Yang Chen | University of Macau (original) (raw)

Papers by Yang Chen

Physical Review Letters, 2017

In this paper, we study the large N behavior of the smallest eigenvalue N of the (N + 1) ⇥ (N + 1... more In this paper, we study the large N behavior of the smallest eigenvalue N of the (N + 1) ⇥ (N + 1) Hankel matrix, H N = (µ j+k) 0 j,kN , generated by the dependent Jacobi weight w(z,) = e z z ↵ (1 z) , z 2 [0, 1], 2 R, ↵ > 1, > 1. Applying the arguments of Szegö, Widom and Wilf, we obtain the asymptotic representation of the orthonormal polynomials P N (z), z 2 C\[0, 1], with the weight w(z,) = e z z ↵ (1 z). Using the polynomials P N (z), we obtain the theoretical expression of N , for large N. We also display the smallest eigenvalue N for su ciently large N, computed numerically.

The "Diophantine" property of the zeros of certain polynomials in the Askey scheme, recently disc... more The "Diophantine" property of the zeros of certain polynomials in the Askey scheme, recently discovered by Calogero and his collaborators, is explained, with suitably chosen parameter values, in terms of the summation theorem of hypergeometric series. Here the Diophantine property refers to integer valued zeros. It turns out that the same procedure can also be applied to polynomials arising from the basic hypergeometric series. We found, with suitably chosen parameters and certain q−analogue of the summation theorems, zeros of these polynomials explicitly, which are no longer integer valued. This goes beyond the results obtained by the Authors mentioned above.

Gaussian Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé IV System, 2023

We study the Hankel determinant generated by the Gaussian weight with jump discontinuities at t 1... more We study the Hankel determinant generated by the Gaussian weight with jump discontinuities at t 1 , • • • , tm. 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 differential equation which is reduced to the σ-form of a Painlevé IV equation when m = 1. Moreover, under the assumption that t k − t 1 is fixed for k = 2, • • • , m, by considering the Riemann-Hilbert problem for the orthogonal polynomials, we construct direct relationships between the auxiliary quantities introduced in the ladder operators and solutions of a coupled Painlevé IV system.

This paper gives an in-depth study of a multiple-antenna wireless communication scenario in which... more This paper gives an in-depth study of a multiple-antenna wireless communication scenario in which a weak signal received at an intermediate relay station is amplified and then forwarded to the final destination. The key quantity determining system performance is the statistical properties of the signal-to-noise ratio (SNR) γ at the destination. Under certain assumptions on the encoding structure, recent work has characterized the SNR distribution through its moment generating function, in terms of a certain Hankel determinant generated via a deformed Laguerre weight. Here, we employ two different methods to describe the Hankel determinant. First, we make use of ladder operators satisfied by orthogonal polynomials to give an exact characterization in terms of a "double-time" Painlevé differential equation, which reduces to Painlevé V under certain limits. Second, we employ Dyson's Coulomb fluid method to derive a closed form approximation for the Hankel determinant. The two characterizations are used to derive closed-form expressions for the cumulants of γ , and to compute performance quantities of engineering interest.

Nuclear Physics B, 2011

We present a new technique for computing Hilbert series of N = 1 supersymmetric QCD in four dimen... more We present a new technique for computing Hilbert series of N = 1 supersymmetric QCD in four dimensions with unitary and special unitary gauge groups. We show that the Hilbert series of this theory can be written in terms of determinants of Toeplitz matrices. Applying related theorems from random matrix theory, we compute a number of exact Hilbert series as well as asymptotic formulae for large numbers of colours and flavours-many of which have not been derived before.

