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Peter J. Forrester

Publications and source records attributed to Peter J. Forrester.

At least 19 recordsLinked to original sources
arXiv2026

Large $N$ expansions of the partition function for Coulomb systems on the surface of a cylinder

The two-dimensional one-component plasma forms a droplet in the case that the one-body potential is proportional to the number of charges. Our interest is in studying the large $N$ form of the corresponding partition function, with the plasma confined to the surface of the cylinder. The one-body potential is (up to technical restrictions) allowed to be arbitrary in the direction of the axis of the cylinder, whereas the coupling is restricted to the exactly solvable value $β= 2$. It is demonstrated that a change of variables can be made to an annular droplet plasma model, up to a certain Jacobian factor. The latter can be interpreted as the characteristic function of a certain linear statistic, which generalises the original aim of our study to also analysing the large $N$ form of the characteristic function for the annular droplet plasma model. While this is well known in the case of soft wall boundary conditions, our analysis (based on Laplace's method applied to certain integrals, and the Euler-Maclaurin summation formula) covers the case of one or two hard walls at the boundary, or inside, of the droplet for which the soft edge formulas are shown to no longer hold. Use of this to study the large $N$ expansion of the cylinder partition function requires knowledge of the same expansion in the annular case. This is not available in the case of two hard walls inside the droplet, which we treat instead via a direct analysis. In distinction to the case of the annular partition function, the large $N$ expansion of the logarithm of the cylinder partition function is found to always have no term proportional to $\log N$, independent of the presence of hard walls.

math-ph↗
arXiv2026

Difference equations of average entropies

Exact cumulants of entanglement entropies of random state ensembles have traditionally been studied within the random matrix framework. In this work, we propose an alternative approach based on the intrinsic connection to integrable systems. The central idea is to embed entropic quantities into tau functions satisfying Toda-type lattice equations, which in turn yield linear difference equations for their averages. Directly solving the difference equations recovers exact entropy formulas in the literature. The integrable systems approach bypasses the case-by-case, ensemble-dependent derivations required by random matrix methods. The approach also suggests a possible route towards unified and more efficient higher-order cumulant calculations by exploring integrable hierarchies.

math-ph↗
arXiv2026

$q$-Selberg extensions of Gasper's two stage $q$-beta integrals

Gasper gave a two-step extension of the Askey-Roy formula for a beta-integral type defined on the unit circle. This involved increasing the number of parameters and adding a balancing condition. We derive a multidimensional generalization of the final formula in Gasper's extension. By considering a two-step degeneration involving parameters our generalization becomes Tarasov-Varchenko's formula -- itself a multidimensional generalization of the original Askey-Roy formula. The proof is done by Aomoto's method. This generalized integration by parts strategy makes use of a class of functions known as the fundamental invariants of type $BC$.

math.CV↗
arXiv2026

Gegenbauer polynomials and fluctuation properties of the one-dimensional Riesz gas

The Riesz gas in one-dimension consists of particles interacting via a pair potential, ${\rm sgn}(s) |x - x'|^{-s}$, $s \ne 0$ and $-\log | x - x'|$ for $s=0$. In the infinite density limit, with the particle support the interval $[-1,1]$, we apply a functional derivative method due to Beenakker to compute the covariance of two smooth linear statistics for the Riesz gas with exponent $s \in (-1,1)$, $s \ne 0$. This we give in terms of a sum over Fourier components of the linear statistics with respect to a Gegenbauer polynomial $\{C_n^{(s/2)}(x) \}$ basis, which generalises a known form in the case $s=0$ involving a cosine expansion. For the power sum linear statistic, our general formula can be reduced to a product of gamma function form, and compared against recent exact results in the literature for this case.

