
Yang ChenUniversity of Macau 2012 to 2023
Yang Chen
Ph. D. Physics
Looking for collaborators. If you are a student and would like to study PhD with me email chenyayang57@gmail.com
About
260
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Introduction
I joined Math. Dept. University of Macau,as Prof of Math., August 2012. I was Lecturer, Reader and Professor(Personal Chair) of Mathematical Physics,1992 -- 2012, Imperial College London. Research area: Random Matrix Theory, Information theory of Wireless Comm.,Multivariate Statistics,Painleve eqns.,counting problems in QCD. Submit papers on RMT and related topics to yayangchen@um.edu.mo; I am the Founding Editor of Random Matrices:Theory and Applications, World Scientific Publication.
Additional affiliations
Education
April 1983 - June 1986
March 1981 - February 1983
February 1978 - March 1981
Publications
Publications (260)
The purpose of this paper is to compute asymptotically Hankel determinants for weights that are supported in a semi-infinite interval. The main idea is to reduce the problem to determinants of other operators whose determinant asymptotics are well known.
In this paper we study a particular Painlevé V (denoted P V) that arises from Multi-Input-Multi-Output (MIMO) wireless communication systems. Such a P V appears through its intimate relation with the Hankel determinant that describes the moment generating function (MGF) of the Shannon capacity. This originates through the multiplication of the Lagu...
We investigate the large N behavior of the smallest eigenvalue, λ N , of an (N + 1) × (N + 1) Hankel (or moments) matrix H N , generated by the weight w(x) = x α (1 − x) β , x ∈ [0, 1], α > −1, β > −1. By applying the arguments of Szegö, Widom and Wilf, we establish the asymptotic formula for the orthonormal polynomials P n (z), z ∈ C \ [0, 1], ass...
In this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble, namely the probability that the interval (−a, a) (0 < a < 1) is free of eigenvalues. Using the ladder operator technique for orthogonal polynomials and the associated supplementary conditions, we derive three quantities instrumental in the gap probabilit...
In this paper, we study the recurrence coe�fficients of a deformed or semi-classical
Laguerre polynomials orthogonal with respect to the weight
w(x; s) = w(x; �; s) := x�eN(x+s(x2x)); 0 � x < 1:
Here � > 1, 0 � s � 1 and N > 0. We will describe this problem in terms of the
ratio r := n
N ; where ultimately r is bounded away from 0, and close to...
We study the Hankel determinant generated by the Gaussian weight with jump discontinuities at t 1 t_1 , …, t m t_m . By making use of a pair of ladder operators satisfied by the associated monic orthogonal polynomials and three supplementary conditions, we show that the logarithmic derivative of the Hankel determinant satisfies a second-order parti...
In this paper, we investigate the Hankel determinant generated by a singular Laguerre weight with two parameters. Using ladder operators adapted to monic orthogonal polynomials associated with the weight, we show that one of the auxiliary quantities is a solution to the Painlevé III′ equation and derive the discrete σ-forms of two logarithmic parti...
We study the monic polynomials Pn(x; t), orthogonal with respect to a symmetric perturbed Gaussian weight function w(x)=w(x;t)≔e−x21+tx2λ,x∈R, with t>0,λ∈R. This problem is related to single-user multiple-input multiple-output systems in information theory. It is shown that the recurrence coefficient βn(t) is related to a particular Painlevé V tran...
We discuss the monic polynomials of degree n orthogonal with respect to the perturbed Gaussian weight w(z,t)=|z|α(z2+t)λe−z2,z∈R,t>0,α>−1,λ>0, which arises from a symmetrization of a semi-classical Laguerre weight wLag(z,t)=zγ(z+t)ρe−z,z∈R+,t>0,γ>−1,ρ>0. The weight wLag(z) has been widely investigated in multiple-input multi-output antenna wireless...
In this paper, we study the asymptotic behavior of the smallest eigenvalue λ N \lambda _N , of the ( N + 1 ) × ( N + 1 ) (N+1)\times (N+1) Hankel matrix M N = ( μ j + k ) 0 ≤ j , k ≤ N \mathcal {M}_N=(\mu _{j+k})_{0\le j,k\le N} generated by the semi-classical Hermite weight w ( z , t ) = | z | λ exp ( − z 2 + t z ) , z , t ∈ R , λ > − 1 w(z,t)=|...