Nuclear Physics B 860[PM](2012)421-463, 2012

We present a novel approach for computing the Hilbert series of 4d N = 1 supersymmetric QCD with ... more We present a novel approach for computing the Hilbert series of 4d N = 1 supersymmetric QCD with SO(N c) and Sp(N c) gauge groups. It is shown that such Hilbert series can be recast in terms of determinants of Hankel matrices. With the aid of results from random matrix theory, such Hankel determinants can be evaluated both exactly and asymptotically. Several new results on Hilbert series for general numbers of colours and flavours are thus obtained in this paper. We show that the Hilbert series give rise to families of rational solutions, with palindromic numerators, to the Painlevé VI equations. Due to the presence of such Painlevé equations, there exist integrable Hamiltonian systems that describe the moduli spaces of SO(N c) and Sp(N c) SQCD. To each system, we explicitly state the corresponding Hamiltonian and family of elliptic curves. It turns out that such elliptic curves take the same form as the Seiberg-Witten curves for 4d N = 2 SU(2) gauge theory with 4 flavours.

We study the moduli space of 4d N = 1 supersymmetric QCD in the Veneziano limit using Hilbert ser... more We study the moduli space of 4d N = 1 supersymmetric QCD in the Veneziano limit using Hilbert series. In this limit, the numbers of colours and flavours are taken to be large with their ratio fixed. It is shown that the Hilbert series, which is a partition function of an ensemble of gauge invariant quantities parametrising the moduli space, can also be realised as a partition function of a system of interacting Coulomb gas in two dimensions. In the electrostatic equilibrium, exact and asymptotic analyses reveal that such a system exhibits two possible phases. Physical quantities, such as charge densities, free energies, and Hilbert series, associated with each phase, are computed explicitly and discussed in detail. We then demonstrate the existence of the third order phase transition in this system.

This paper gives an in-depth study of a multiple-antenna wireless communication scenario in which... more This paper gives an in-depth study of a multiple-antenna wireless communication scenario in which a weak signal received at an intermediate relay station is amplified and then forwarded to the final destination. The key quantity determining system performance is the statistical properties of the signal-to-noise ratio (SNR) γ at the destination. Under certain assumptions on the encoding structure, recent work has characterized the SNR distribution through its moment generating function, in terms of a certain Hankel determinant generated via a deformed Laguerre weight. Here, we employ two different methods to describe the Hankel determinant. First, we make use of ladder operators satisfied by orthogonal polynomials to give an exact characterization in terms of a "double-time" Painlevé differential equation, which reduces to Painlevé V under certain limits. Second, we employ Dyson's Coulomb Fluid method to derive a closed form approximation for the Hankel determinant. The two characterizations are used to derive closed-form expressions for the cumulants of γ, and to compute performance quantities of engineering interest.

E. Heine in the 19th century studied a system of orthogonal polynomials associated with the weigh... more E. Heine in the 19th century studied a system of orthogonal polynomials associated with the weight [x(x − α)(x − β)] − 1 2 , x ∈ [0, α], 0 < α < β. A related system was studied by C. J. Rees in 1945, associated with the weight  (1 − x 2)(1 − k 2 x 2)  − 1 2 , x ∈ [−1, 1], k 2 ∈ (0, 1). These are also known as elliptic orthogonal polynomials, since the moments of the weights may be expressed in terms of elliptic integrals. Such orthogonal polynomials are of great interest because the corresponding Hankel determinant, depending on a parameter k 2 , where 0 < k 2 < 1 is the τ function of a particular Painlevé VI, the special cases of which are related to enumerative problems arising from string theory. We show that the recurrence coefficients, denoted by β n (k 2), n = 1, 2,. . .; and p 1 (n, k 2), the coefficients of x n−2 of the monic polynomials orthogonal with respect to a generalized version of the weight studied by Rees, (1 − x 2) α (1 − k 2 x 2) β , x ∈ [−1, 1], α > −1, β ∈ R, satisfy second order non-linear difference equations. The large n expansion based on the difference equations when combined with known asymptotics of the leading terms of the associated Hankel determinant yields a complete asymptotic expansion of the Hankel determinant. The Painlevé equation

