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Peter J. Forrester

Publications and source records attributed to Peter J. Forrester.

At least 37 records · Page 2Linked to original sources
arXiv2025

Dualities in random matrix theory

Duality identities in random matrix theory for products and powers of characteristic polynomials, and for moments, are reviewed. The structure of a typical duality identity for the average of a positive integer power $k$ of the characteristic polynomial for particular ensemble of $N \times N$ matrices is that it is expressed as the average of the power $N$ of the characteristic polynomial of some other ensemble of random matrices, now of size $k \times k$. With only a few exceptions, such dualities involve (the $β$ generalised) classical Gaussian, Laguerre and Jacobi ensembles Hermitian ensembles, the circular Jacobi ensemble, or the various non-Hermitian ensembles relating to Ginibre random matrices. In the case of unitary symmetry in the Hermitian case, they can be studied using the determinantal structure. The $β$ generalised case requires the use of Jack polynomial theory, and in particular Jack polynomial based hypergeometric functions. Applications to the computation of the scaling limit of various $β$ ensemble correlation and distribution functions are also reviewed. The non-Hermitian case relies on the particular cases of Jack polynomials corresponding to zonal polynomials, and their integration properties when their arguments are eigenvalues of certain matrices. The main tool to study dualities for moments of the spectral density, and generalisations, is the loop equations.

math-ph↗
arXiv2025

Properties of the one-component Coulomb gas on a sphere with two macroscopic external charges

The one-component Coulomb gas on the sphere, consisting on $N$ unit charges interacting via a logarithmic potential, and in the presence of two external charges each of strength proportional to $N$, is considered. There are two spherical caps naturally associated with the external charges, giving rise to two distinct phases depending on them not overlapping (post-critical) or overlapping (pre-critical). The equilibrium measure in the post-critical phase is known from earlier work. We determine the equilibrium measure in the pre-critical phase using a particular conformal map, with the parameters therein specified in terms of a root of a certain fourth order polynomial. This is used to determine the exact form of the electrostatic energy for the pre-critical phase. Using a duality relation from random matrix theory, the partition function for the Coulomb gas at the inverse temperature $β= 2$ can be expanded for large $N$ in the post-critical phase, and in a scaling region of the post and pre-critical boundary. For the pre-critical phase, the duality identity implies a relation between two electrostatic energies, one for the present sphere system, and the other for a certain constrained log-gas relating to the Jacobi unitary ensemble.

math-ph↗
arXiv2024

Computing marginal eigenvalue distributions for the Gaussian and Laguerre orthogonal ensembles

The Gaussian and Laguerre orthogonal ensembles are fundamental to random matrix theory, and the marginal eigenvalue distributions are basic observable quantities. Notwithstanding a long history, a formulation providing high precision numerical evaluations for $N$ large enough to probe asymptotic regimes, has not been provided. An exception is for the largest eigenvalue, where there is a formalism due to Chiani which uses a combination of the Pfaffian structure underlying the ensembles, and a recursive computation of the matrix elements. We augment this strategy by introducing a generating function for the conditioned gap probabilities. A finite Fourier series approach is then used to extract the sequence of marginal eigenvalue distributions as a linear combination of Pfaffians, with the latter then evaluated using an efficient numerical procedure available in the literature. Applications are given to illustrating various asymptotic formulas, local central limit theorems, and central limit theorems, as well as to probing finite size corrections. Further, our data indicates that the mean values of the marginal distributions interlace with the zeros of the Hermite polynomial (Gaussian ensemble) and a Laguerre polynomial (Laguerre ensemble).

math-ph↗
arXiv2024

Dualities for characteristic polynomial averages of complex symmetric and self dual non-Hermitian random matrices

Ensembles of complex symmetric, and complex self dual random matrices are known to exhibit local statistical properties distinct from those of the non-Hermitian Ginibre ensembles. On the other hand, in distinction to the latter, the joint eigenvalue probability density function of these two ensembles are not known. Nonetheless, as carried out in the recent works of Liu and Zhang, Akemann et al.~and Kulkarni et al., by considering averages of products of characteristic polynomials, analytic progress can be made. Here we show that an approach based on the theory of zonal polynomials provides an alternative to the diffusion operator or supersymmetric Grassmann integrations methods of these works. It has the advantage of not being restricted to a Gaussian unitary invariant measure on the matrix spaces. To illustrate this, as an extension, we consider averages of products and powers of characteristic polynomials for complex symmetric, and complex self dual random matrices subject to a spherical measure. In the case of powers, when comparing against the corresponding real Ginibre, respectively quaternion Ginibre averages with a spherical measure, one finds the qualitative feature of a decreasing (increasing) profile as the magnitude of the argument of the characteristic polynomial increases. This is analogous to the findings of the second two of the cited works in the Gaussian case.

