arXiv subjects

Peter J. Forrester

Publications and source records attributed to Peter J. Forrester.

At least 55 records · Page 3Linked to original sources

arXiv2023

Progress on the study of the Ginibre ensembles II: GinOE and GinSE

This is part II of a review relating to the three classes of random non-Hermitian Gaussian matrices introduced by Ginibre in 1965. While part I restricted attention to the GinUE (Ginibre unitary ensemble) case of complex elements, in this part the cases of real elements (GinOE, denoting Ginibre orthogonal ensemble) and quaternion elements represented as $2 \times 2$ complex blocks (GinSE, denoting Ginibre symplectic ensemble) are considered. The eigenvalues of both GinOE and GinSE form Pfaffian point processes, which are more complicated than the determinantal point processes resulting from GinUE. Nevertheless, many of the obstacles that have slowed progress on the development of traditional aspects of the theory have now been overcome, while new theoretical aspects and new applications have been identified. This permits a comprehensive account of themes addressed too in the complex case: eigenvalue probability density functions and correlation functions, limit formulas for correlation functions, fluctuation formulas, sum rules, gap probabilities and eigenvector statistics, among others. Distinct from the complex case is the need to develop a theory of skew orthogonal polynomials corresponding to the skew inner product associated with the Pfaffian. Another distinct theme is the statistics of real eigenvalues, which is unique to GinOE. These appear in a number of applications of the theory, coming from areas as diverse as diffusion processes and persistence in statistical physics, topologically driven parametric energy level crossings for certain quantum dots, and equilibria counting for a system of random nonlinear differential equations.

math-ph↗

arXiv2022

Comment on "Sum of squares of uniform random variables" by I. Weissman

The recent paper by I. Weissman, "Sum of squares of uniform random variables",[Statist. Probab. Lett. 129 (2017), 147-154] is compared to earlier work of B. Tibken and D. Constales relating to the area of the intersection of a centred ball and cube in $\mathbb R^n$, published in the Problems and Solutions section of SIAM Review in 1997. Some recent applications of explicit formulas for the corresponding probabilities from these references, to problems in lattice reduction, and to the study of Lyapunov exponents of products of random matrices, are noted.

math.PR↗

arXiv2022

Finite size corrections relating to distributions of the length of longest increasing subsequences

Considered are the large $N$, or large intensity, forms of the distribution of the length of the longest increasing subsequences for various models. Earlier work has established that after centring and scaling, the limit laws for these distributions relate to certain distribution functions at the hard edge known from random matrix theory. By analysing the hard to soft edge transition, we supplement and extend results of Baik and Jenkins for the Hammersley model and symmetrisations, which give that the leading correction is proportional to $z^{-2/3}$, where $z^2$ is the intensity of the Poisson rate, and provides a functional form as derivates of the limit law. Our methods give the functional form both in terms of Fredholm operator theoretic quantities, and in terms of Painlevé transcendents. For random permutations and their symmetrisations, numerical analysis of exact enumerations and simulations gives compelling evidence that the leading corrections are proportional to $N^{-1/3}$, and moreover provides an approximation to their graphical forms.

math-ph↗

arXiv2022

Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics

The framework of spherical transforms and Pólya ensembles is of utility in deriving structured analytic results for sums and products of random matrices in a unified way. In the present work, we will carry over this framework to study products of unitary matrices. Those are not distributed via the Haar measure, but still are drawn from distributions where the eigenvalue and eigenvector statistics factorise. They include the circular Jacobi ensemble, known in relation to the Fisher-Hartwig singularity in the theory of Toeplitz determinants, as well as the heat kernel for Brownian motion on the unitary group. We define cyclic Pólya frequency functions and show their relation to the cyclic Pólya ensembles, give a uniqueness statement for the corresponding weights, and derive the determinantal point processes of the eigenvalue statistics at fixed matrix dimension. An outline is given of problems one may encounter when investigating the local spectral statistics.

math.PR↗

arXiv2022

A review of exact results for fluctuation formulas in random matrix theory

Covariances and variances of linear statistics of a point process can be written as integrals over the truncated two-point correlation function. When the point process consists of the eigenvalues of a random matrix ensemble, there are often large $N$ universal forms for this correlation after smoothing, which results in particularly simple limiting formulas for the fluctuation of the linear statistics. We review these limiting formulas, derived in the simplest cases as corollaries of explicit knowledge of the truncated two-point correlation. One of the large $N$ limits is to scale the eigenvalues so that limiting support is compact, and the linear statistics vary on the scale of the support. This is a global scaling. The other, where a thermodynamic limit is first taken so that the spacing between eigenvalues is of order unity, and then a scale imposed on the test functions so they are slowly varying, is the bulk scaling. The latter was already identified as a probe of random matrix characteristics for quantum spectra in the pioneering work of Dyson and Mehta.

