arXiv subjects

Peter J. Forrester

Publications and source records attributed to Peter J. Forrester.

At least 73 records · Page 4Linked to original sources

arXiv2020

Asymptotic correlations with corrections for the circular Jacobi $β$-ensemble

Previous works have considered the leading correction term to the scaled limit of various correlation functions and distributions for classical random matrix ensembles and their $β$ generalisations at the hard and soft edge. It has been found that the functional form of this correction is given by a derivative operation applied to the leading term. In the present work we compute the leading correction term of the correlation kernel at the spectrum singularity for the circular Jacobi ensemble with Dyson indices $β= 1,2$ and 4, and also to the spectral density in the corresponding $β$-ensemble with $β$ even. The former requires an analysis involving the Routh-Romanovski polynomials, while the latter is based on multidimensional integral formulas for generalised hypergeometric series based on Jack polynomials. In all cases this correction term is found to be related to the leading term by a derivative operation.

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arXiv2020

Rate of convergence at the hard edge for various Pólya ensembles of positive definite matrices

The theory of Pólya ensembles of positive definite random matrices provides structural formulas for the corresponding biorthogonal pair, and correlation kernel, which are well suited to computing the hard edge large $N$ asymptotics. Such an analysis is carried out for products of Laguerre ensembles, the Laguerre Muttalib-Borodin ensemble, and products of Laguerre ensembles and their inverses. The latter includes as a special case the Jacobi unitary ensemble. In each case the hard edge scaled kernel permits an expansion in powers of $1/N$, with the leading term given in a structured form involving the hard edge scaling of the biorthogonal pair. The Laguerre and Jacobi ensembles have the special feature that their hard edge scaled kernel -- the Bessel kernel -- is symmetric and this leads to there being a choice of hard edge scaling variables for which the rate of convergence of the correlation functions is $O(1/N^2)$.

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arXiv2020

Differential identities for the structure function of some random matrix ensembles

The structure function of a random matrix ensemble can be specified as the covariance of the linear statistics $\sum_{j=1}^N e^{i k_1 λ_j}$, $\sum_{j=1}^N e^{-i k_2 λ_j}$ for Hermitian matrices, and the same with the eigenvalues $λ_j$ replaced by the eigenangles $θ_j$ for unitary matrices. As such it can be written in terms of the Fourier transform of the density-density correlation $ρ_{(2)}$. For the circular $β$-ensemble of unitary matrices, and with $β$ even, we characterise the bulk scaling limit of $ρ_{(2)}$ as the solution of a linear differential equation of order $β+ 1$ -- a duality relates $ρ_{(2)}$ with $β$ replaced by $4/β$ to the same equation. Asymptotics obtained in the case $β= 6$ from this characterisation are combined with previously established results to determine the explicit form of the degree 10 palindromic polynomial in $β/2$ which determines the coefficient of $|k|^{11}$ in the small $|k|$ expansion of the structure function for general $β> 0$. For the Gaussian unitary ensemble we give a reworking of a recent derivation and generalisation, due to Okuyama, of an identity relating the structure function to simpler quantities in the Laguerre unitary ensemble first derived in random matrix theory by Brézin and Hikami. This is used to determine various scaling limits, many of which relate to the dip-ramp-plateau effect emphasised in recent studies of many body quantum chaos, and allows too for rates of convergence to be established.

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arXiv2020

Linear Differential Equations for the Resolvents of the Classical Matrix Ensembles

The spectral density for random matrix $β$ ensembles can be written in terms of the average of the absolute value of the characteristic polynomial raised to the power of $β$, which for even $β$ is a polynomial of degree $β(N-1)$. In the cases of the classical Gaussian, Laguerre, and Jacobi weights, we show that this polynomial, and moreover the spectral density itself, can be characterised as the solution of a linear differential equation of degree $β+1$. This equation, and its companion for the resolvent, are given explicitly for $β=2$ and $4$ for all three classical cases, and also for $β=6$ in the Gaussian case. Known dualities for the spectral moments relating $β$ to $4/β$ then imply corresponding differential equations in the case $β=1$, and for the Gaussian ensemble, the case $β=2/3$. We apply the differential equations to give a systematic derivation of recurrences satisfied by the spectral moments and by the coefficients of their $1/N$ expansions, along with first-order differential equations for the coefficients of the $1/N$ expansions of the corresponding resolvents. We also present the form of the differential equations when scaled at the hard or soft edges.

math-ph↗

arXiv2020

Computable structural formulas for the distribution of the $β$-Jacobi edge eigenvalues

The Jacobi ensemble is one of the classical ensembles of random matrix theory. Prominent in applications are properties of the eigenvalues at the spectrum edge, specifically the distribution of the largest (e.g. Roy's largest root test in multivariate statistics) and smallest (e.g.~condition numbers of linear systems) eigenvalues. We identify three ranges of parameter values for which the gap probability determining these distributions is a finite sum with respect to particular bases, and moreover make use of a certain differential-difference system fundamental in the theory of the Selberg integral to provide a recursive scheme to compute the corresponding coefficients.