math-ph↗
arXiv2026

Edge density expansions for the classical Gaussian and Laguerre ensembles

Recent work of Bornemann has uncovered hitherto hidden integrable structures relating to the asymptotic expansion of quantities at the soft edge of Gaussian and Laguerre random matrix ensembles. These quantities are spacing distributions and the eigenvalue density, and the findings cover the cases of the three symmetry classes orthogonal, unitary and symplectic. In this work we give a different viewpoint on these results in the case of the soft edge scaled density, and in the Laguerre case we initiate an analogous study at the hard edge. Our tool is the scalar differential equation satisfied by the latter, known from earlier work. Unlike integral representations, these differential equations in soft edge scaling variables isolate the function of $N$ which is the expansion variable. Moreover, they give information on the correction terms which supplements the findings from the work of Bornemann. In the case of the Gaussian ensemble, we can demonstrate analogous features for Dyson index $β= 6$, which suggests a broader class of models, namely the classical $β$ ensembles, with asymptotic expansions exhibiting integrable features. For the Laguerre ensembles at the hard edge, we give the explicit form of the correction at second order for unitary symmetry, and at first order in the orthogonal and symplectic cases. Various differential relations are demonstrated.

math-ph↗
arXiv2026

Electrostatic computations for statistical mechanics and random matrix applications

Although for the most part classical, the topic of electrostatics finds to this day new applications. In this review we highlight several theoretical results on electrostatics, chosen to both illustrate general principles, and for their application in statistical mechanics and random matrix settings. The theoretical results include electrostatic potentials and energies associated with balls and hyperellipsoids in general dimension, equilibrium measures associated with surfaces of the latter, the use of conformal mappings in two-dimensions, and the balayage measure. A number of explicit examples of their use in predicting the leading asymptotic form of certain configuration integrals and particle density in particular statistical mechanical systems are given, as well as with regards to questions relating to fluctuation formulas and (conditioned) gap probabilities.

math-ph↗
arXiv2026

Higher-Order Linear Differential Equations for Unitary Matrix Integrals: Applications and Generalisations

In this paper, we consider characterisations of the class of unitary matrix integrals $\big\langle (\det U)^q {\rm e}^{s^{1/2} \operatorname{Tr}(U + U^\dagger)} \big\rangle_{U(l)}$ in terms of a first-order matrix linear differential equation for a vector function of size $l+1$, and in terms of a scalar linear differential equation of degree ${l+1}$. It will be shown that the latter follows from the former. The matrix linear differential equation provides an efficient way to compute the power series expansion of the matrix integrals, which with $q=0$ and $q=l$ are of relevance to the enumeration of longest increasing subsequences for random permutations, and to the question of the moments of the first and second derivative of the Riemann zeta function on the critical line, respectively. This procedure is compared against that following from known characterisations involving the $σ$-Painlev&é III$'$ second-order nonlinear differential equation. We show too that the natural $β$ generalisation of the unitary group integral permits characterisation by the same classes of linear differential equations.

math-ph↗
arXiv2026

Equilibrium measures for higher dimensional rotationally symmetric Riesz gases

We study equilibrium measures for Riesz gases in dimension $d$ with pairwise interaction kernel $|x-y|^{-s}$, subject to radially symmetric external fields. We characterise broad classes of confining potentials for which the equilibrium measure is supported on the unit ball and admits an explicit density. Our main contribution is a converse construction: starting from a prescribed radially symmetric equilibrium density given as a power series in the squared radius, we determine the associated external potential and establish the corresponding Euler-Lagrange variational conditions. A key ingredient in the proof is an identity between two ${}_3F_2$ hypergeometric functions evaluated at unit argument, which is of independent interest. As applications, we identify the external potentials corresponding to equilibrium densities proportional to $(1-|x|^2)^α$, $α>-1$, and show that these potentials can be expressed in terms of Gauss hypergeometric functions ${}_2F_1$, reducing to polynomials for special values of $α$. We also determine the equilibrium measure associated with purely power-type external potentials, often referred to as Freud or Mittag--Leffler potentials in the context of log gases, for which the equilibrium density admits an explicit ${}_2F_1$ representation. Furthermore, we apply our framework to a Coulomb gas in dimension $d+1$ confined by a harmonic potential to the half-space. We derive a necessary condition under which the equilibrium measure is fully supported on the boundary hyperplane of dimension $d$, with the induced density corresponding to that of a Riesz gas with exponent $s=d-1$.

math-ph↗
arXiv2026

A note on "Higher order linear differential equations for unitary matrix integrals: applications and generalisations"