Preprint, 2023

In this paper we study the simplest deformation on a sequence of orthogonal polynomials. This in ... more In this paper we study the simplest deformation on a sequence of orthogonal polynomials. This in turn induces a deformation on the moment matrix of the polynomials and associated Hankel determinant. We replace the original (or reference) weight w 0 (x) (supported on R or subsets of R) by w 0 (x) e −tx. It is a well-known fact that under such a deformation the recurrence coefficients denoted as α n and β n evolve in t according to the Toda equations, giving rise to the time-dependent orthogonal polynomials and time-dependent determinants, using Sogo's terminology. If w 0 is the normal density e −x 2 , x ∈ R, or the gamma density x α e −x , x ∈ R + , α > −1, then the initial value problem of the Toda equations can be trivially solved. This is because under elementary scaling and translation the orthogonality relations reduce to the original ones. However, if w 0 is the beta density (1 − x) α (1 + x) β , x ∈ [−1, 1], α, β > −1, the resulting 'time-dependent' Jacobi polynomials will again satisfy a linear second-order ode, but no longer in the Sturm-Liouville form, which is to be expected. This deformation induces an irregular singular point at infinity in addition to three regular singular points of the hypergeometric equation satisfied by the Jacobi polynomials. We will show that the coefficients of this ode, as well as the Hankel determinant, are intimately related to a particular Painlevé V. In particular we show that p 1 (n, t), where p 1 (n, t) is the coefficient of z n−1 of the monic orthogonal polynomials associated with the 'time-dependent' Jacobi weight, satisfies, up to a translation in t, the Jimbo-Miwa σ-form of the same P V ; while a recurrence coefficient α n (t) is up to a translation in t and a linear fractional transformation P V (α 2 /2, −β 2 /2, 2n + 1 + α + β, −1/2). These results are found from combining a pair of nonlinear difference equations and a pair of Toda equations. This will in turn allow us to show that a certain Fredholm determinant related to a class of Toeplitz plus Hankel operators has a connection to a Painlevé equation. The case with α = β = −1/2

The ground state or moduli space of super-symmetric QCD as described by a matrix integral, the β ... more The ground state or moduli space of super-symmetric QCD as described by a matrix integral, the β = 2 case.

The purpose of this paper is to compute asymptotically Hankel determinants for weights that are s... more 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 investigate the smallest eigenvalue, denoted as λ N , of a (N +1)×(N +1) Hankel ... more In this paper we investigate the smallest eigenvalue, denoted as λ N , of a (N +1)×(N +1) Hankel or moments matrix, associated with the weight, w(x) = exp(−x β), x > 0, β > 0, in the large N limit. Using a previous result, the asymptotics for the polynomials, P n (z), z / ∈ [0, ∞), orthonormal with respect to w, which are required in the determination of λ N are found. Adopting an argument of Szegö the asymptotic behaviour of λ N , for β > 1 2 where the related moment problem is determinate, is derived. This generalizes the result given by Szegö for β = 1. It is shown that for β > 1 2 the smallest eigenvalue of the infinite Hankel matrix is zero, while for 0 < β < 1 2 it is greater then a positive constant. This shows a phase transition in the corresponding Hermitian random matrix model as the parameter β varies with β = 1 2 identified as the critical point. The smallest eigenvalue at this point is conjectured.

Hankel determinants which occur in problems associated with orthogonal polynomials, integrable sy... more Hankel determinants which occur in problems associated with orthogonal polynomials, integrable systems and random matrices are computed asymptotically for weights that are supported in an semi-in nite or in nite interval. The main idea is to turn the determinant computation into a random matrix \linear statistics" type problem where the Coulomb uid approach can be applied.