math-ph↗
arXiv2024

Ensemble inter-relations in random matrix theory

The ensemble inter-relations to be considered are special features of classical cases, where the joint eigenvalue probability density can be computed explicitly. Attention will be focussed too on the consequences of these inter-relations, most often in relation to gap probabilities. A highlight from this viewpoint is an evaluation formula for the gap probability generating function for the circular orthogonal and circular symplectic ensembles (examples of Pfaffian point processes), in terms of the gap probability generating function for real orthogonal matrices chosen with Haar measure ensembles (examples of determinantal point processes). The classes of inter-relations which lead to this result involve the superposition of the eigenvalue point processes of two independent classical ensembles, or consideration of the singular values of the classical ensembles with an evenness symmetry, and the decimation operation of integrating over every second eigenvalue. It turns out that the probability density functions encountered through the consideration of superposition and decimation can in some cases be obtained via the consideration of various rank one matrix perturbations. These allow for different understandings of some of the inter-relations in this class. We make note too of inter-relations -- which we consider generally as an identity or evaluation formula of a statistic linking two distinct random matrix ensemble -- that fall outside of this class, applying to the spectral form factor, and ones relating to discrete determinantal point processes. Not part of our review are the so-called duality relations of random matrix theory. Their consideration warrants a separate discussion.

math-ph↗
arXiv2024

Power spectra of Dyson's circular ensembles

The power spectrum is a Fourier series statistic associated with the covariances of the displacement from average positions of the members of an eigen-sequence. When this eigen-sequence has rotational invariance, as for the eigen-angles of Dyson's circular ensembles, recent work of Riser and Kanzieper has uncovered an exact identity expressing the power spectrum in terms of the generating function for the conditioned gap probability of having $k$ eigenvalues in an interval. These authors moreover showed how for the circular unitary ensemble integrability properties of the generating function, via a particular Painlevé VI system, imply a computational scheme for the corresponding power spectrum, and allow for the determination of its large $N$ limit. In the present work, these results are extended to the case of the circular orthogonal ensemble and circular symplectic ensemble, where the integrability is expressed through four particular Painlevé VI systems for finite $N$, and two Painlevé III$'$ systems for the limit $N\to\infty$, and also via corresponding Fredholm determinants. The relation between the limiting power spectrum $S_\infty(ω)$, where $ω$ denotes the Fourier variable, and the limiting generating function for the conditioned gap probabilities is particular direct, involving just a single integration over the gap endpoint in the latter. Interpreting this generating function as the characteristic function of a counting statistic allows for it to be shown that $S_\infty(ω) \mathop{\sim} \limits_{ω\to 0} {1 \over πβ| ω|}$, where $β$ is the Dyson index.

math-ph↗
arXiv2024

Spectral Moments of the real Ginibre ensemble

The moments of the real eigenvalues of real Ginibre matrices are investigated from the viewpoint of explicit formulas, differential and difference equations, and large $N$ expansions. These topics are inter-related. For example, a third order differential equation can be derived for the density of the real eigenvalues, and this can be used to deduce a second order difference equation for the general complex moments $M_{2p}^{\rm r}$. The latter are expressed in terms of the ${}_3 F_2$ hypergeometric functions, with a simplification to the ${}_2 F_1$ hypergeometric function possible for $p=0$ and $p=1$, allowing for the large $N$ expansion of these moments to be obtained. The large $N$ expansion involves both integer and half integer powers of $1/N$. The three term recurrence then provides the large $N$ expansion of the full sequence $\{ M_{2p}^{\rm r} \}_{p=0}^\infty$. Fourth and third order linear differential equations are obtained for the moment generating function and for the Stieltjes transform of the real density, respectively, and properties of the large $N$ expansion of these quantities are determined.

math-ph↗
arXiv2024

$q$-deformed Gaussian unitary ensemble: spectral moments and genus-type expansions

The eigenvalue probability density function of the Gaussian unitary ensemble permits a $q$-extension related to the discrete $q$-Hermite weight and corresponding $q$-orthogonal polynomials. A combinatorial counting method is used to specify a positive sum formula for the spectral moments of this model. The leading two terms of the scaled $1/N^2$ genus-type expansion of the moments are evaluated explicitly in terms of the incomplete beta function. Knowledge of these functional forms allows for the smoothed leading eigenvalue density and its first correction to be determined analytically.

math.PR↗
arXiv2024

Computation of marginal eigenvalue distributions in the Laguerre and Jacobi $β$ ensembles