In this note, we briefly introduce the background and motivation of the collaborative work [arXiv:2508.20797], and provide an outline of the main results. The latter relates to matrix and higher order scalar differential equations satisfied by certain Hankel and Toeplitz determinants involving I-Bessel functions, or equivalently certain unitary matrix integrals, and moreover puts this property in a broader context. We also investigate large gaps between zeros of the derivatives of the Hardy $\mathsf{Z}$-function, assuming the validity of a certain joint moments conjecture in random matrix theory.

math-ph↗
arXiv2026

Integrability enabled computations relating to the fixed trace Laguerre ensemble

Studies of density matrices for random quantum states lead naturally to the fixed trace Laguerre ensemble in random matrix theory. Previous studies have uncovered explicit rational function formulas for moments of purity statistic (trace of the squared density matrix), and also a third order linear differential equation satisfied by the eigenvalue density. We further probe the origin of these results from the viewpoint of integrability, which is taken here to mean wider classes of recursions and differential equations, and give extensions. Prominent in our study are first order linear matrix differential equations. One application given is to the derivation of the third order scalar equation for the density. Another is to obtain the explicit rational function formula for the variance of the purity statistic in the $β$ generalised fixed trace Laguerre ensemble. In the original case ($β= 2$), the purity cumulants are expressed in terms of the large argument expansion of a particular $σ$-Painlevé IV transcendent. In a different but related direction, the exact computation of the two-point correlation for the fixed determinant circular unitary ensemble SU$(N)$ is given the Appendix.

math-ph↗
arXiv2025

Finite size corrections in the bulk for circular $β$ ensembles

The circular $β$ ensemble for $β=1,2$ and 4 corresponds to circular orthogonal, unitary and symplectic ensemble respectively as introduced by Dyson. The statistical state of the eigenvalues is then a determinantal point process ($β= 2$) and Pfaffian point process ($β= 1,4$). The explicit functional forms of the correlation kernels then imply that the general $n$-point correlation functions exhibit an asymptotic expansion in $1/N^2$, which moreover can be lifted to an asymptotic in $1/N^2$ for the spacing distributions and their generating function. We use $σ$-Painlevé characterisations to show that the functional form of the first correction is related to the leading term via a second derivative. In the case $β= 2$ this finding has immediate consequence in interpreting the empirical Riemann zeros spacing distribution at large height, and that of their thinning. Explicit functional forms are used to show that the spectral form factors for $β=1,2$ and 4 also admit an asymptotic expansion in $1/N^2$. Differential relations are identified expressing the first and second correction in terms of the limiting functional form, and evidence is presented that they hold for general $β$. For even $β$ it is proved that the two-point correlation function permits an asymptotic expansion in $1/N^2$, and moreover that the leading correction relates to the limiting functional form via a second derivative.

math-ph↗
arXiv2025

Asymptotics of the real eigenvalue distribution for the real spherical ensemble

The real Ginibre spherical ensemble consists of random matrices of the form $A B^{-1}$, where $A,B$ are independent standard real Gaussian $N \times N$ matrices. The expected number of real eigenvalues is known to be of order $\sqrt{N}$. We consider the probability $p_{N.M}^{\rm r}$ that there are $M$ real eigenvalues in various regimes. These are when $M$ is proportional to $N$ (large deviations), when $N$ is proportional to $\sqrt{N}$ (intermediate deviations), and when $M$ is in the neighbourhood of the mean (local central limit theorem). This is done using a Coulomb gas formalism in the large deviations case, and by determining the leading asymptotic form of the generating function for the probabilities in the case of intermediate deviations (the local central limit regime was known from earlier work). Moreover a matching of the left tail asymptotics of the intermediate deviation regime with that of the right tail of the large deviation regime is exhibited, as is a matching of the right tail intermediate deviation regime with the leading order form of the probabilities in the local central limit regime. We also give the leading asymptotic form of $p_{N,0}^{\rm r}$, i.e. the probability of no real eigenvalues.

