arXiv subjects

Shuai-Xia Xu

Publications and source records attributed to Shuai-Xia Xu.

At least 19 recordsLinked to original sources

arXiv2026

Hard to soft edge transition for the Muttalib-Borodin ensembles with integer parameter $θ$

We find the universal limiting correlation kernels of the Muttalib-Borodin (MB) ensembles with integer parameter $θ\geq 2$ at $0$ in the transitive regime between the hard edge regime and the soft edge regime. This generalizes the previously studied hard edge to soft edge transition in unitarily invariant random matrix theory by Its, Kuijlaars and Östensson, which is the $θ= 1$ special case of our MB ensemble. The derivation is based on the vector Riemann-Hilbert (RH) problems for the biorthogonal polynomials associated with the MB ensemble. In the analysis of the RH problems, we construct matrix-valued model RH problems of size $(θ+ 1) \times (θ+ 1)$, and prove the solvability of the model RH problems by a vanishing lemma. The new limiting correlation kernels are proved to be universal for a large class of potential functions, and they interpolate the Meijer G-kernels for the hard edge regime and the Airy kernel for the soft edge regime. We observe that the new limiting correlation kernels have the integrability that is not seen in previous studies in random matrix theory and determinantal point processes. In the $θ= 2$ case, we give a detailed analysis of the Lax pair associated with the model RH problem, show that it results in the Chazy I equation, which has a Painlevé IV reduction, and find that the Lax pair is in the Drinfeld-Sokolov hierarchies.

math-ph↗

arXiv2025

Critical Hermitian matrix model with external source and Boussinesq hierarchy

We consider the random Hermitian matrix model of dimension $2n$, with external source, defined by the probability density function \begin{equation*} \frac{1}{Z_{2n}} \lvert \det(M) \rvert^α e^{-2n\mathrm{Tr} (V(M) - AM)}, \quad V(x) = \frac{x^4}{4} - t\frac{x^2}{2}, \end{equation*} where the external source $A$ has two eigenvalues $\pm a$ of equal multiplicity. We investigate the limiting local statistics of the eigenvalues of $M$ around $0$ in certain critical regimes as $n \to \infty$. When the parameters $t$ and $a$ lie on a critical curve along which the limiting mean eigenvalue density vanishes as $|x|^{1/3}$, the double scaling limit of the correlation kernel is constructed from functions associated with the Boussinesq equation. This new limiting kernel reduces to the classical Pearcey kernel when $α= 0$. Furthermore, in the multi-critical case where the limiting mean eigenvalue density vanishes as $|x|^{5/3}$, the limiting kernel is built from the second member of the Boussinesq hierarchy. We derive the results by transforming the random matrix model into biorthogonal ensembles that are analogous to the Muttalib-Borodin ensemble, and then analyzing its asymptotic behavior via a vector Riemann-Hilbert problem.

math-ph↗

arXiv2025

Large time and distance asymptotics of the one-dimensional impenetrable Bose gas and Painlevé IV transition

In the present paper, we study the time-dependent correlation function of the one-dimensional impenetrable Bose gas, which can be expressed in terms of the Fredholm determinant of a time-dependent sine kernel and the solutions of the separated NLS equations. We derive the large time and distance asymptotic expansions of this determinant and the solutions of the separated NLS equations in both the space-like region and time-like region of the $(x,t)$-plane. Furthermore, we observe a phase transition between the asymptotic expansions in these two different regions. The phase transition is then shown to be described by a particular solution of the Painlevé IV equation.

math-ph↗

arXiv2025

Asymptotics of the partition function of the perturbed Gross-Witten-Wadia unitary matrix model

We consider the asymptotics of the partition function of the extended Gross-Witten-Wadia unitary matrix model by introducing an extra logarithmic term in the potential. The partition function can be written as a Toeplitz determinant with entries expressed in terms of the modified Bessel functions of the first kind and furnishes a $τ$-function sequence of the Painlevé III' equation. We derive the asymptotic expansions of the Toeplitz determinant up to and including the constant terms as the size of the determinant tends to infinity. The constant terms therein are expressed in terms of the Riemann zeta-function and the Barnes $G$-function. A third-order phase transition in the leading terms of the asymptotic expansions is also observed.

math-ph↗

arXiv2025

Asymptotics of Fredholm determinant solutions of the noncommutative Painlevé II equation