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arXiv2019

Recursion scheme for the largest $β$-Wishart-Laguerre eigenvalue and Landauer conductance in quantum transport

The largest eigenvalue distribution of the Wishart-Laguerre ensemble, indexed by Dyson parameter $β$ and Laguerre parameter $a$, is fundamental in multivariate statistics and finds applications in diverse areas. Based on a generalization of the Selberg integral, we provide an effective recursion scheme to compute this distribution explicitly in both the original model, and a fixed-trace variant, for $a,β$ non-negative integers and finite matrix size. For $β= 2$ this circumvents known symbolic evaluation based on determinants which become impractical for large dimensions. Our exact results have immediate applications in the areas of multiple channel communication and bipartite entanglement. Moreover, we are also led to the exact solution of a long standing problem of finding a general result for Landauer conductance distribution in a chaotic mesoscopic cavity with two ideal leads. Thus far, exact closed-form results for this were available only in the Fourier-Laplace space or could be obtained on a case-by-case basis.

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arXiv2019

Co-rank 1 projections and the randomised Horn problem

Let $\hat{\boldsymbol x}$ be a normalised standard complex Gaussian vector, and project an Hermitian matrix $A$ onto the hyperplane orthogonal to $\hat{\boldsymbol x}$. In a recent paper Faraut [Tunisian J. Math. \textbf{1} (2019), 585--606] has observed that the corresponding eigenvalue PDF has an almost identical structure to the eigenvalue PDF for the rank 1 perturbation $A + b \hat{\boldsymbol x} \hat{\boldsymbol x}^\dagger$, and asks for an explanation. We provide this by way of a common derivation involving the secular equations and associated Jacobians. This applies too in related setting, for example when $\hat{\boldsymbol x}$ is a real Gaussian and $A$ Hermitian, and also in a multiplicative setting $A U B U^\dagger$ where $A, B$ are fixed unitary matrices with $B$ a multiplicative rank 1 deviation from unity, and $U$ is a Haar distributed unitary matrix. Specifically, in each case there is a dual eigenvalue problem giving rise to a PDF of almost identical structure.

math-ph↗

arXiv2019

Schwinger-Dyson and loop equations for a product of square Ginibre random matrices

In this paper, we study the product of two complex Ginibre matrices and the loop equations satisfied by their resolvents (i.e. the Stieltjes transform of the correlation functions). We obtain using Schwinger-Dyson equation (SDE) techniques the general loop equations satisfied by the resolvents. In order to deal with the product structure of the random matrix of interest, we consider SDEs involving the integral of higher derivatives. One of the advantage of this technique is that it bypasses the reformulation of the problem in terms of singular values. As a byproduct of this study we obtain the large $N$ limit of the Stieltjes transform of the $2$-point correlation function, as well as the first correction to the Stieltjes transform of the density, giving us access to corrections to the smoothed density. In order to pave the way for the establishment of a topological recursion formula we also study the geometry of the corresponding spectral curve. This paper also contains explicit results for different resolvents and their corrections.

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arXiv2019

Finite size corrections at the hard edge for the Laguerre $β$ ensemble

A fundamental question in random matrix theory is to quantify the optimal rate of convergence to universal laws. We take up this problem for the Laguerre $β$ ensemble, characterised by the Dyson parameter $β$, and the Laguerre weight $x^a e^{-βx/2}$, $x > 0$ in the hard edge limit. The latter relates to the eigenvalues in the vicinity of the origin in the scaled variable $x \mapsto x/4N$. Previous work has established the corresponding functional form of various statistical quantities --- for example the distribution of the smallest eigenvalue, provided that $a \in \mathbb Z_{\ge 0}$. We show, using the theory of multidimensional hypergeometric functions based on Jack polynomials, that with the modified hard edge scaling $x \mapsto x/4(N+a/β)$, the rate of convergence to the limiting distribution is $O(1/N^2)$, which is optimal. In the case $β= 2$, general $a> -1$ the explicit functional form of the distribution of the smallest eigenvalue at this order can be computed, as it can for $a=1$ and general $β> 0$. An iterative scheme is presented to numerically approximate the functional form for general $a \in \mathbb Z_{\ge 2}$.

