School of Mathematics and Statistics - Research Publications
Permanent URI for this collection
Search Results
1 - 10 of 35
-
ItemExpanding the Fourier Transform of the Scaled Circular Jacobi β Ensemble Density(SPRINGER, 2023-10-18)Abstract The family of circular Jacobi $$\beta $$ 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 $$N \rightarrow \infty $$ N → ∞ bulk scaled spectral density about this singularity, expanded as a series in the Fourier variable. Various integrability aspects of the circular Jacobi$$\beta $$ β ensemble are used for this purpose. These include linear differential equations satisfied by the scaled spectral density for $$\beta = 2$$ β = 2 and $$\beta = 4$$ β = 4 , and the loop equation hierarchy. The polynomials in the variable $$u=2/\beta $$ 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 $$\beta $$ β ensemble, specifically in relation to the zeros lying on the unit circle $$|u|=1$$ | 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.
-
ItemNo Preview AvailableRank 1 perturbations in random matrix theory - A review of exact results(WORLD SCIENTIFIC PUBL CO PTE LTD, 2023-10)A number of random matrix ensembles permitting exact determination of their eigenvalue and eigenvector statistics maintain this property under a rank [Formula: see text] perturbation. Considered in this review are the additive rank [Formula: see text] perturbation of the Hermitian Gaussian ensembles, the multiplicative rank [Formula: see text] perturbation of the Wishart ensembles, and rank [Formula: see text] 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 [Formula: see text] perturbation.
-
ItemNo Preview AvailableComputable structural formulas for the distribution of the beta-Jacobi edge eigenvalues(SPRINGER, 2023-05-01)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.
-
ItemNo Preview AvailableHigh-low temperature dualities for the classical β -ensembles(World Scientific Pub Co Pte Ltd, 2022-01-01)The loop equations for the [Formula: see text]-ensembles are conventionally solved in terms of a [Formula: see text] expansion. We observe that it is also possible to fix N and expand in inverse powers of [Formula: see text]. At leading order, for the one-point function [Formula: see text] corresponding to the average of the linear statistic [Formula: see text] and after specialising to 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 [Formula: see text] in the case of classical weights — which are particular Riccati equations — are simply related to the differential equations satisfied by [Formula: see text] in the high temperature scaled limit [Formula: see text] ([Formula: see text] fixed, [Formula: see text]), implying a certain high–low temperature duality. A generalisation of this duality, valid without any limiting procedure, is shown to hold for [Formula: see text] and all its higher point analogues in the classical [Formula: see text]-ensembles.
-
ItemNo Preview AvailableGlobal and local scaling limits for the β=2 Stieltjes-Wigert random matrix ensemble(WORLD SCIENTIFIC PUBL CO PTE LTD, 2022-04)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 [Formula: see text], confined by a one-body harmonic potential. A generalization is to replace the pair potential by [Formula: see text]. The resulting PDF first appeared in the statistical physics literature in relation to non-intersecting Brownian walkers, equally spaced at time [Formula: see text], 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 ([Formula: see text]-Airy) function. We show how in a particular [Formula: see text] scaling limit, this reduces to the Airy kernel.
-
ItemNo Preview Available
-
ItemNo Preview Availableq-Pearson pair and moments in q-deformed ensembles(SPRINGER, 2023-01)
-
ItemNo Preview AvailableCyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics(Springer Science and Business Media LLC, 2023-06-01)
-
ItemFinite Size Effects for Spacing Distributions in Random Matrix Theory: Circular Ensembles and Riemann Zeros(Wiley, 2017-05-01)According to Dyson's threefold way, from the viewpoint of global time reversal symmetry, there are three circular ensembles of unitary random matrices relevant to the study of chaotic spectra in quantum mechanics. These are the circular orthogonal, unitary, and symplectic ensembles, denoted COE, CUE, and CSE, respectively. For each of these three ensembles and their thinned versions, whereby each eigenvalue is deleted independently with probability 1 − ξ, we take up the problem of calculating the first two terms in the scaled large N expansion of the spacing distributions. It is well known that the leading term admits a characterization in terms of both Fredholm determinants and Painlevé transcendents. We show that modifications of these characterizations also remain valid for the next to leading term, and that they provide schemes for high precision numerical computations. In the case of the CUE, there is an application to the analysis of Odlyzko's data set for the Riemann zeros, and in that case, some further statistics are similarly analyzed.
-
ItemSingular Values of Products of Ginibre Random Matrices(Wiley, 2017-02-01)The squared singular values of the product of M complex Ginibre matrices form a biorthogonal ensemble, and thus their distribution is fully determined by a correlation kernel. The kernel permits a hard edge scaling to a form specified in terms of certain Meijer G-functions, or equivalently hypergeometric functions0FM, also referred to as hyper-Bessel functions. In the case M = 1, it is well known that the corresponding gap probability for no squared singular values in (0, s) can be evaluated in terms of a solution of a particular sigma form of the Painlevé III’ system. One approach to this result is a formalism due to Tracy and Widom, involving the reduction of a certain integrable system. Strahov has generalized this formalism to general M ≥ 1, but has not exhibited its reduction. After detailing the necessary working in the case M = 1, we consider the problem of reducing the 12 coupled differential equations in the case M = 2 to a single differential equation for the resolvent. An explicit fourth-order nonlinear is found for general hard edge parameters. For a particular choice of parameters, evidence is given that this simplifies to a much simpler third-order nonlinear equation. The small and large s asymptotics of the fourth-order equation are discussed, as is a possible relationship of the M = 2 systems to so-called four-dimensional Painlevé-type equations.