In this paper, we study the large $n$ asymptotics of Hankel determinants generated by some nontrivial Freud weight functions. These are obtained from the integral representations of the Hankel determinants in terms of the recurrence coefficients for the corresponding orthogonal polynomials and with the aid of a recent result of Claeys, Krasovsky an...
This paper studies the monic semi‐classical Laguerre polynomials based on previous work by Boelen and Van Assche, Filipuk et al., and Clarkson and Jordaan. Filipuk et al. proved that the diagonal recurrence coefficient αn(t)$$ {\alpha}_n(t) $$ satisfies the fourth Painlevé equation. In this paper, we show that the off‐diagonal recurrence coefficien...
We study the Hankel determinant generated by the Gaussian weight with jump dis-continuities at t_1 , · · · , t_m. By making use of a pair of ladder operators satisfied by the associated monic orthogonal polynomials and three supplementary conditions, we show that the logarithmic derivative of the Hankel determinant satisfies a second order partial...
We study the Hankel determinant generated by the Laguerre weight with jump discontinuities attk, k = 1, . . . , m. By employing the ladder operator approach to establish Riccati equations, we showthat σn(t1, . . . , tm), the logarithmic derivative of the n-dimensional Hankel determinant, satisfies anm-variable generalization of the σ -form of a Pai...
We study the Hankel determinant generated by a two-parameter deformation of the Laguerre weight function. The second parameter introduces a "stronger" zero at the origin. By using a pair of ladder operators satisfied by the associated monic orthogonal polynomials and three compatibility conditions, we establish coupled of second order partial diffe...
In this paper we study one modification of the Jacobi weight. We show that the coefficients in the three term recurrence relation for polynomials orthogonal with respect to this weight satisfy a complicated system of nonlinear difference equations.
Let W ∈ C<sup> n × n </sup> be a single-spiked Wishart matrix in the class W ~ CW<sub>n</sub> ( m , I<sub> n </sub> + θvv†) with m ≥ n , where I<sub> n </sub> is the n × n identity matrix, v ∈ C<sup> n ×1</sup> is an arbitrary vector with unit Euclidean norm, θ ≥ 0 is a non-random parameter, and (·)† represents the conjugate-transpose operator. Let...
We are concerned with the monic orthogonal polynomials with respect to a singularly perturbed Laguerre-type weight. By using the ladder operator approach, we derive a complicated system of nonlinear second-order difference equations satisfied by the recurrence coefficients. This allows us to derive the large n n asymptotic expansions of the recurre...
Let
$\mathbf {X}\in \mathbb {C}^{n\times m}$
(
$m\geq n$
) be a random matrix with independent columns each distributed as complex multivariate Gaussian with zero mean and
single-spiked
covariance matrix
$\mathbf {I}_{n}+ \eta \mathbf {u}\mathbf {u}^{*}$
, where
$\mathbf {I}_{n}$
is the
$n\times n$
identity matrix,
$\mathbf {u}\in \math...
We are concerned with the monic orthogonal polynomials with respect to a singularly perturbed Laguerre-type weight. By using the ladder operator approach, we derive a complicated system of nonlinear second-order difference equations satisfied by the recurrence coefficients. This allows us to derive the large $n$ asymptotic expansions of the recurre...
This paper studies the monic semi-classical Laguerre polynomials based on previous work by Boelen and Van Assche \cite{Boelen}, Filipuk et al. \cite{Filipuk} and Clarkson and Jordaan \cite{Clarkson}. Filipuk, Van Assche and Zhang proved that the diagonal recurrence coefficient $\alpha_n(t)$ satisfies the fourth Painlev\'{e} equation. In this paper...
We study the moment-generating functions (MGF) for linear eigenvalue statistics of Jacobi unitary, symplectic and orthogonal ensembles. By expressing the MGF as Fredholm determinants of kernels of finite rank, we show that the mean and variance of the suitably scaled linear statistics in these Jacobi ensembles are related to the sine kernel in the...
We study recurrence coefficients of semi-classical Laguerre orthogonal polynomials and the associated Hankel determinant generated by a semi-classical Laguerre weight [Formula: see text]. If t = 0, it is reduced to the classical Laguerre weight. For t > 0, this weight tends to zero faster than the classical Laguerre weight as x → ∞. In the finite n...