math-ph↗

arXiv2022

Rank $1$ perturbations in random matrix theory -- a review of exact results

A number of random matrix ensembles permitting exact determination of their eigenvalue and eigenvector statistics maintain this property under a rank $1$ perturbation. Considered in this review are the additive rank $1$ perturbation of the Hermitian Gaussian ensembles, the multiplicative rank $1$ perturbation of the Wishart ensembles, and rank $1$ perturbations of Hermitian and unitary matrices giving rise to a two-dimensional support for the eigenvalues. The focus throughout is on exact formulas, which are typically the result of various integrable structures. The simplest is that of a determinantal point process, with others relating to partial differential equations implied by a formulation in terms of certain random tridiagonal matrices. Attention is also given to eigenvector overlaps in the setting of a rank $1$ perturbation.

math-ph↗

arXiv2021

Harmonic analysis for rank-1 Randomised Horn Problems

The randomised Horn problem, in both its additive and multiplicative version, has recently drawn increasing interest. Especially, closed analytical results have been found for the rank-1 perturbation of sums of Hermitian matrices and products of unitary matrices. We will generalise these results to rank-1 perturbations for products of positive definite Hermitian matrices and prove the other results in a new unified way. Our ideas work along harmonic analysis for matrix groups via spherical transforms that have been successfully applied in products of random matrices in the past years. In order to achieve the unified derivation of all three cases, we define the spherical transform on the unitary group and prove its invertibility.

math-ph↗

arXiv2021

Joint moments of a characteristic polynomial and its derivative for the circular $β$-ensemble

The problem of calculating the scaled limit of the joint moments of the characteristic polynomial, and the derivative of the characteristic polynomial, for matrices from the unitary group with Haar measure first arose in studies relating to the Riemann zeta function in the thesis of Hughes. Subsequently, Winn showed that these joint moments can equivalently be written as the moments for the distribution of the trace in the Cauchy unitary ensemble, and furthermore relate to certain hypergeometric functions based on Schur polynomials, which enabled explicit computations. We give a $β$-generalisation of these results, where now the role of the Schur polynomials is played by the Jack polynomials. This leads to an explicit evaluation of the scaled moments for all $β> 0$, subject to the constraint that a particular parameter therein is equal to a non negative integer. Consideration is also given to the calculation of the moments of the singular statistic $\sum_{j=1}^N 1/x_j$ for the Jacobi $β$-ensemble.

math-ph↗

arXiv2021

Circulant $L$-ensembles in the thermodynamic limit

$L$-ensembles are a class of determinantal point processes which can be viewed as a statistical mechanical systems in the grand canonical ensemble. Circulant $L$-ensembles are the subclass which are locally translationally invariant and furthermore subject to periodic boundary conditions. Existing theory can very simply be specialised to this setting, allowing for the derivation of formulas for the system pressure, and the correlation kernel, in the thermodynamic limit. For a one-dimensional domain, this is possible when the circulant matrix is both real symmetric, or complex Hermitian. The special case of the former having a Gaussian functional form for the entries is shown to correspond to free fermions at finite temperature, and be generalisable to higher dimensions. A special case of the latter is shown to be the statistical mechanical model introduced by Gaudin to interpolate between Poisson and unitary symmetry statistics in random matrix theory. It is shown in all cases that the compressibility sum rule for the two-point correlation is obeyed, and the small and large distance asymptotics of the latter are considered. Also, a conjecture relating the asymptotic form of the hole probability to the pressure is verified.

math-ph↗

arXiv2021

Quantifying dip-ramp-plateau for the Laguerre unitary ensemble structure function

The ensemble average of $| \sum_{j=1}^N e^{i k λ_j} |^2$ is of interest as a probe of quantum chaos, as is its connected part, the structure function. Plotting this average for model systems of chaotic spectra reveals what has been termed a dip-ramp-plateau shape. Generalising earlier work of Brézin and Hikami for the Gaussian unitary ensemble, it is shown how the average in the case of the Laguerre unitary ensemble can be reduced to an expression involving the spectral density of the Jacobi unitary ensemble. This facilitates studying the large $N$ limit, and so quantifying the dip-ramp-plateau effect. When the parameter $a$ in the Laguerre weight $x^a e^{-x}$ scales with $N$, quantitative agreement is found with the characteristic features of this effect known for the Gaussian unitary ensemble. However, for the parameter $a$ fixed, the bulk scaled structure function is shown to have the simple functional form ${2 \over π} {\rm Arctan} \, k$, and so there is no ramp-plateau transition.

