arXiv subjects

Dan Dai

Publications and source records attributed to Dan Dai.

At least 19 recordsLinked to original sources

arXiv2025

On The Eigenvalue Rigidity of the Jacobi Unitary Ensemble

In this paper, we prove an optimal global rigidity estimate for the eigenvalues of the Jacobi unitary ensemble. Our approach begins by constructing a random measure defined through the eigenvalue counting function. We then prove its convergence to a Gaussian multiplicative chaos measure, which leads to the desired rigidity result. To establish this convergence, we apply a sufficient condition from Claeys et al. \cite{CFL2021} and conduct an asymptotic analysis of the related exponential moments.

math.PR↗

arXiv2025

Nonisospectral deformations of noncommutative Laurent biorthogonal polynomials and matrix discrete Painlev\'e-type equations

In this paper, we establishes a connection between noncommutative Laurent biorthogonal polynomials (bi-OPs) and matrix discrete Painlev\'e (dP) equations. We first apply nonisospectral deformations to noncommutative Laurent bi-OPs to obtain the noncommutative nonisospectral mixed relativistic Toda lattice and its Lax pair. Then, we perform a stationary reduction on this Lax pair to obtain a matrix dP-type equation. The validity of this reduction is demonstrated through a specific choice of weight function and the application of quasideterminant properties. In the scalar case, our matrix dP equation reduces to the known alternate dP II equation.

nlin.SI↗

arXiv2025

The multiplicative constant in asymptotics of higher-order analogues of the Tracy-Widom distribution

In this paper, we are concerned with higher-order analogues of the Tracy-Widom distribution, which describe the eigenvalue distributions in unitary random matrix models near critical edge points. The associated kernels are constructed by functions related to the even members of the Painlev\'{e} I hierarchy $\mathrm{P_{I}^{2k}}, k\in\mathbb{N}^{+}$, and are regarded as higher-order analogues of the Airy kernel. We present a novel approach to establish the multiplicative constant in the large gap asymptotics of the distribution, resolving an open problem in the work of Clayes, Its and Krasovsky. An important new feature of the expression is the involvement of an integral of the Hamiltonian associated with a special, real, pole-free solution for $\mathrm{P_{I}^{2k}}$. In addition, we show that the total integral of the Hamiltonian vanishes for all $k$, and establish a transition from the higher-order Tracy-Widom distribution to the classical one in the asymptotic regime. Our approach can also be adapted to calculate similar critical constants in other problems arising from mathematical physics.

math-ph↗

arXiv2024

Orthogonal polynomials with periodic recurrence coefficients

In this paper, we study a class of orthogonal polynomials defined by a three-term recurrence relation with periodic coefficients. We derive explicit formulas for the generating function, the associated continued fraction, the orthogonality measure of these polynomials, as well as the spectral measure for the associated doubly infinite tridiagonal Jacobi matrix. Notably, while the orthogonality measure may include discrete mass points, the spectral measure(s) of the doubly infinite Jacobi matrix are absolutely continuous. Additionally, we uncover an intrinsic connection between these new orthogonal polynomials and Chebyshev polynomials through a nonlinear transformation of the polynomial variables.

math.CA↗

arXiv2024

Asymptotics of the confluent hypergeometric process with a varying external potential in the super-exponential region

In this paper, we investigate a determinantal point process on the interval $(-s,s)$, associated with the confluent hypergeometric kernel. Let $\mathcal{K}^{(\alpha,\beta)}_s$ 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 $\det(I-\gamma \mathcal{K}^{(\alpha,\beta)}_s)$ as $s \to +\infty$, while simultaneously $\gamma \to 1^-$ in a super-exponential region. In this regime of double scaling limit, our asymptotic result also gives us asymptotics of the eigenvalues $\lambda^{(\alpha, \beta)}_k(s)$ of the integral operator $\mathcal{K}^{(\alpha,\beta)}_s$ 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.

math.PR↗

arXiv2023

Asymptotics and total integrals of the $\mathrm{P}_{\rm I}^{2}$ tritronqu\'{e}e solution and its Hamiltonian

We study the tritronqu\'{e}e solution $u(x,t)$ of the $\mathrm{P}_{\rm I}^{2}$ equation, the second member of the Painlev\'{e} I hierarchy. This solution is pole-free on the real line and has various applications in mathematical physics. We obtain a full asymptotic expansion of $u(x,t)$ as $x\to\pm \infty$, uniformly for the parameter $t$ in a large interval. Based on this result, we successfully derive the total integrals of $u(x,t)$ and the associated Hamiltonian.

math.CA↗

arXiv2022

Asymptotics of the deformed Fredholm determinant of the confluent hypergeometric kernel