We consider the problem of the exact computation of the marginal eigenvalue distributions in the Laguerre and Jacobi $β$ ensembles. In the case $β=1$ this is a question of long standing in the mathematical statistics literature. A recursive procedure to accomplish this task is given for $β$ a positive integer, and the parameter $λ_1$ a non-negative integer. This case is special due to a finite basis of elementary functions, with coefficients which are polynomials. In the Laguerre case with $β= 1$ and $λ_1 + 1/2$ a non-negative integer some evidence is given of their again being a finite basis, now consisting of elementary functions and the error function multiplied by elementary functions. Moreover, from this the corresponding distributions in the fixed trace case permit a finite basis of power functions, as also for $λ_1$ a non-negative integer. The fixed trace case in this setting is relevant to quantum information theory and quantum transport problem, allowing particularly the exact determination of Landauer conductance distributions in a previously intractable parameter regime. Our findings also aid in analyzing zeros of the generating function for specific gap probabilities, supporting the validity of an associated large $N$ local central limit theorem.

math-ph↗
arXiv2024

On Efficient Sampling Schemes for the Eigenvalues of Complex Wishart Matrices

The paper "An efficient sampling scheme for the eigenvalues of dual Wishart matrices", by I.~Santamaría and V.~Elvira, [\emph{IEEE Signal Processing Letters}, vol.~28, pp.~2177--2181, 2021] \cite{SE21}, poses the question of efficient sampling from the eigenvalue probability density function of the $n \times n$ central complex Wishart matrices with variance matrix equal to the identity. Underlying such complex Wishart matrices is a rectangular $R \times n$ $(R \ge n)$ standard complex Gaussian matrix, requiring then $2Rn$ real random variables for their generation. The main result of \cite{SE21} gives a formula involving just two classical distributions specifying the two eigenvalues in the case $n=2$. The purpose of this Letter is to point out that existing results in the literature give two distinct ways to efficiently sample the eigenvalues in the general $n$ case. One is in terms of the eigenvalues of a tridiagonal matrix which factors as the product of a bidiagonal matrix and its transpose, with the $2n+1$ nonzero entries of the latter given by (the square root of) certain chi-squared random variables. The other is as the generalised eigenvalues for a pair of bidiagonal matrices, also containing a total of $2n+1$ chi-squared random variables. Moreover, these characterisation persist in the case of that the variance matrix consists of a single spike, and for the case of real Wishart matrices.

math.ST↗
arXiv2023

On the gamma difference distribution

The gamma difference distribution is defined as the difference of two gamma distributions, with in general different shape and rate parameters. Starting with knowledge of the corresponding characteristic function, a second order linear differential equation characterisation of the probability density function is given. This is used to derive a Stein-type differential identity relating to the expectation with respect to the gamma difference distribution of a general twice differentiable function $g(x)$. Choosing $g(x) = x^k$ gives a second order recurrence for the positive integer moments, which are also shown to permit evaluations in terms of ${}_2 F_1$ hypergeometric polynomials. A hypergeometric function evaluation is given for the absolute continuous moments. Specialising the gamma difference distribution gives the variance gamma distribution. Results of the type obtained herein have previously been obtained for this distribution, allowing for comparisons to be made.

math.ST↗
arXiv2023

Relations between generalised Wishart matrices, the Muttalib--Borodin model and matrix spherical functions

Generalised uncorrelated Wishart matrices are formed out of rectangular standard Gaussian data matrices with a certain pattern of zero entries. Development of the theory in the real and complex cases has proceeded along separate line. For example, emphasis in the real case has been placed on the Bellman and Riesz distributions, while that in the complex case has been shown to be closely related to the Muttalib-Borodin model. In this work, as well as uniting the lines of development, a tie in with matrix spherical functions is identified in the context of deducing the eigenvalue probability density function from the joint element probability density function.

math-ph↗
arXiv2023

Expanding the Fourier transform of the scaled circular Jacobi $β$ ensemble density

The family of circular Jacobi $β$ ensembles has a singularity of a type associated with Fisher and Hartwig in the theory of Toeplitz determinants. Our interest is in the Fourier transform of the corresponding bulk scaled spectral density about this singularity, expanded as a series in the Fourier variable. Various integrability aspects of the circular Jacobi $β$ ensemble are used for this purpose. These include linear differential equations satisfied by the scaled spectral density for $β= 2$ and $β= 4$, and the loop equation hierarchy. The polynomials in the variable $u=2/β$ which occur in the expansion coefficents are found to have special properties analogous to those known for the structure function of the circular $β$ ensemble, specifically in relation to the zeros lying on the unit circle $|u|=1$ and interlacing. Comparison is also made with known results for the expanded Fourier transform of the density about a guest charge in the two-dimensional one-component plasma.