We propose a type of hybrid Seesaw model that combines Type-1 and Type-2 Seesaw mechanism in multiplicative way to generate tree level Majorana neutrino mass and provides a Dark Matter candidate. The model extends the Standard Model by extra gauge symmetry U(1)D and hidden sector consisted of chiral fermions and additional scalar fields. After spon...
We discuss the recurrence coefficients of the three-term recurrence relation for the orthogonal polynomials with a singularly perturbed Gaussian weight [Formula: see text]. Based on the ladder operator approach, two auxiliary quantities are defined. We show that the auxiliary quantities and the recurrence coefficients satisfy some equations with th...
We propose a type of hybrid Seesaw model that combines Type-1 and Type-2 Seesaw mechanism in multiplicative way to generate tree level Majorana neutrino mass and provides a Dark Matter candidate. The model extends the Standard Model by extra gauge symmetry $U(1)_{D}$ and hidden sector consisted of chiral fermions and additional scalar fields. After...
We study the monic polynomials orthogonal with respect to a symmetric perturbed Gaussian weight $$ w(x;t):=\mathrm{e}^{-x^2}\left(1+t\: x^2\right)^\lambda,\qquad x\in \mathbb{R}, $$ where $t> 0,\;\lambda\in \mathbb{R}$. This weight is related to the single-user MIMO systems in information theory. It is shown that the recurrence coefficient $\beta_n...
In this paper, two families of exact solutions to two-dimensional incompressible rotational Euler equations are constructed by connecting the Euler equations with the Laplace equation via a stream function. The constituent solutions in the first family are smooth, orthogonal, and conjugate harmonic solutions, while their constituent velocities are...
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 $\sigma_n(t_1,\cdots,t_m)$, the logarithmic derivative of the $n$-dimensional Hankel determinant, satisfies a generalization of the $\sigma$-from of...
We study orthogonal polynomials and Hankel determinants generated by a symmetric semi-classical Jacobi weight. By using the ladder operator technique, we derive the second-order nonlinear difference equations satisfied by the recurrence coefficient \(\beta _n(t)\) and the sub-leading coefficient \(\mathrm {p}(n,t)\) of the monic orthogonal polynomi...
In this paper, we study those polynomials, orthogonal with respect to a particularweight, over the union of disjoint intervals, first introduced by N. I. Akhiezer, via areformulation as a matrix factorization or Riemann–Hilbert problem. This approachcomplements the method proposed in a previous paper, which involves theconstruction of a certain mer...
In this paper we study one modification of the Jacobi weight. We show that the coefficients in the three term recurrence relation for polynomials orthogonal with respect to this weight satisy a complicated system of nonlinear difference equations.
We study the Hankel determinant generated by a singularly perturbed Jacobi weight w(x,s):=(1−x)α(1+x)βe−s1−x,x∈[−1,1],α>0,β>0s≥0. If s=0, it is reduced to the classical Jacobi weight. For s>0, the factor e−s1−x induces an infinitely strong zero at x=1. For the finite n case, we obtain four auxiliary quantities Rn(s), rn(s), R˜n(s), and r˜n(s) by us...
We study the large $N$ behavior of the moment-generating function (MGF) for linear eigenvalue statistics of Jacobi unitary, symplectic and orthogonal ensembles. We show that the mean and variance of the suitably scaled linear statistics in these Jacobi ensembles are related to the sine kernel in the bulk of the spectrum, whereas they are related to...
We study orthogonal polynomials and Hankel determinants generated by a symmetric semi-classical Jacobi weight. By using the ladder operator technique, we derive the second-order nonlinear difference equations satisfied by the recurrence coefficient $\beta_n(t)$ and the sub-leading coefficient $\mathrm{p}(n,t)$ of the monic orthogonal polynomials. T...
Let W ∈ C n×n be a single-spiked Wishart matrix in the class W ∼ CW n (m, I n + θvv †) with m ≥ n, where I n is the n × n identity matrix, v ∈ C n×1 is an arbitrary vector with unit Euclidean norm, θ ≥ 0 is a non-random parameter, and (·) † represents the conjugate-transpose operator. Let u 1 and u n denote the eigenvectors corresponding to the sam...
In 2010, Basor, Chen and Ehrhardt [J. Phys. A: Math. Theor. 43 (2010) 015204 (25pp)] studied the (monic) time-dependent Jacobi polynomials. They proved that the diagonal recurrence coefficient α n (t) satisfies a particular Painlevé V equation and the sub-leading coefficient p(n, t) satisfies the Jimbo-Miwa-Okamoto σ-form of Painlevé V under suitab...
