arXiv subjects

P. J. Forrester

Publications and source records attributed to P. J. Forrester.

At least 37 records · Page 2Linked to original sources

arXiv2010

Edge effects in some perturbations of the GUE

A bordering of GUE matrices is considered, in which the bordered row consists of zero mean complex Gaussians N$[0,σ/2] + i {\rm N}[0,σ/2]$ off the diagonal, and the real Gaussian N$[μ,σ/\sqrt{2}]$ on the diagonal. We compute the explicit form of the eigenvalue probability function for such matrices, as well as that for matrices obtained by repeating the bordering. The correlations are in general determinantal, and in the single bordering case the explicit form of the correlation kernel is computed. In the large $N$ limit it is shown that $μ$ and/or $σ$ can be tuned to induce a separation of the largest eigenvalue. This effect is shown to be controlled by a single parameter, universal correlation kernel.

math-ph↗

arXiv2006

Random Matrix Theory and the Sixth Painlevé Equation

A feature of certain ensembles of random matrices is that the corresponding measure is invariant under conjugation by unitary matrices. Study of such ensembles realised by matrices with Gaussian entries leads to statistical quantities related to the eigenspectrum, such as the distribution of the largest eigenvalue, which can be expressed as multidimensional integrals or equivalently as determinants. These distributions are well known to be $τ$-functions for Painlevé systems, allowing for the former to be characterised as the solution of certain nonlinear equations. We consider the random matrix ensembles for which the nonlinear equation is the $σ$ form of \PVI. Known results are reviewed, as is their implication by way of series expansions for the distributions. New results are given for the boundary conditions in the neighbourhood of the fixed singularities at $t=0,1,\infty$ of $σ$\PVI displayed by a generalisation of the generating function for the distributions. The structure of these expansions is related to Jimbo's general expansions for the $τ$-function of $σ$\PVI in the neighbourhood of its fixed singularities, and this theory is itself put in its context of the linear isomonodromy problem relating to \PVI.

math.CA↗

arXiv2006

Analytic solutions of the 1D finite coupling delta function Bose gas

An intensive study for both the weak coupling and strong coupling limits of the ground state properties of this classic system is presented. Detailed results for specific values of finite $N$ are given and from them results for general $N$ are determined. We focus on the density matrix and concomitantly its Fourier transform, the occupation numbers, along with the pair correlation function and concomitantly its Fourier transform, the structure factor. These are the signature quantities of the Bose gas. One specific result is that for weak coupling a rational polynomial structure holds despite the transcendental nature of the Bethe equations. All these new results are predicated on the Bethe ansatz and are built upon the seminal works of the past.

cond-mat.stat-mech↗

arXiv2006

Hard and soft edge spacing distributions for random matrix ensembles with orthogonal and symplectic symmetry

Inter-relations between random matrix ensembles with different symmetry types provide inter-relations between generating functions for the gap probabilites at the spectrum edge. Combining these in the scaled limit with the exact evaluation of the gap probabilities for certain superimposed ensembles with orthogonal symmetry allows for the exact evaluation of the gap probabilities at the hard and soft spectrum edges in the cases of orthogonal and symplectic symmetry. These exact evaluations are given in terms of Painlevé transcendents, and in terms of Fredholm determinants.

math-ph↗

arXiv2006

Eigenvalue distributions for some correlated complex sample covariance matrices

The distributions of the smallest and largest eigenvalues for the matrix product $Z^\dagger Z$, where $Z$ is an $n \times m$ complex Gaussian matrix with correlations both along rows and down columns, are expressed as $m \times m$ determinants. In the case of correlation along rows, these expressions are computationally more efficient than those involving sums over partitions and Schur polynomials reported recently for the same distributions.

math-ph↗

arXiv2006

Quantum conductance problems and the Jacobi ensemble

In one dimensional transport problems the scattering matrix $S$ is decomposed into a block structure corresponding to reflection and transmission matrices at the two ends. For $S$ a random unitary matrix, the singular value probability distribution function of these blocks is calculated. The same is done when $S$ is constrained to be symmetric, or to be self dual quaternion real, or when $S$ has real elements, or has real quaternion elements. Three methods are used: metric forms; a variant of the Ingham-Seigel matrix integral; and a theorem specifying the Jacobi random matrix ensemble in terms of Wishart distributed matrices.

math-ph↗

arXiv2005

Particles in a magnetic field and plasma analogies: doubly periodic boundary conditions

The $N$-particle free fermion state for quantum particles in the plane subject to a perpendicular magnetic field, and with doubly periodic boundary conditions, is written in a product form. The absolute value of this is used to formulate an exactly solvable one-component plasma model, and further motivates the formulation of an exactly solvable two-species Coulomb gas. The large $N$ expansion of the free energy of both these models exhibits the same O(1) term. On the basis of a relationship to the Gaussian free field, this term is predicted to be universal for conductive Coulomb systems in doubly periodic boundary conditions.