math-ph↗
arXiv2023

Dip-ramp-plateau for Dyson Brownian motion from the identity on $U(N)$

In a recent work the present authors have shown that the eigenvalue probability density function for Dyson Brownian motion from the identity on $U(N)$ is an example of a newly identified class of random unitary matrices called cyclic Pólya ensembles. In general the latter exhibit a structured form of the correlation kernel. Specialising to the case of Dyson Brownian motion from the identity on $U(N)$ allows the moments of the spectral density, and the spectral form factor $S_N(k;t)$, to be evaluated explicitly in terms of a certain hypergeometric polynomial. Upon transformation, this can be identified in terms of a Jacobi polynomial with parameters $(N(μ- 1),1)$, where $μ= k/N$ and $k$ is the integer labelling the Fourier coefficients. From existing results in the literature for the asymptotics of the latter, the asymptotic forms of the moments of the spectral density can be specified, as can $\lim_{N \to \infty} {1 \over N} S_N(k;t) |_{μ= k/N}$. These in turn allow us to give a quantitative description of the large $N$ behaviour of the average $ \langle | \sum_{l=1}^N e^{ i k x_l} |^2 \rangle$. The latter exhibits a dip-ramp-plateau effect, which is attracting recent interest from the viewpoints of many body quantum chaos, and the scrambling of information in black holes.

math-ph↗
arXiv2023

Spherical Induced Ensembles with Symplectic Symmetry

We consider the complex eigenvalues of the induced spherical Ginibre ensemble with symplectic symmetry and establish the local universality of these point processes along the real axis. We derive scaling limits of all correlation functions at regular points both in the strong and weak non-unitary regimes as well as at the origin having spectral singularity. A key ingredient of our proof is a derivation of a differential equation satisfied by the correlation kernels of the associated Pfaffian point processes, thereby allowing us to perform asymptotic analysis.

math.PR↗
arXiv2023

Local central limit theorem for real eigenvalue fluctuations of elliptic GinOE matrices

Random matrices from the elliptic Ginibre orthogonal ensemble (GinOE) are a certain linear combination of a real symmetric, and real anti-symmetric, real Gaussian random matrices and controlled by a parameter $τ$. Our interest is in the fluctuations of the number of real eigenvalues, for fixed $τ$ when the expected number is proportional to the square root of the matrix size $N$, and for $τ$ scaled to the weakly non-symmetric limit, when the number of eigenvalues is proportional to $N$. By establishing that the generating function for the probabilities specifying the distribution of the number of real eigenvalues has only negative real zeros, and using too the fact that variances in both circumstancesof interest tends to infinity as $N \to \infty$, the known central limit theorem for the fluctuations is strengthened to a local central limit theorem, and the rate of convergence is discussed.

math.PR↗
arXiv2023

High-low temperature dualities for the classical $β$-ensembles

The loop equations for the $β$-ensembles are conventionally solved in terms of a $1/N$ expansion. We observe that it is also possible to fix $N$ and expand in inverse powers of $β$. At leading order, for the one-point function $W_1(x)$ corresponding to the average of the linear statistic $A = \sum_{j=1}^N 1/(x - λ_j)$, and specialising the classical weights, this reclaims well known results of Stieltjes relating the zeros of the classical polynomials to the minimum energy configuration of certain log-gas potential energies. Moreover, it is observed that the differential equations satisfied by $W_1(x)$ in the case of classical weights -- which are particular Riccati equations -- are simply related to the differential equations satisfied by $W_1(x)$ in the high temperature scaled limit $β= 2α/N$ ($α$ fixed, $N \to \infty$), implying a certain high-low temperature duality. A generalisation of this duality, valid without any limiting procedure, is shown to hold for $W_1(x)$ and all its higher point analogues in the classical $β$-ensembles.

math-ph↗
arXiv2023

Progress on the study of the Ginibre ensembles I: GinUE

The Ginibre unitary ensemble (GinUE) consists of $N \times N$ random matrices with independent complex standard Gaussian entries. This was introduced in 1965 by Ginbre, who showed that the eigenvalues form a determinantal point process with an explicit correlation kernel, and after scaling they are supported on the unit disk with constant density. For some time now it has been appreciated that GinUE has a fundamental place within random matrix theory, both for its applications and for the richness of its theory. Here we review the progress on a number of themes relating to the study of GinUE. These are eigenvalue probability density functions and correlation functions, fluctuation formulas, sum rules and asymptotic behaviours of correlation functions, and normal matrix models. We discuss too applications in quantum many body physics and quantum chaos, and give an account of some statistical properties of the eigenvectors.

math-ph↗