In this paper, we present the characteristic of a certain discontinuous linear statistic of the semi-classical Laguerre unitary ensembles $$\begin{aligned} w(z,t)=A\theta (z-t)e^{-z^2+tz}, \end{aligned}$$here \(\theta (x)\) is the Heaviside function, where \(A> 0\), \(t>0\), and \(z\in [0,\infty )\). We derive the ladder operators and its interrela...
In this paper, we study the orthogonal polynomials with respect to a singularly perturbed Pollaczek-Jacobi type weight
$$
w(x,t):=(1-x^2)^\alpha\mathrm{e}^{-\frac{t}{1-x^{2}}}, x\in[-1,1], \alpha>0, t>0.
$$
By using the ladder operator approach, we establish the second-order difference equations satisfied by the recurrence coefficient $\beta_n(t)$...
We study the probability that all the eigenvalues of n × n Hermitian matrices, from the Laguerre unitary ensemble with the weight xγe−4nx,x∈0,∞,γ>−1, lie in the interval [0, α]. By using previous results for finite n obtained by the ladder operator approach of orthogonal polynomials, we derive the large n asymptotics of the largest eigenvalue distr...
Let $\mathbf{X}\in\mathbb{C}^{m\times n}$ ($m\geq n$) be a random matrix with independent rows each distributed as complex multivariate Gaussian with zero mean and {\it single-spiked} covariance matrix $\mathbf{I}_n+ \eta \mathbf{u}\mathbf{u}^*$, where $\mathbf{I}_n$ is the $n\times n$ identity matrix, $\mathbf{u}\in\mathbb{C}^{n\times n}$ is an ar...
We study the dynamics of a two-level crossing model with a polynomial modification of the linear Landau–Zener tunneling. For a cubic modification, we express the non-adiabatic transition amplitudes analytically via the bi-confluent Heun functions. We find a closed-form series expression of the transition probability at the long time limit, and deri...
In this paper, we study the orthogonal polynomials with respect to a singularly perturbed Pollaczek-Jacobi type weight $$ w(x,t):=(1-x^2)^\alpha\mathrm{e}^{-\frac{t}{1-x^{2}}},\qquad x\in[-1,1],\;\;\alpha>0,\;\;t>0. $$ By using the ladder operator approach, we establish the second-order difference equations satisfied by the recurrence coefficient $...
In the hard edge scaling limit of the Jacobi unitary ensemble generated by the weight xα(1 − x)β, x ∈ [0, 1], α, β > −1, the probability that all eigenvalues of Hermitian matrices from this ensemble lie in the interval [t, 1] is given by the Fredholm determinant of the Bessel kernel. We derive the constant in the asymptotics of this Bessel kernel d...
We study the Hankel determinant generated by a deformed Hermite weight with one jump w(z, t, γ) = e −z 2 +tz |z − t| γ (A + Bθ(z − t)), where A ≥ 0, A + B ≥ 0, t ∈ R, γ > −1 and z ∈ R. By using the ladder operators for the corresponding monic orthogonal polynomials, and their relative compatibility conditions, we obtain a series of difference and d...
In this paper, we focus on the relationship between the fifth Painlevé equation and a Jacobi weight perturbed with random singularities, w(z) = 1 − z 2 α e − t z 2 −k 2 , z, k ∈ [−1, 1], α, t > 0. By using the ladder operator approach, we obtain that an auxiliary quantity R n (t), which is closely related to the recurrence coefficients of monic pol...
In this paper, we focus on the relationship between the fifth Painlev\'{e} equation and a Jacobi weight perturbed with random singularities, \begin{equation*} w(z)=\left(1-z^2\right)^{\alpha}{\rm e}^{-\frac{t}{z^2-k^2}},~~~z,k\in[-1,1],~\alpha,t>0. \end{equation*} By using the ladder operator approach, we obtain that an auxiliary quantity $R_n(t)$,...
We study the Hankel determinant generated by a deformed Hermite weight with one jump $w(z,t,\gamma)=e^{-z^2+tz}|z-t|^{\gamma}(A+B\theta(z-t))$, where $A\geq 0$, $A+B\geq 0$, $t\in\textbf{R}$, $\gamma>-1$ and $z\in\textbf{R}$. By using the ladder operators for the corresponding monic orthogonal polynomials, and their relative compatibility condition...