math-ph↗

arXiv2005

Boundary conditions associated with the Painlevé III' and V evaluations of some random matrix averages

In a previous work a random matrix average for the Laguerre unitary ensemble, generalising the generating function for the probability that an interval $ (0,s) $ at the hard edge contains $ k $ eigenvalues, was evaluated in terms of a Painlevé V transcendent in $ σ$-form. However the boundary conditions for the corresponding differential equation were not specified for the full parameter space. Here this task is accomplished in general, and the obtained functional form is compared against the most general small $ s $ behaviour of the Painlevé V equation in $ σ$-form known from the work of Jimbo. An analogous study is carried out for the the hard edge scaling limit of the random matrix average, which we have previously evaluated in terms of a Painlevé \IIId transcendent in $ σ$-form. An application of the latter result is given to the rapid evaluation of a Hankel determinant appearing in a recent work of Conrey, Rubinstein and Snaith relating to the derivative of the Riemann zeta function.

math.CA↗

arXiv2005

Asymptotic form of the density profile for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry

In a recent study we have obtained correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N by N matrices, both in the bulk and at the soft edge of the spectrum. In the present study these results are used to similarly analyze the eigenvalue density for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry. As in the case of unitary symmetry, a matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion of the edge density, expanded into the bulk. In addition, aspects of the asymptotic expansion of the smoothed density, which involves delta functions at the endpoints of the support, are interpreted microscopically.

math-ph↗

arXiv2005

Asymptotic corrections to the eigenvalue density of the GUE and LUE

We obtain correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N by N matrices, both in the bulk of the spectrum and near the spectral edge. This is achieved by using the well known orthogonal polynomial expression for the kernel to construct a double contour integral representation for the density, to which we apply the saddle point method. The main correction to the bulk density is oscillatory in N and depends on the distribution function of the limiting density, while the corrections to the Airy kernel at the soft edge are again expressed in terms of the Airy function and its first derivative. We demonstrate numerically that these expansions are very accurate. A matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion of the edge density, expanded into the bulk.

math-ph↗

arXiv2004

Spacing distributions in random matrix ensembles

This paper is my contribution to the planned publication Recent Perspectives in Random Matrix Theory (Cambridge University Press). Addressed is the problem of computing spacing distributions in the bulk for the three symmetry classes orthogonal, unitary and symplectic. Emphasis is placed on the role of Painleve transcendents.

math-ph↗

arXiv2004

Bi-orthogonal Polynomials on the Unit Circle, regular semi-classical Weights and Integrable Systems

The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference equations of certain coefficient functions appearing in the theory. A natural formulation of the Riemann-Hilbert problem is presented which has as its solution the above system of bi-orthogonal polynomials and associated functions. In particular for the case of regular semi-classical weights on the unit circle $ w(z) = \prod^m_{j=1}(z-z_j(t))^{ρ_j} $, consisting of $ m \in \mathbb{Z}_{> 0} $ finite singularities, difference equations with respect to the bi-orthogonal polynomial degree $ n $ (Laguerre-Freud equations or discrete analogs of the Schlesinger equations) and differential equations with respect to the deformation variables $ z_j(t) $ (Schlesinger equations) are derived completely characterising the system.

math.CA↗

arXiv2004

Discrete Painlevé equations for a class of \PVI $τ$-functions given as U(N) averages

In a recent work difference equations (Laguerre-Freud equations) for the bi-orthogonal polynomials and related quantities corresponding to the weight on the unit circle $ w(z)=\prod^m_{j=1}(z-z_j(t))^{ρ_j} $ were derived.Here it is shown that in the case $ m=3 $ these difference equations, when applied to the calculation of the underlying U(N) average, reduce to a coupled system identifiable with that obtained by Adler and van Moerbeke using methods of the Toeplitz lattice and Virasoro constraints. Moreover it is shown that this coupled system can be reduced to yield the discrete fifth Painlevé equation \dPV as it occurs in the theory of the sixth Painlevé system. Methods based on affine Weyl group symmetries of Bäcklund transformations have previously yielded the \dPV equation but with different parameters for the same problem. We find the explicit mapping between the two forms. Applications of our results are made to give recurrences for the gap probabilities and moments in the circular unitary ensemble of random matrices, and to the diagonal spin-spin correlation function of the square lattice Ising model.

math-ph↗

arXiv2004

Singularity dominated strong fluctuations for some random matrix averages

The circular and Jacobi ensembles of random matrices have their eigenvalue support on the unit circle of the complex plane and the interval $(0,1)$ of the real line respectively. The averaged value of the modulus of the corresponding characteristic polynomial raised to the power $2 μ$ diverges, for $2μ\le -1$, at points approaching the eigenvalue support. Using the theory of generalized hypergeometric functions based on Jack polynomials, the functional form of the leading asymptotic behaviour is established rigorously. In the circular ensemble case this confirms a conjecture of Berry and Keating.