math-ph↗

arXiv2018

Optimal soft edge scaling variables for the Gaussian and Laguerre even $β$ ensembles

The $β$ ensembles are a class of eigenvalue probability densities which generalise the invariant ensembles of classical random matrix theory. In the case of the Gaussian and Laguerre weights, the corresponding eigenvalue densities are known in terms of certain $β$ dimensional integrals. We study the large $N$ asymptotics of the density with a soft edge scaling. In the Laguerre case, this is done with both the parameter $a$ fixed, and with $a$ proportional to $N$. It is found in all these cases that by appropriately centring the scaled variable, the leading correction term to the limiting density is $O(N^{-2/3})$. A known differential-difference recurrence from the theory of Selberg integrals allows for a numerical demonstration of this effect.

math-ph↗

arXiv2018

Comment on "Finite size effects in the averaged eigenvalue density of Wigner random-sign real symmetric matrices" by G.S. Dhesi and M. Ausloos

The recent paper "Finite size effects in the averaged eigenvalue density of Wigner random-sign real symmetric matrices" by G.S. Dhesi and M. Ausloos [Phys. Rev. E 93 (2016), 062115] uses the replica method to compute the $1/N$ correction to the Wigner semi-circle law for the ensemble of real symmetric random matrices with $0$'s down the diagonal, and upper triangular entries independently chosen from the values $\pm w$ with equal probability. We point out that the results obtained are inconsistent with known results in the literature, as well as with known large $N$ series expansions for the trace of powers of these random matrices. An incorrect assumption relating to the role of the diagonal terms at order $1/N$ appears to be the cause for the inconsistency. Moreover, results already in the literature can be used to deduce the $1/N$ correction to the Wigner semi-circle law for real symmetric random matrices with entries drawn independently from distributions $\mathcal D_1$ (diagonal entries) and $\mathcal D_2$ (upper triangular entries) assumed to be even and have finite moments. Large $N$ expansions for the trace of the $2k$-th power ($k=1,2,3$) for these matrices can be computed and used as checks.

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arXiv2018

A generalisation of the relation between zeros of the complex Kac polynomial and eigenvalues of truncated unitary matrices

The zeros of the random Laurent series $1/μ- \sum_{j=1}^\infty c_j/z^j$, where each $c_j$ is an independent standard complex Gaussian, is known to correspond to the scaled eigenvalues of a particular additive rank 1 perturbation of a standard complex Gaussian matrix. For the corresponding random Maclaurin series obtained by the replacement $z \mapsto 1/z$, we show that these same zeros correspond to the scaled eigenvalues of a particular multiplicative rank 1 perturbation of a random unitary matrix. Since the correlation functions of the latter are known, by taking an appropriate limit the correlation functions for the random Maclaurin series can be determined. Only for $|μ| \to \infty$ is a determinantal point process obtained. For the one and two point correlations, by regarding the Maclaurin series as the limit of a random polynomial, a direct calculation can also be given.

math-ph↗

arXiv2018

Meet Andréief, Bordeaux 1886, and Andreev, Kharkov 1882-83

The paper `Note sur une relation entre les intégrales définies des produits des fonctions' by C. Andréief is an often cited paper in random matrix theory, due to it containing what is now referred to as Andréief's integration formula. Nearly all citing works state the publication year as 1883. However the journal containing the paper, Mémories de la Societé des Sciences physiques et naturelles de Bordeaux, issue 3 volume 2 actually appeared in 1886. In addition to clarifying this point, some historical information relating to C. Andréief (better known as K.A. Andreev) and the lead up to this work is given, as is a review of some of the context of Andréief's integration formula.