In this paper, we study the asymptotic behavior of a family of pole-free solutions to the noncommutative Painlevé II equation. These particular solutions can be expressed in terms of the Fredholm determinant of the matrix version of the classical Airy operator, which are analogous to the Hastings-McLeod solution and the Ablowitz-Segur solution of the classical Painlevé II equation. Using the Riemann-Hilbert approach, we derive the asymptotics of the Fredholm determinant and the associated particular solutions $β(\vec{s})$ to the noncommutative Painlevé II equation in the regime $\vec{s}=\left(s+\fracτ{\sqrt{-s}},s-\fracτ{\sqrt{-s}}\right)$ with $τ\ge 0$ and $s\to-\infty$. The solutions depend on a two by two Hermitian matrix with eigenvalues in the interval $(-1,1)$. The asymptotics are expressed in terms of one parameter family of special solutions of the classical Painlevé V equation. Furthermore, we derive the asymptotics, including the connection formulas, for this one parameter family of solutions of the Painlevé V equation both as $ix\to -\infty$ and $x\to 0$.

math-ph↗

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 Painlevé 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

Asymptotics of the finite-temperature sine kernel determinant

In the present paper, we study the asymptotics of the Fredholm determinant $D(x,s)$ of the finite-temperature deformation of the sine kernel, which represents the probability that there is no particles on the interval $(-x/π,x/π)$ in the bulk scaling limit of the finite-temperature fermion system. The variable $s$ in $D(x,s)$ is related to the temperature. The determinant also corresponds to the finite-temperature correlation function of one dimensional Bose gas. We derive the asymptotics of $D(x,s)$ in several different regimes in the $(x,s)$-plane. A third-order phase transition is observed in the asymptotic expansions as both $x$ and $s$ tend to positive infinity at certain related speed. The phase transition is then shown to be described by an integral involving the Hastings-McLeod solution of the second Painlevé equation.

math-ph↗

arXiv2024

On the Fredholm determinant of the confluent hypergeometric kernel with discontinuities

We consider the determinantal point process with the confluent hypergeometric kernel. This process is a universal point process in random matrix theory and describes the distribution of eigenvalues of large random Hermitian matrices near the Fisher-Hartwig singularity. Applying the Riemann-Hilbert method, we study the generating function of this process on any given number of intervals. It can be expressed as the Fredholm determinant of the confluent hypergeometric kernel with $n$ discontinuities. In this paper, we derive an integral representation for the determinant by using the Hamiltonian of the coupled Painlevé V system. By evaluating the total integral of the Hamiltonian, we obtain the asymptotics of the determinant as the $n$ discontinuities tend to infinity up to and including the constant term. Here the constant term is expressed in terms of the Barnes $G$-function.

math-ph↗

arXiv2024

Asymptotics of the determinant of the modified Bessel functions and the second Painlevé equation

In the paper, we consider the extended Gross-Witten-Wadia unitary matrix model by introducing a logarithmic term in the potential. The partition function of the model can be expressed equivalently in terms of the Toeplitz determinant with the $(i,j)$-entry being the modified Bessel functions of order $i-j-ν$, $ν\in\mathbb{C}$. When the degree $n$ is finite, we show that the Toeplitz determinant is described by the isomonodromy $τ$-function of the Painlevé III equation. As a double scaling limit, %In the double scaling limit as the degree $n\to\infty$, we establish an asymptotic approximation of the logarithmic derivative of the Toeplitz determinant, expressed in terms of the Hastings-McLeod solution of the inhomogeneous Painlevé II equation with parameter $ν+\frac{1}{2}$. The asymptotics of the leading coefficient and recurrence coefficient of the associated orthogonal polynomials are also derived. We obtain the results by applying the Deift-Zhou nonlinear steepest descent method to the Riemann-Hilbert problem for orthogonal polynomials on the Hankel loop. The main concern here is the construction of a local parametrix at the critical point $z=-1$, where the $ψ$-function of the Jimbo-Miwa Lax pair for the inhomogeneous Painlevé II equation is involved.

math-ph↗

arXiv2023

Asymptotics of the deformed higher order Airy-kernel determinants and applications