In this paper, we consider the deformed Fredholm determinant of the confluent hypergeometric kernel. This determinant represents the gap probability of the corresponding determinantal point process where each particle is removed independently with probability $1- \gamma$, $0 \leq \gamma <1$. We derive asymptotics of the deformed Fredholm determinant when the gap interval tends to infinity, up to and including the constant term. As an application of our results, we establish a central limit theorem for the eigenvalue counting function and a global rigidity upper bound for its maximum deviation.

math-ph↗

arXiv2022

Gap probability for the hard edge Pearcey process

The hard edge Pearcey process is universal in random matrix theory and many other stochastic models. This paper deals with the gap probability for the thinned/unthinned hard edge Pearcey process over the interval $(0,s)$ by working on the relevant Fredholm determinants. We establish an integral representation of the gap probability via a Hamiltonian related a system of coupled differential equations. Together with some remarkable differential identities for the Hamiltonian, we derive the large gap asymptotics for the thinned case, up to and including the constant term. As an application, we also obtain the asymptotic statistical properties of the counting function for the hard edge Pearcey process.

math-ph↗

arXiv2022

On the deformed Pearcey determinant

In this paper, we are concerned with the deformed Pearcey determinant $\det\left(I-γK^{\mathrm{Pe}}_{s,ρ}\right)$, where $0 \leq γ<1$ and $K^{\mathrm{Pe}}_{s,ρ}$ stands for the trace class operator acting on $L^2\left(-s, s\right)$ with the classical Pearcey kernel arising from random matrix theory. This determinant corresponds to the gap probability for the Pearcey process after thinning, which means each particle in the Pearcey process is removed independently with probability $1-γ$. We establish an integral representation of the deformed Pearcey determinant involving the Hamiltonian associated with a family of special solutions to a system of nonlinear differential equations. Together with some remarkable differential identities for the Hamiltonian, this allows us to obtain the large gap asymptotics, including the exact calculation of the constant term, which complements our previous work on the undeformed case (i.e., $γ=1$). It comes out that the deformed Pearcey determinant exhibits a significantly different asymptotic behavior from the undeformed case, which suggests a transition will occur as the parameter $γ$ varies. As an application of our results, we obtain the asymptotics for the expectation and variance of the counting function for the Pearcey process, and a central limit theorem as well.

math-ph↗

arXiv2021

Inner-Imaging Networks: Put Lenses into Convolutional Structure

Despite the tremendous success in computer vision, deep convolutional networks suffer from serious computation costs and redundancies. Although previous works address this issue by enhancing diversities of filters, they have not considered the complementarity and the completeness of the internal structure of the convolutional network. To deal with these problems, a novel Inner-Imaging architecture is proposed in this paper, which allows relationships between channels to meet the above requirement. Specifically, we organize the channel signal points in groups using convolutional kernels to model both the intra-group and inter-group relationships simultaneously. The convolutional filter is a powerful tool for modeling spatial relations and organizing grouped signals, so the proposed methods map the channel signals onto a pseudo-image, like putting a lens into convolution internal structure. Consequently, not only the diversity of channels is increased, but also the complementarity and completeness can be explicitly enhanced. The proposed architecture is lightweight and easy to be implemented. It provides an efficient self-organization strategy for convolutional networks so as to improve their efficiency and performance. Extensive experiments are conducted on multiple benchmark image recognition data sets including CIFAR, SVHN and ImageNet. Experimental results verify the effectiveness of the Inner-Imaging mechanism with the most popular convolutional networks as the backbones.

cs.CV↗

arXiv2021

Graph-based Visual-Semantic Entanglement Network for Zero-shot Image Recognition

Zero-shot learning uses semantic attributes to connect the search space of unseen objects. In recent years, although the deep convolutional network brings powerful visual modeling capabilities to the ZSL task, its visual features have severe pattern inertia and lack of representation of semantic relationships, which leads to severe bias and ambiguity. In response to this, we propose the Graph-based Visual-Semantic Entanglement Network to conduct graph modeling of visual features, which is mapped to semantic attributes by using a knowledge graph, it contains several novel designs: 1. it establishes a multi-path entangled network with the convolutional neural network (CNN) and the graph convolutional network (GCN), which input the visual features from CNN to GCN to model the implicit semantic relations, then GCN feedback the graph modeled information to CNN features; 2. it uses attribute word vectors as the target for the graph semantic modeling of GCN, which forms a self-consistent regression for graph modeling and supervise GCN to learn more personalized attribute relations; 3. it fuses and supplements the hierarchical visual-semantic features refined by graph modeling into visual embedding. Our method outperforms state-of-the-art approaches on multiple representative ZSL datasets: AwA2, CUB, and SUN by promoting the semantic linkage modelling of visual features.

cs.CV↗

arXiv2021

Applications in random matrix theory of a PIII$'$ $τ$-function sequence from Okamoto's Hamiltonian formulation