math-ph↗

arXiv2018

Volumes and distributions for random unimodular complex and quaternion lattices

Two themes associated with invariant measures on the matrix groups ${\rm SL}_N(\mathbb F)$, with $\mathbb F = \mathbb R, \mathbb C$ or $\mathbb H$, and their corresponding lattices parametrised by ${\rm SL}_N(\mathbb F)/{\rm SL}_N(\mathbb O)$, $\mathbb O$ being an appropriate Euclidean ring of integers, are considered. The first is the computation of the volume of the subset of ${\rm SL}_N(\mathbb F)$ with bounded 2-norm or Frobenius norm. Key here is the decomposition of measure in terms of the singular values. The form of the volume, for large values of the bound, is relevant to asymptotic counting problems in ${\rm SL}_N(\mathcal O)$. The second is the problem of lattice reduction in the case $N=2$. A unified proof of the validity of the appropriate analogue of the Lagrange--Gauss algorithm for computing the shortest basis is given. A decomposition of measure corresponding to the QR decomposition is used to specify the invariant measure in the coordinates of the shortest basis vectors. With $\mathbb F = \mathbb C$ this allows for the exact computation of the PDF of the first minimum (for $\mathcal O = \mathbb Z[i]$ and $\mathbb Z[(1+\sqrt{-3})/2]$), and the PDF of the second minimum and that of the angle between the minimal basis vectors (for $\mathcal O = \mathbb Z[i]$). It also encodes the specification of fundamental domains of the corresponding quotient spaces. Integration over the latter gives rise to certain number theoretic constants, which are also present in the asymptotic forms of the PDFs of the lengths of the shortest basis vectors. Siegel's mean value gives an alternative method to compute the arithmetic constants, allowing in particular the computation of the leading form of the PDF of the first minimum for $\mathbb F = \mathbb H$ and $\mathcal O$ the Hurwitz integers, for which direct integration was not possible.

math.NT↗

arXiv2017

Multiplicative Convolution of Real Asymmetric and Real Antisymmetric Matrices

The singular values of products of standard complex Gaussian random matrices, or sub-blocks of Haar distributed unitary matrices, have the property that their probability distribution has an explicit, structured form referred to as a polynomial ensemble. It is furthermore the case that the corresponding bi-orthogonal system can be determined in terms of Meijer G-functions, and the correlation kernel given as an explicit double contour integral. It has recently been shown that the Hermitised product $X_M \cdots X_2 X_1A X_1^T X_2^T \cdots X_M^T$, where each $X_i$ is a standard real complex Gaussian matrix, and $A$ is real anti-symmetric shares exhibits analogous properties. Here we use the theory of spherical functions and transforms to present a theory which, for even dimensions, includes these properties of the latter product as a special case. As an example we show that the theory also allows for a treatment of this class of Hermitised product when the $X_i$ are chosen as sub-blocks of Haar distributed real orthogonal matrices.

math.PR↗

arXiv2017

Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

In this paper, we highlight the role played by orthogonal and symplectic Harish-Chandra integrals in the study of real-valued matrix product ensembles. By making use of these integrals and the matrix-valued Fourier-Laplace transform, we find the explicit eigenvalue distributions for particular Hermitian anti-symmetric matrices and particular Hermitian anti-self dual matrices, involving both sums and products. As a consequence of these results, the eigenvalue probability density function of the random product structure $X_M \cdots X_1( iA) X_1^T \cdots X_M^T$, where each $X_i$ is a standard real Gaussian matrix, and $A$ is a real anti-symmetric matrix can be determined. For $M=1$ and $A$ the bidiagonal anti-symmetric matrix with 1's above the diagonal, this reclaims results of Defosseux. For general $M$, and this choice of $A$, or $A$ itself a standard Gaussian anti-symmetric matrix, the eigenvalue distribution is shown to coincide with that of the squared singular values for the product of certain complex Gaussian matrices first studied by Akemann et al. As a point of independent interest, we also include a self-contained diffusion equation derivation of the orthogonal and symplectic Harish-Chandra integrals.

math-ph↗

arXiv2017

Large $N$ expansions for the Laguerre and Jacobi $β$ ensembles from the loop equations

The $β$-ensembles of random matrix theory with classical weights have many special properties. One is that the loop equations specifying the resolvent and corresponding multipoint correlators permit a derivation at general order of the correlator via Aomoto's method from the theory of the Selberg integral. We use Aomoto's method to derive the full hierarchy of loop equations for Laguerre and Jacobi $β$ ensembles, and use these to systematically construct the explicit form of the $1/N$ expansion at low orders. This allows us to give the explicit form of corrections to the global density, and allows various moments to be computed, complementing results available in the literature motivated by problems in quantum transport.

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arXiv2017

The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices

The probability that all eigenvalues of a product of $m$ independent $N \times N$ sub-blocks of a Haar distributed random real orthogonal matrix of size $(L_i+N) \times (L_i+N)$, $(i=1,\dots,m)$ are real is calculated as a multi-dimensional integral, and as a determinant. Both involve Meijer G-functions. Evaluation formulae of the latter, based on a recursive scheme, allow it to be proved that for any $m$ and with each $L_i$ even the probability is a rational number. The formulae furthermore provide for explicit computation in small order cases.

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