math-ph↗
arXiv2025

Moments of characteristic polynomials for classical $β$ ensembles

For random matrix ensembles with unitary symmetry, there is interest in the large $N$ form of the moments of the absolute value of the characteristic polynomial for their relevance to the Riemann zeta function on the critical line, and to Fisher-Hartwig asymptotics in the theory of Toeplitz determinants. The constant (with respect to $N$) in this asymptotic expansion, involving the Barnes $G$ function, is most relevant to the first of these, while the algebraic term (in $N$) and the functional dependence on the power are of primary interest in the latter. Desrosiers and Liu [20] have obtained the analogous expansions for the classical Gaussian, Laguerre and Jacobi $β$ ensembles in the case of even moments. We give simplified working of these results -- which requires the use of duality formulas and the use of steepest descents for multidimensional integrals -- providing too an error bound on the resulting asymptotic expressions. The universality of the constant term with respect to an earlier result known for the circular $β$ ensemble is established, which requires writing it in a Barnes $G$ function form, while the functional dependence on the powers is related to that appearing in Gaussian fluctuation formulas for linear statistics. In the Laguerre and Jacobi cases our working can be extended to the circumstance when the exponents in the weight function are (strictly) proportional to $N$, giving results not previously available in the literature.

math-ph↗
arXiv2025

Partition function of 2D Coulomb gases with radially symmetric potentials and a hard wall

The large $N$ asymptotic expansion of the partition function for the normal matrix model is predicted to have special features inherited from its interpretation as a two-dimensional Coulomb gas. However for the latter, it is most natural to include a hard wall at the boundary of the droplet. We probe how this affects the asymptotic expansion in the solvable case that the potential is radially symmetric and the droplet is a disk or an annulus. We allow too for the hard wall to be strictly inside the boundary of the droplet. It is observed the term of order $\log N$, has then a different rational number prefactor to that when the hard wall is at the droplet boundary. Also found are certain universal (potential independent) numerical constants given by definite integrals, both at order $\sqrt{N}$, and in the constant term.

math.PR↗
arXiv2025

Orthogonal polynomials in the spherical ensemble with two insertions

We consider asymptotics of planar orthogonal polynomials $P_{n,N}$ (where $\mathrm{deg}P_{n,N}=n$) with respect to the weight $$\frac{|z-w|^{2NQ_1}}{(1+|z|^2)^{N(1+Q_0+Q_1)+1}}, \quad(Q_0,Q_1 > 0)$$ in the whole complex plane. With $n, N\rightarrow\infty$ and $N-n$ fixed, we obtain the strong asymptotics of the polynomials, asymptotics for the weighted $L^2$ norm and the limiting zero counting measure. These results apply to the pre-critical phase of the underlying two-dimensional Coulomb gas system, when the support of the equilibrium measure is simply connected. Our method relies on specifying the mother body of the two-dimensional potential problem. It relies too on the fact that the planar orthogonality can be rewritten as a non-Hermitian contour orthogonality. This allows us to perform the Deift-Zhou steepest descent analysis of the associated $2\times 2$ Riemann-Hilbert problem.

math.CA↗
arXiv2025

On the superintegrability of the Gaussian $β$ ensemble and its $(q,t)$ generalisation

In the present context, superintegrability is a property of certain probability density functions coming from matrix models, which relates to the average over a distinguished basis of symmetric functions, typically the Jack or Macdonald polynomials. It states that the average can be computed according a certain combination of those same polynomials, now specialised by specific substitutions when expressed in terms of the power sum basis. For a particular $(q,t)$ generalisation of the Gaussian $β$ ensemble from random matrix theory, known independently from the consideration of certain integrable gauge theories, we use results developed in a theory of multivariable Al-Salam and Carlitz polynomials based on Macdonald polynomials to prove the superintegrability identity. This then is used to deduce a duality formula for these same averages, which in turn allows for a derivation of a functional equation for the spectral moments.

math-ph↗
arXiv2025

In memoriam: aspects of Santosh Kumar's work on exact results in RMT

Santosh Kumar was an active researcher on the topic of exact results in random matrix theory and their various applications, particularly to quantum chaos and information theory. Barely entering his mid-career, he died unexpectedly on the 18th October 2024. The present article gives an account of some of his research directions and findings. As well as serving as a tribute to his work, this is done also for the purpose of providing a resource for those who may continue along related lines in the future.

math-ph↗