We consider the singular linear statistic of the Laguerre unitary ensemble consisting of the sum of the reciprocal of the eigenvalues. It is observed that the exponential generating function for this statistic can be written as a Toeplitz determinant with entries given in terms of particular $K$ Bessel functions. Earlier studies have identified the same determinant, but with the $K$ Bessel functions replaced by $I$ Bessel functions, as relating to the hard edge scaling limit of a generalized gap probability for the Laguerre unitary ensemble, in the case of non-negative integer Laguerre parameter. We show that the Toeplitz determinant formed from an arbitrary linear combination of these two Bessel functions occurs as a $τ$-function sequence in Okamoto's Hamiltonian formulation of Painlevé III$'$, and consequently the logarithmic derivative of both Toeplitz determinants satisfies the same $σ$-form Painlevé III$'$ differential equation, giving an explanation of a fact which can be observed from earlier results. In addition, some insights into the relationship between this characterization of the generating function, and its characterization in the $n \to \infty$ limit, both with the Laguerre parameter $α$ fixed, and with $α= n$ (this latter circumstance being relevant to an application to the distribution of the Wigner time delay statistic), are given.

math-ph↗

arXiv2021

Uniform asymptotics for the discrete Laguerre polynomials

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.

math.CA↗

arXiv2020

Asymptotics of Fredholm determinant associated with the Pearcey kernel

The Pearcey kernel is a classical and universal kernel arising from random matrix theory, which describes the local statistics of eigenvalues when the limiting mean eigenvalue density exhibits a cusp-like singularity. It appears in a variety of statistical physics models beyond matrix models as well. We consider the Fredholm determinant of a trace class operator acting on $L^2\left(-s, s\right)$ with the Pearcey kernel. Based on a steepest descent analysis for a $3\times 3$ matrix-valued Riemann-Hilbert problem, we obtain asymptotics of the Fredholm determinant as $s\to +\infty$, which is also interpreted as large gap asymptotics in the context of random matrix theory.

math-ph↗

arXiv2020

Multiple Attentional Pyramid Networks for Chinese Herbal Recognition

Chinese herbs play a critical role in Traditional Chinese Medicine. Due to different recognition granularity, they can be recognized accurately only by professionals with much experience. It is expected that they can be recognized automatically using new techniques like machine learning. However, there is no Chinese herbal image dataset available. Simultaneously, there is no machine learning method which can deal with Chinese herbal image recognition well. Therefore, this paper begins with building a new standard Chinese-Herbs dataset. Subsequently, a new Attentional Pyramid Networks (APN) for Chinese herbal recognition is proposed, where both novel competitive attention and spatial collaborative attention are proposed and then applied. APN can adaptively model Chinese herbal images with different feature scales. Finally, a new framework for Chinese herbal recognition is proposed as a new application of APN. Experiments are conducted on our constructed dataset and validate the effectiveness of our methods.

cs.CV↗

arXiv2020

On integrals of the tronquée solutions and the associated Hamiltonians for the Painlevé II equation

We consider a family of tronquée solutions of the Painelvé II equation \begin{equation*} q''(s)=2q(s)^3+sq(s)-(2α+\frac12), \qquad α> -\frac12, \end{equation*} which is characterized by the Stokes multipliers $$s_1=-e^{-2απi },\quad s_2=ω, \quad s_3=-e^{2 απi} $$ with $ω$ being a free parameter. These solutions include the well-known generalized Hastings-McLeod solution as a special case if $ω=0$. We derive asymptotics of integrals of the tronquée solutions and the associated Hamiltonians over the real axis for $α> -1/2$ and $ω\geq 0$, with the constant terms evaluated explicitly. Our results agree with those already known in the literature if the parameters $α$ and $ω$ are chosen to be special values. Some applications of our results in random matrix theory are also discussed.

math-ph↗

arXiv2019

Gaussian unitary ensembles with pole singularities near the soft edge and a system of coupled Painlevé XXXIV equations

In this paper, we study the singularly perturbed Gaussian unitary ensembles defined by the measure \begin{equation*} \frac{1}{C_n} e^{- n\textrm{tr}\, V(M;λ,\vec{t}\;)}dM, \end{equation*} over the space of $n \times n$ Hermitian matrices $M$, where $V(x;λ,\vec{t}\;):= 2x^2 + \sum_{k=1}^{2m}t_k(x-λ)^{-k}$ with $\vec{t}= (t_1, t_2, \ldots, t_{2m})\in \mathbb{R}^{2m-1} \times (0,\infty)$, in the multiple scaling limit where $λ\to 1$ together with $\vec{t} \to \vec{0}$ as $n\to \infty$ at appropriate related rates. We obtain the asymptotics of the partition function, which is described explicitly in terms of an integral involving a smooth solution to a new coupled Painlevé system generalizing the Painlevé XXXIV equation. The large $n$ limit of the correlation kernel is also derived, which leads to a new universal class built out of the $Ψ$-function associated with the coupled Painlevé system.

math-ph↗

arXiv2019

The distribution function for the maximal height of $N$ non-intersecting Bessel paths

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.

math-ph↗