In this paper we compute two important information-theoretic quantities which arise in the applic... more In this paper we compute two important information-theoretic quantities which arise in the application of multiple-input multiple-output (MIMO) antenna wireless communication systems: the distribution of the mutual information of multi-antenna Gaussian channels, and the Gallager random coding upper bound on the error probability achievable by finite-length channel codes. We show that the mathematical problem underpinning both quantities is the computation of certain Hankel determinants generated by deformed versions of classical weight functions. For single-user MIMO systems, it is a deformed

Physical Review Letters, 2017

Gaussian Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé IV System, 2023

Nuclear Physics B, 2011

Nuclear Physics B 860[PM](2012)421-463, 2012

This paper gives an in-depth study of a multiple-antenna wireless communication scenario in which... more This paper gives an in-depth study of a multiple-antenna wireless communication scenario in which a weak signal received at an intermediate relay station is amplified and then forwarded to the final destination. The key quantity determining system performance is the statistical properties of the signal-to-noise ratio (SNR) γ at the destination. Under certain assumptions on the encoding structure, recent work has characterized the SNR distribution through its moment generating function, in terms of a certain Hankel determinant generated via a deformed Laguerre weight. Here, we employ two different methods to describe the Hankel determinant. First, we make use of ladder operators satisfied by orthogonal polynomials to give an exact characterization in terms of a "double-time" Painlevé differential equation, which reduces to Painlevé V under certain limits. Second, we employ Dyson's Coulomb Fluid method to derive a closed form approximation for the Hankel determinant. The two characterizations are used to derive closed-form expressions for the cumulants of γ, and to compute performance quantities of engineering interest.

Preprint, 2023

My published papers, 2024

List of Publications.

Preprint, 2020

In the hard edge scaling limit of the Jacobi unitary ensemble generated by the weight x α (1 − x)... more In the hard edge scaling limit of the Jacobi unitary ensemble generated by the weight x α (1 − x) β , x ∈ [0, 1], α, β > 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 this Bessel-kernel determinant. A specialization of the results gives the constant in the asymptotics of the probability that the interval (−a, a), a > 0, is free of eigenvalues in the Jacobi unitary ensemble with the symmetric weight (1−x 2) β , x ∈ [−1, 1].

The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, ... more 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 [12-14, 30] one finds second order linear ordinary differential equations for associated orthogonal polynomials with coefficients being rational functions. The Painlevé and related functions appear as the residues of these rational functions. We will be interested in the situation when n, the order of the Hankel matrix and also the degree of the polynomials P n (x) orthogonal with respect to the deformed weights, gets large. We show that the second order linear differential equations satisfied by P n (x) are particular cases of Heun equations when n is large. In some sense, monic orthogonal polynomials generated by deformed weights mentioned below are solutions of a variety of Heun equations. Heun equations are of considerable importance in mathematical physics and in the special cases they degenerate to the hypergeometric and confluent hypergeometric equations (see, for instance, [1, 23, 36]). In this paper we look at three type of weights: the Jacobi type, the Laguerre type and the weights deformed by the indicator function of (a, b) χ (a,b) and the step function θ(x). In particular, we consider the following Jacobi type weights: 1.

We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highl... more We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highly ill-conditioned matrices. It is based on the LDLT decomposition and involves finding a k × k sub-matrix of the inverse of the original N × N Hankel matrix H −1 N. The computation involves extremely high precision arithmetic, message passing interface, and shared memory parallelisation. We demonstrate that this approach achieves good scalability on a high performance computing cluster (HPCC) which constitute a major improvement of the earlier approaches. We use this method to study a family of Hankel matrices generated by the weight w(x) = e −x β , supported on [0, ∞) and β > 0. Such weight generates Hankel determinant, a fundamental object in random matrix theory. In the situation where β > 1/2, the smallest eigenvalue tend to 0, exponentially fast. If β < 1/2, the situation where the classical moment problem is indeterminate, the smallest eigenval-ue is bounded from below by a positive number. If β = 1/2, it is conjectured that the smallest eigenvalue tends to 0 algebraically, with a precise exponent. The algorithm run on the HPCC producing fantastic match between the theoretical value of 2/π and the numerical result.