Exceptional points, which are spectral degeneracy points in the complex parameter space, are fundamental to non‐Hermitian quantum systems. The dynamics of non‐Hermitian systems in the presence of exceptional points differ significantly from those of Hermitian ones. Here, non‐adiabatic transitions in non‐Hermitian PT‐symmetric systems are investigat...
We study the Hankel determinant generated by a singularly perturbed Jacobi weightw(x,t):=(1−x2)αe−tx2,x∈[−1,1],α>0,t≥0. If t=0, it is reduced to the classical symmetric Jacobi weight. For t>0, the factor e−tx2 induces an infinitely strong zero at the origin. This Hankel determinant is related to the Wigner time-delay distribution in chaotic cavitie...
We consider the Hankel determinant generated by the Gaussian weight with two jump discontinuities. Utilizing the results of Min and Chen [Math. Methods Appl Sci. 2019;42:301‐321] where a second‐order partial differential equation (PDE) was deduced for the log derivative of the Hankel determinant by using the ladder operators adapted to orthogonal p...
In this paper, we focus on four weights ω(z, s) = z λ e −N(z+s(z 2 −z)) , where z ∈ (0, ∞), λ > −1, 0 ≤ s ≤ 1, N > 0; ω(z, t) = z λ e −z 2 +tz , where z ∈ (0, ∞), λ > −1, t ∈ R; ω(z, t 1) = e −z 2 (A + Bθ(z − t 1)) , with z ∈ R, A ≥ 0, A + B ≥ 0, B = 0, where θ(z) is the Heaviside step function; and ω(z) = |z| α e −N(z 2 +s(z 4 −z 2)) , with z ∈ R,...
In this paper we study a certain recurrence relation, that can be used to generate ladder operators for the Laguerre unitary ensemble, from the point of view of Sakai’s geometric theory of Painlevé equations. On one hand, this gives us one more detailed example of the appearance of discrete Painlevé equations in the theory of orthogonal polynomials...
In this paper we study a certain recurrence relation, that can be used to generate ladder operators for the Laguerre unitary ensemble, from the point of view of Sakai’s geometric theory of Painlev ́e equations. On one hand, this gives us one more detailed example of the appearance of discrete Painleve equations in thetheory of orthogonal polynomial...
It is known from [G. Filipuk and W. Van Assche, Discrete orthogonal polynomials with hypergeometric weights and Painlevé VI, Symmetry Integr. Geom. Methods Appl. 14 (2018), Article ID: 088, 19 pp.] that the recurrence coefficients of discrete orthogonal polynomials on the nonnegative integers with hypergeometric weights satisfy a system of nonlinea...
Exceptional points, the spectral degeneracy points in the complex parameter space, are fundamental to non-Hermitian quantum systems. The dynamics of non-Hermitian systems in the presence of exceptional points differ significantly from those of Hermitian ones. Here we investigate non-adiabatic transitions in non-Hermitian $\mathcal{P}\mathcal{T}$-sy...
This paper studies the Hankel determinant generated by a perturbed Jacobi weight, which is closely related to the largest and smallest eigenvalue distribution of the degenerate Jacobi unitary ensemble. By using the ladder operator approach for the orthogonal polynomials, we find that the logarithmic derivative of the Hankel determinant satisfies a...
We study the Hankel determinant generated by a singularly perturbed Jacobi weight w(x,t):=(1-x^2)^\alpha\mathrm{e}^{-\frac{t}{x^{2}}}, x\in[-1,1], \alpha>0, t\geq 0. If t=0, it is reduced to the classical symmetric Jacobi weight. For t>0, the factor \mathrm{e}^{-\frac{t}{x^{2}}} induces an infinitely strong zero at the origin. This Hankel determina...
We consider the Hankel determinant generated by the Gaussian weight with two jump discontinuities. Utilizing the results of [C. Min and Y. Chen, Math. Meth. Appl. Sci. {\bf 42} (2019), 301--321] where a second order PDE was deduced for the log derivative of the Hankel determinant by using the ladder operators adapted to orthogonal polynomials, we d...
An asymptotic expression of the orthonormal polynomials P N (z) as N → ∞, associated with the singularly perturbed Laguerre weight w α (x; t) = x α e −x− t x , x ∈ [0, ∞), α > −1, t ≥ 0 is derived. Based on this, we establish the asymptotic behavior of the smallest eigenvalue, λ N , of the Hankel matrix generated by the weight w α (x; t).