math-ph↗

arXiv2021

The classical $β$-ensembles with $β$ proportional to $1/N$: from loop equations to Dyson's disordered chain

In the classical $β$-ensembles of random matrix theory, setting $β= 2 α/N$ and taking the $N \to \infty$ limit gives a statistical state depending on $α$. Using the loop equations for the classical $β$-ensembles, we study the corresponding eigenvalue density, its moments, covariances of monomial linear statistics, and the moments of the leading $1/N$ correction to the density. From earlier literature the limiting eigenvalue density is known to be related to classical functions. Our study gives a unifying mechanism underlying this fact, identifying in particular the Gauss hypergeometric differential equation determining the Stieltjes transform of the limiting density in the Jacobi case. Our characterisation of the moments and covariances of monomial linear statistics is through recurrence relations. Also, we extend recent work which begins with the $β$-ensembles in the high temperature limit and constructs a family of tridiagonal matrices referred to as $α$-ensembles, obtaining a random anti-symmetric tridiagonal matrix with i.i.d.~gamma distributed random variables. From this we are able to supplement analytic results obtained by Dyson in the study of the so-called type I disordered chain.

math.PR↗

arXiv2021

Relations between moments for the Jacobi and Cauchy random matrix ensembles

We outline a relation between the densities for the $β$-ensembles with respect to the Jacobi weight $(1-x)^a(1+x)^b$ supported on the interval $(-1,1)$ and the Cauchy weight $(1-\mathrm{i}x)^η(1+\mathrm{i}x)^{\barη}$ by appropriate analytic continuation. This has the consequence of implying that the latter density satisfies a linear differential equation of degree three for $β=2$, and of degree five for $β=1$ and $4$, analogues of which are already known for the Jacobi weight $x^a(1-x)^b$ supported on $(0,1)$. We concentrate on the case $a=b$ (Jacobi weight on $(-1,1)$) and $η$ real (Cauchy weight) since the density is then an even function and the differential equations simplify. From the differential equations, recurrences can be obtained for the moments of the Jacobi weight supported on $(-1,1)$ and/or the moments of the Cauchy weight. Particular attention is paid to the case $β=2$ and the Jacobi weight on $(-1,1)$ in the symmetric case $a=b$, which in keeping with a recent result obtained by Assiotis et al.~for the $β=2$ case of the symmetric Cauchy weight (parameter $η$ real), allows for an explicit solution of the recurrence in terms of particular continuous Hahn polynomials. Also for the symmetric Cauchy weight with $η=-β(N-1)/2-1-α$, after appropriately scaling $α$ proportional to $N$, we use differential equations to compute terms in the $1/N^2$ ($1/N$) expansion of the resolvent for $β=2$ ($β=1,4$).

math-ph↗

arXiv2021

Applications in random matrix theory of a PIII$'$ $τ$-function sequence from Okamoto's Hamiltonian formulation

We consider the singular linear statistic of the Laguerre unitary ensemble consisting of the sum of the reciprocal of the eigenvalues. It is observed that the exponential generating function for this statistic can be written as a Toeplitz determinant with entries given in terms of particular $K$ Bessel functions. Earlier studies have identified the same determinant, but with the $K$ Bessel functions replaced by $I$ Bessel functions, as relating to the hard edge scaling limit of a generalized gap probability for the Laguerre unitary ensemble, in the case of non-negative integer Laguerre parameter. We show that the Toeplitz determinant formed from an arbitrary linear combination of these two Bessel functions occurs as a $τ$-function sequence in Okamoto's Hamiltonian formulation of Painlevé III$'$, and consequently the logarithmic derivative of both Toeplitz determinants satisfies the same $σ$-form Painlevé III$'$ differential equation, giving an explanation of a fact which can be observed from earlier results. In addition, some insights into the relationship between this characterization of the generating function, and its characterization in the $n \to \infty$ limit, both with the Laguerre parameter $α$ fixed, and with $α= n$ (this latter circumstance being relevant to an application to the distribution of the Wigner time delay statistic), are given.