math-ph↗

arXiv2004

Applications and generalizations of Fisher-Hartwig asymptotics

Fisher-Hartwig asymptotics refers to the large $n$ form of a class of Toeplitz determinants with singular generating functions. This class of Toeplitz determinants occurs in the study of the spin-spin correlations for the two-dimensional Ising model, and the ground state density matrix of the impenetrable Bose gas, amongst other problems in mathematical physics. We give a new application of the original Fisher-Hartwig formula to the asymptotic decay of the Ising correlations above $T_c$, while the study of the Bose gas density matrix leads us to generalize the Fisher-Hartwig formula to the asymptotic form of random matrix averages over the classical groups and the Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our generalizations is that they extend to Hankel determinants the Fisher-Hartwig asymptotic form known for Toeplitz determinants.

math-ph↗

arXiv2003

Discrete Painlevé equations, Orthogonal Polynomials on the Unit Circle and $N$-recurrences for averages over U(N) -- \PVI $τ$-functions

The theory of orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference equations of certain coefficient functions appearing in the theory. A natural formulation of the Riemann-Hilbert problem is presented which has as its solution the above system of orthogonal polynomials and associated functions. In particular for the case of regular semi-classical weights on the unit circle $ w(z) = \prod^m_{j=1}(z-z_j(t))^{ρ_j} $, consisting of $ m \in \mathbb{Z}_{> 0} $ singularities, difference equations with respect to the orthogonal polynomial degree $ n $ (Laguerre-Freud equations) and differential equations with respect to the deformation variables $ z_j(t) $ (Schlesinger equations) are derived completely characterising the system. It is shown in the simplest non-trivial case of $ m=3 $ that quite generally and simply the difference equations are equivalent to the discrete Painlevé equation associated with the degeneration of the rational surface $ D^{(1)}_4 \to D^{(1)}_5 $ and no other. In a three way comparison with other methods employed on this problem - the Toeplitz lattice and Virasoro constraints, the isomonodromic deformation of $ 2\times 2 $ linear Fuchsian differential equations, and the algebraic approach based upon the affine Weyl group symmetry - it is shown all are entirely equivalent, when reduced in order by exact summation, to the above discrete Painlevé equation through explicit transformation formulae. The fundamental matrix integrals over the unitary group $ U(N) $ arising in the theory are given by the generalised hypergeometric function $ {{}^{\vphantom{(1)}}_2}F^{(1)}_1 $.

math-ph↗

arXiv2003

Random matrix averages and the impenetrable Bose gas in Dirichlet and Neumann boundary conditions

The density matrix for the impenetrable Bose gas in Dirichlet and Neumann boundary conditions can be written in terms of $<\prod_{l=1}^n| \cosϕ_1-\cosθ_l| |\cosϕ_2-\cosθ_l|>$, where the average is with respect to the eigenvalue probability density function for random unitary matrices from the classical groups $Sp(n)$ and $O^+(2n)$ respectively. In the large $n$ limit log-gas considerations imply that the average factorizes into the product of averages of the form $<\prod_{l=1}^n|\cosϕ-\cosθ_l>$. By changing variables this average in turn is a special case of the function of $t$ obtained by averaging $\prod_{l=1}^n| t-x_l|^{2q}$ over the Jacobi unitary ensemble from random matrix theory. The latter task is accomplished by a duality formula from the theory of Selberg correlation integrals, and the large $n$ asymptotic form is obtained. The corresponding large $n$ asymptotic form of the density matrix is used, via the exact solution of a particular integral equation, to compute the asymptotic form of the low lying effective single particle states and their occupations, which are proportional to $\sqrt{N}$.

math-ph↗

arXiv2003

Application of the $τ$-function theory of Painlevé equations to random matrices: \PVI, the JUE, CyUE, cJUE and scaled limits

Okamoto has obtained a sequence of $τ$-functions for the \PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be re-expressed as multi-dimensional integrals, and these in turn can be identified with averages over the eigenvalue probability density function for the Jacobi unitary ensemble (JUE), and the Cauchy unitary ensemble (CyUE) (the latter being equivalent to the circular Jacobi unitary ensemble (cJUE)). Hence these averages, which depend on four continuous parameters and the discrete parameter $N$, can be characterised as the solution of the second order second degree equation satisfied by the Hamiltonian in the \PVI theory. Applications are given to the evaluation of the spacing distribution for the circular unitary ensemble (CUE) and its scaled counterpart, giving formulas more succinct than those known previously; to expressions for the hard edge gap probability in the scaled Laguerre orthogonal ensemble (LOE) (parameter $a$ a non-negative integer) and Laguerre symplectic ensemble (LSE) (parameter $a$ an even non-negative integer) as finite dimensional combinatorial integrals over the symplectic and orthogonal groups respectively; to the evaluation of the cumulative distribution function for the last passage time in certain models of directed percolation; to the $τ$-function evaluation of the largest eigenvalue in the finite LOE and LSE with parameter $a=0$; and to the characterisation of the diagonal-diagonal spin-spin correlation in the two-dimensional Ising model.

math-ph↗