We study the one-parameter family of Fredholm determinants $\det(I-ρ^2\mathcal{K}_{n,x})$, $ρ\in\mathbb{R}$, where $\mathcal{K}_{n,x}$ stands for the integral operator acting on $L^2(x,+\infty)$ with the higher order Airy kernel. This family of determinants represents a new universal class of distributions which is a higher order analogue of the classical Tracy-Widom distribution. Each of the determinants admits an integral representation in terms of a special real solution to the $n$-th member of the Painlevé II hierarchy. Using the Riemann-Hilbert approach, we establish asymptotics of the determinants and the associated higher order Painlevé II transcendents as $x\to -\infty$ for $0<|ρ|<1$ and $|ρ|>1$, respectively. In the case of $0<|ρ|<1$, we are able to calculate the constant term in the asymptotic expansion of the determinants, while for $|ρ|>1$, the relevant asymptotics exhibit singular behaviors. Applications of our results are also discussed, which particularly include asymptotic statistical properties of the counting function for the random point process defined by the higher order Airy kernel.

math-ph↗

arXiv2023

Clarkson-McLeod solutions of the fourth Painlevé equation and the parabolic cylinder-kernel determinant

The Clarkson-McLeod solutions of the fourth Painlevé equation behave like $κD_{α-\frac{1}{2}}^2(\sqrt{2}x)$ as $x\rightarrow +\infty$, where $κ$ is some real constant and $D_{α-\frac{1}{2}}(x)$ is the parabolic cylinder function. Using the Deift-Zhou nonlinear steepest descent method, we derive the asymptotic behaviors for this class of solutions as $x\to-\infty$. This completes a proof of Clarkson and McLeod's conjecture on the asymptotics of this family of solutions. The total integrals of the Clarkson-McLeod solutions and the asymptotic approximations of the $σ$-form of this family of solutions are also derived. Furthermore, we find a determinantal representation of the $σ$-form of the Clarkson-McLeod solutions via an integrable operator with the parabolic cylinder kernel.

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

Singular asymptotics for the Clarkson-McLeod solutions of the fourth Painlevé equation

We consider the Clarkson-McLeod solutions of the fourth Painlevé equation. This family of solutions behave like $κD_{α-\frac{1}{2}}^2(\sqrt{2}x)$ as $x\rightarrow +\infty$, where $κ$ is an arbitrary real constant and $D_{α-\frac{1}{2}}(x)$ is the parabolic cylinder function. Using the Deift-Zhou nonlinear steepest descent method, we obtain the singular asymptotics of the solutions as $x\to-\infty$ when $κ\left( κ-κ^*\right )>0$ for some real constant $κ^*$. The connection formulas are also explicitly evaluated. This proves and extends Clarkson and McLeod's conjecture that when the parameter $κ>κ^*>0$, the Clarkson-McLeod solutions have infinitely many simple poles on the negative real axis.

math-ph↗

arXiv2022

Laguerre Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé V System

We study the Hankel determinant generated by the Laguerre weight with jump discontinuities at $t_k, k=1,\cdots,m$. By employing the ladder operator approach to establish Riccati equations, we show that $σ_n(t_1,\cdots,t_m)$, the logarithmic derivative of the $n$-dimensional Hankel determinant, satisfies a generalization of the $σ$-from of Painlevé V equation. Through investigating the Riemann-Hilbert problem for the associated orthogonal polynomials and via Lax pair, we express $σ_n$ in terms of solutions of a coupled Painlevé V system. We also build relations between the auxiliary quantities introduced in the above two methods, which provides connections between the Riccati equations and Lax pair. In addition, when each $t_k$ tends to the hard edge of the spectrum and $n$ goes to $\infty$, the scaled $σ_n$ is shown to satisfy a generalized Painlevé III equation.

nlin.SI↗

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

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

Isomonodromy sets of accessory parameters for Heun class equations

In this paper, we consider the monodromy and, in particularly, the isomonodromy sets of accessory parameters for the Heun class equations. We show that the Heun class equations can be obtained as limits of the linear systems associated with the Painlevé equations when the Painlevé transcendents go to one of the actual singular points of the linear systems. While the isomonodromy sets of accessory parameters for the Heun class equations are described by the Taylor or Laurent coefficients of the corresponding Painlevé functions, or the associated tau functions, at the positions of the critical values. As an application of these results, we derive some asymptotic approximations for the isomonodromy sets of accessory parameters in the Heun class equations, including the confluent Heun equation, the doubly-confluent Heun equation and the reduced biconfluent Heun equation.

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↗