An asymptotic expression of the orthonormal polynomials $\mathcal{P}_{N}(z)$ as $N\rightarrow\infty$, associated with the singularly perturbed Laguerre weight $w_{\alpha}(x;t)=x^{\alpha}{\rm e}^{-x-\frac{t}{x}},~x\in[0,\infty),~\alpha>-1,~t\geq0$ is derived. Based on this, we establish the asymptotic behavior of the smallest eigenvalue, $\lambda_{N...
A Hermitian random matrix ensembles, on the line with Fisher-Hartwick type singularity.
Focusing on the weight function ω(x,t) = xα e-1/3 x3 + tx, x ϵ [0, ∞) α > -1, t > 0, we state its asymptotic orthogonal polynomials. Through Toda evolution, differential equations of αn(t) and βn(t) have been worked. Consequently, we also talk about the approximate value of αn(t). Basing on the asymptotic value of αn(t), the asymptotic of second or...
In this paper, we study the Hankel determinant associated with the degenerate Laguerre unitary ensemble (dLUE). This problem originates from the largest or smallest eigenvalue distribution of the dLUE. We derive the ladder operators and its compatibility condition with respect to a general perturbed weight function. By applying the ladder operators...
In the hard edge scaling limit of the Jacobi unitary ensemble generated by the weight $x^{\alpha}(1-x)^{\beta},~x\in[0,1],~\alpha,\beta>0$, the probability that all eigenvalues of Hermitian matrices from this ensemble lie in the interval $[t,1]$ is given by the Fredholm determinant of the Bessel kernel. We derive the constant in the asymptotics of...
In this paper, we study the recurrence coefficients of a deformed Hermite polynomials orthogonal with respect to the weight [Formula: see text] where [Formula: see text] and [Formula: see text].
It is an extension of Chen and Feigin [J. Phys. A., Math. Gen. 39 (2006) 12381–12393]. By using the ladder operator technique, we show that the recurrence...
The recursion relationship: zP n (z) = P n+1 (z) + β n P n−1 (z), n = 0, 1, 2..., is satisfied by all monic orthogonal polynomials in regard to an arbitrary Freud-type weight function. In current paper, one focuses on the weight function ω(z) = |z| α e −z 6 +tz 2 , z ∈ R, t ∈ R, α > −1, to analyse its relative β n and P n (z). Through above equatio...
Let w be a semiclassical weight that is generic in Magnus’s sense, and ({p_n})_{n = 0}^\infty the corresponding sequence of orthogonal polynomials. We express the Christoffel–Darboux kernel as a sum of products of Hankel integral operators. For ψ ∈ L ∞ ( i ℝ), let W (ψ) be the Wiener-Hopf operator with symbol ψ. We give sufficient conditions on ψ s...
We study the dynamics of a two-level crossing model with a parabolic separation of the diabatic energies. The solutions are expressed in terms of the tri-confluent Heun equations - the generalization of the confluent hypergeometric equations. We obtain analytical approximations for the state populations in terms of Airy and Bessel functions. Applic...
In this paper, we study the large $N$ behavior of the moment-generating function (MGF) of the linear statistics of $N\times N$ Hermitian matrices in the Gaussian unitary, symplectic, orthogonal ensembles (GUE, GSE, GOE) and Laguerre unitary, symplectic, orthogonal ensembles (LUE, LSE, LOE) at the edge of the spectrum. From the finite $N$ Fredholm d...
We study the probability that all the eigenvalues of $n\times n$ Hermitian matrices, from the Laguerre unitary ensemble with the weight $x^{\gamma}\mathrm{e}^{-4nx},\;x\in[0,\infty),\;\gamma>-1$, lie in the interval $[0,\alpha]$. By using previous results for finite $n$ obtained by the ladder operator approach of orthogonal polynomials, we derive t...
In this paper we study a certain recurrence relation, that can be used to generate ladder operators for the Laguerre Unitary ensemble, from the point of view of Sakai's geometric theory of Painlev\'e equations. On one hand, this gives us one more detailed example of the appearance of discrete Painlev\'e equations in the theory of orthogonal polynom...
We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highly ill-conditioned matrices. It is based on the {\it LDLT} decomposition and involves finding a $k \times k$ sub-matrix of the inverse of the original $N \times N$ Hankel matrix $H_N^{-1}$ . The computation involves
extremely high precision arithmetic, m...