In this paper, we study the recurrence coefficients of a deformed or semi-classical Laguerre poly... more In this paper, we study the recurrence coefficients of a deformed or semi-classical Laguerre polynomials orthogonal with respect to the weight w(x, s) = w(x; α, s) := x α e −N (x+s(x 2 −x)) , 0 ≤ x < ∞. Here α > −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 1. We show that the recurrence coefficients satisfy a second order ordinary differential equation (ODE) when viewed as functions of the parameter s in the weight. Then we work out the large-degree asymptotics of the recurrence coefficients. We also discuss the associated Hankel determinant, or the normalization constant. As the logarithmic derivative of D n (s) can be expressed in terms of the recurrence coefficients, ultimately we compute the large-degree asymptotics of ln[D n (1)]. Based on this result, we compute the probability that an n by n (or rN ×rN) random matrix from a generalised Gaussian Unitary Ensemble (gGUE) is positive definite.

In this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble, n... more 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 probability, denoted by H n (a), R n (a) and r n (a). We find that each one satisfies a second order differential equation. We show that after a double scaling, the large second order differential equation in the variable a with n as parameter satisfied by H n (a), can be reduced to the Jimbo-Miwa-Okamoto σ form of the Painlevé V equation.

In this paper, we address a class of problems in unitary ensembles. Specifically, we study the pr... more In this paper, we address a class of problems in unitary ensembles. Specifically, we study the probability that a gap symmetric about 0, i.e. (−a, a) is found in the Gaussian unitary ensembles (GUE) and the Jacobi unitary ensembles (JUE) (where in the JUE, we take the parameters α = β). We show the probability that the interval (−a, a) is free of eigenvalues, for an n × n random matrix chosen from GUE, viewed as a function of a, satisfies a large second order ordinary differential equation in the variable a with n as parameter. The finite n version was mentioned in various publications but was not written down, and what we found appears not to be a Painlevé equation. In order to go to the thermodynamic limit, we consider a scenario where there is finite density of eigenvalues, namely, n, the order of random matrix is infinitely large, such that a suitable combination of n and a is finite: in the Gaussian case, take τ := 2 √ 2n a. After such a double scaling, the large equation " travels down " to the JMMS (M. Jimbo, T. Miwa, Y. Môri and M. Sato, Physica 1D (1980), 80–158) σ form of the Painlevé V equation. * lvshulin1989@163.com † yangbrookchen@yahoo.co.uk ‡ faneg@fudan.edu.cn 1 We also obtain the asymptotic expansions for the smallest eigenvalue distributions of the Laguerre unitary and Jacobi unitary ensembles after appropriate double scalings, including the constants. An elementary 'doubling' of the intervals, and with special choices of the α parameter in the Gamma density x α e −x and the Beta density x α (1 − x) β identifies the constant term in the asymptotic expansion of the gap probability in GUE (i.e. Widom-Dyson constant) and JUE respectively.

In this paper we study a particular Painlevé V (denoted P V) that arises from Multi-Input-Multi-O... more 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 Laguerre weight or the Gamma density x α e −x , x > 0, for α > −1 by (1 + x/t) λ with t > 0 a scaling parameter. Here the λ parameter " generates " the Shannon capacity; see Yang Chen and Matthew McKay, IEEE Trans. IT, vol.(58) (2012) 4594–4634. It was found that the MGF has an integral representation as a functional of y(t) and y (t) , where y(t) satisfies the " classical form " of P V. In this paper, we consider the situation where n, the number of transmit antenna , (or the size of the random matrix), tends to infinity, and the signal-to-noise ratio (SNR) P tends to infinity, such that s = 4n 2 /P is finite. Under such double scaling the MGF, effectively an infinite determinant, has an integral representation in terms of a " lesser " P III. We also consider the situation where α = k + 1/2, k ∈ N, and α ∈ {0, 1, 2,. .. } λ ∈ {1, 2,. .. }, linking relevant the quantity to a solution of the two dimensional sine-Gordon equation in radial coordinates and a certain discrete Painlevé-II. From the large n asymptotic of the orthogonal polynomials, that appears naturally, we obtain the double scaled MGF for small and large s , together with the constant term in the large s expansion. With the aid of these, we derive a number of cumulants and find that the capacity distribution function is non-Gaussian.