.
We study the Fredholm determinant of an integral operator associated to the hard edge Pearcey kernel. This determinant appears in a variety of random matrix and non-intersecting paths models. By relating the logarithmic derivatives of the Fredholm determinant to a $3 \times 3$ Riemann-Hilbert problem, we obtain asymptotics of the determinant, which is also known as the large gap asymptotics for the corresponding point process.
In this paper, we consider the discrete Laguerre polynomials $P_{n, N}(z)$ orthogonal with respect to the weight function $w(x) = x^α e^{-N cx}$ supported on the infinite nodes $L_N = \{ x_{k,N} = \frac{k^2}{N^2}, k \in \mathbb{N} \}$. We focus on the "band-saturated region" situation when the parameter $c > \frac{π^2}{4}$. As $n \to \infty$, uniform expansions for $P_{n, n}(z)$ are achieved for $z$ in different regions in the complex plane. Typically, the Airy-function expansions and Gamma-function expansions are derived for $z$ near the endpoints of the band and the origin, respectively. The asymptotics for the normalizing coefficient $h_{n, N}$, recurrence coefficients $\mathscr{B}_{n, N}$ and $\mathscr{A}_{n, N}^2$, are also obtained. Our method is based on the Deift-Zhou steepest descent method for Riemann-Hilbert problems.
In this paper, we consider $N$ non-intersecting Bessel paths starting at $x=a\geq 0$, and conditioned to end at the origin $x=0$. We derive the explicit formula of the distribution function for the maximum height. Depending on the starting point $a>0$ or $a=0$, the distribution functions are also given in terms of the Hankel determinants associated with the multiple discrete orthogonal polynomials or discrete orthogonal polynomials, respectively.