We study the dynamics of a nonlinear two-level crossing model with a cubic modification of the linear Landau-Zener diabatic energies. The solutions are expressed in terms of the bi-confluent Heun functions --- the generalization of the confluent hypergeometric functions. We express the finial transition probability as a convergent series of the par...
The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of “classical” weights multiplied by suitable “deformation factors,” usually dependent on a “time variable” t. From ladder operators [see A. Magnus, J. Comput. Appl. Math. 57(1-2), 215–237 (1995)], one finds se...
This paper studies the Hankel determinant generated by a perturbed Jacobi weight, which is closely related to the largest and smallest eigenvalue distribution of the degenerate Jacobi unitary ensemble. By using the ladder operator approach for the orthogonal polynomials, we find that the logarithmic derivative of the Hankel determinant satisfies a...
It is known [11] that the recurrence coecients of discrete orthogonal polyno-mials on the non-negative integers with hypergeometric weights satisfy a system of non-linear dierence equations. There is also a connection to the solutions of the-form of the sixth Painlev e equation (one of the parameters of the weights being an independent variable in...
We study the dynamics of a two-level crossing model with a parabolic separation of the diabatic energies. The solutions are expressed in terms of the tri-confluent Heun equations-the generalization of the confluent hyper-geometric equations. We obtain analytical approximations for the state populations in terms of Airy and Bessel functions. Applica...
We study the dynamics of a two-level crossing model with a parabolic separation of the diabatic energies. The solutions are expressed in terms of the tri-confluent Heun equations --- the generalization of the confluent hypergeometric equations. We obtain analytical approximations for the state populations in terms of Airy and Bessel functions. Appl...
In this paper, we consider the perturbed Laguerre unitary ensemble described by the weight function of w(x,t)=(x+t)λxαe−x with x≥0,t>0,α>0,α+λ+1>0. The Deift–Zhou nonlinear steepest descent approach is used to analyze the limit of the eigenvalue correlation kernel. It was found that under the double scaling s=4nt, n→∞, t→0 such that s is positive a...
The Painlev\'{e} equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of ``classical" weights multiplied by suitable ``deformation factors", usually dependent on a ``time variable'' $t$. From ladder operators one finds second order linear ordinary differential equations for a...
We study the asymptotic behavior of the smallest eigenvalue, λN, of the Hankel (or moments) matrix denoted by , with respect to the weight . An asymptotic expression of the polynomials orthogonal with w(x) is established. Using this, we obtain the specific asymptotic formulas of λN in this paper.
Applying a parallel numerical algorithm, we get a va...
In this paper, we study polynomials orthogonal with respect to a Pollaczek–Jacobi type weight
The uniform asymptotic expansions for the monic orthogonal polynomials on the interval (0,1) and outside this interval are obtained. Moreover, near , the uniform asymptotic expansion involves Airy function as , and Bessel function of order α as in the nei...
Let $w$ be a semiclassical weight which is generic in Magnus's sense, and $(p_n)_{n=0}^\infty$ the corresponding sequence of orthogonal polynomials. The paper expresses the Christoffel--Darboux kernel as a sum of products of Hankel integral operators. For $\psi\in L^\infty (i{\mathbb R})$, let $W(\psi )$ be the Wiener-Hopf operator with symbol $\ps...
We are concerned with the probability that all the eigenvalues of a unitary ensemble with the weight function , are greater than s. This probability is expressed as the quotient of Dn(s,t) and its value at s = 0, where Dn(s,t) denotes the determinant of the n dimensional Hankel matrices generated by the moments of w(x;t) on x ∈ [s, ∞). In this pape...
In this paper, we study the Hankel determinant associated with the degenerate Laguerre unitary ensemble. This problem originates from the largest or smallest eigenvalue distribution of the degenerate Laguerre unitary ensemble. We derive the ladder operators and its compatibility condition with respect to a general perturbed weight. By applying the...
Figure S1 The distribution of aligned reads within SPS gene among different libraries.
Figure S2 Overview of single‐dose SNPs calling based on merged alignments RNA‐Seq libraries.
Figure S3 (a and b) Distribution of single dose SNPs for parents based on qualified plant numbers and segregation ratios. (a) Distribution of single dose SNPs for femal...
Table S1 Genetic maps for S. officinarum.
Table S2 Genetic maps for S. robustum.
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You have asked me to send you the ``smallest eigenvalues paper" and here it is.