math-ph↗

arXiv2021

Dyson's disordered linear chain from a random matrix theory viewpoint

The first work of Dyson relating to random matrix theory, "The dynamics of a disordered linear chain", is reviewed. Contained in this work is an exact solution of a so-called Type I chain in the case of the disorder variables being given by a gamma distribution. The exact solution exhibits a singularity in the density of states about the origin, which has since been shown to be universal for one-dimensional tight binding models with off diagonal disorder. We discuss this context and also point out some universal features of the weak disorder expansion of the exact solution near the band edge. Further, a link between the exact solution, and a tridiagonal formalism of anti symmetric Gaussian $β$-ensembles with $β$ proportional to $1/N$, is made.

math-ph↗

arXiv2020

Phase transitions for products of characteristic polynomials under Dyson Brownian motion

We study the averaged products of characteristic polynomials for the Gaussian and Laguerre $β$-ensembles with external source, and prove Pearcey-type phase transitions for particular full rank perturbations of source. The phases are characterised by determining the explicit functional forms of the scaled limits of the averaged products of characteristic polynomials, which are given as certain multidimensional integrals, with dimension equal to the number of products.

math.PR↗

arXiv2020

Differential recurrences for the distribution of the trace of the $β$-Jacobi ensemble

Examples of the $β$-Jacobi ensemble specify the joint distribution of the transmission eigenvalues in scattering problems. In this context, there has been interest in the distribution of the trace, as the trace corresponds to the conductance. Earlier, in the case $β= 1$, the trace statistic was isolated in studies of covariance matrices in multivariate statistics, where it is referred to as Pillai's $V$ statistic. In this context, Davis showed that for $β= 1$ the trace statistic, and its Fourier-Laplace transform, can be characterised by $(N+1) \times (N+1)$ matrix differential equations. For the Fourier-Laplace transform, this leads to a vector recurrence for the moments. However, for the distribution itself the characterisation provided was incomplete, as the connection problem of determining the linear combination of Frobenius type solutions that correspond to the statistic was not solved. We solve this connection problem for Jacobi parameter $b$ and Dyson index $β$ non-negative integers. For the other Jacobi parameter $a$ also a non-negative integer, the power series portion of each Frobenius solution terminates to a polynomial, and the matrix differential equation gives a recurrence for their computation.

math-ph↗

arXiv2020

Global and local scaling limits for the $β= 2$ Stieltjes--Wigert random matrix ensemble

The eigenvalue probability density function (PDF) for the Gaussian unitary ensemble has a well known analogy with the Boltzmann factor for a classical log-gas with pair potential $- \log | x - y|$, confined by a one-body harmonic potential. A generalisation is to replace the pair potential by $- \log |\sinh (π(x-y)/L) |$. The resulting PDF first appeared in the statistical physics literature in relation to non-intersecting Brownian walkers, equally spaced at time $t=0$, and subsequently in the study of quantum many body systems of the Calogero-Sutherland type, and also in Chern-Simons field theory. It is an example of a determinantal point process with correlation kernel based on the Stieltjes--Wigert polynomials. We take up the problem of determining the moments of this ensemble, and find an exact expression in terms of a particular little $q$-Jacobi polynomial. From their large $N$ form, the global density can be computed. Previous work has evaluated the edge scaling limit of the correlation kernel in terms of the Ramanujan ($q$-Airy) function. We show how in a particular $L \to \infty$ scaling limit, this reduces to the Airy kernel.

math-ph↗

arXiv2020

Moments of the ground state density for the $d$-dimensional Fermi gas in an harmonic trap

We consider properties of the ground state density for the $d$-dimensional Fermi gas in an harmonic trap. Previous work has shown that the $d$-dimensional Fourier transform has a very simple functional form. It is shown that this fact can be used to deduce that the density itself satisfies a third order linear differential equation, previously known in the literature but from other considerations. It is shown too how this implies a closed form expression for the $2k$-th non-negative integer moments of the density, and a second order recurrence. Both can be extended to general Re$\, k > -d/2$. The moments, and the smoothed density, permit expansions in $1/\tilde{M}^2$, where $\tilde{M} = M + (d+1)/2$, with $M$ denoting the shell label. The moment expansion substituted in the second order recurrence gives a generalisation of the Harer--Zagier recurrence, satisfied by the coefficients of the $1/N^2$ expansion of the moments of the spectral density for the Gaussian unitary ensemble in random matrix theory.

math-ph↗