arXiv subjects

N. S. Witte

Publications and source records attributed to N. S. Witte.

At least 19 recordsLinked to original sources

arXiv2023

An Infinite Product of the Incomplete Beta Function-type Hypergeometric Function and its Probabilistic Origins

Recently it has been shown that the $α$-Sun density $h(x)$ [{\it J. Math. Anal. Appl.}, {\bf 527} (2023), p. 127371] which interpolates between the Fr{é}chet density and that of the positive, stable distributions whose density is given by a Fox $H$-function, has a Mellin transform involving an infinite product of ratios of Incomplete Beta functions. We develop systematic, but asymptotic, approximations for such products and consequently for the behaviour of the density as $ x\to 0+$ which complement the recent exact form for this by Simon [{\it Electron. Commun. Probab.}, {\bf 28} (2023) p. 1 - 13]. The systematic expansion is an example of a Power Product Expansion, and in our case we derive bounds and estimates which show that this expansion is not convergent and thus only yields an asymptotic expansion.

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arXiv2023

On the Density arising from the Domain of Attraction between Sum and Supremum: the $α$-Sun operator

We explore the analytic properties of the density function $ h(x;γ,α) $, $ x \in (0,\infty) $, $ γ> 0 $, $ 0 < α< 1 $ which arises from the domain of attraction problem for a statistic interpolating between the supremum and sum of random variables. The parameter $ α$ controls the interpolation between these two cases, while $ γ$ parametrises the type of extreme value distribution from which the underlying random variables are drawn from. For $ α= 0 $ the Fréchet density applies, whereas for $ α= 1 $ we identify a particular Fox H-function, which are a natural extension of hypergeometric functions into the realm of fractional calculus. In contrast for intermediate $ α$ an entirely new function appears, which is not one of the extensions to the hypergeometric function considered to date. We derive series, integral and continued fraction representations of this latter function.

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arXiv2022

Gap probabilities for the Bures-Hall Ensemble and the Cauchy-Laguerre Two-Matrix Model

The Bures metric and the associated Bures-Hall measure is arguably the best choice for studying the spectrum of the quantum mechanical density matrix with no apriori knowledge of the system. We investigate the probability of a gap in the spectrum of this model, either at the bottom $ [0,s) $ or at the top $ (s,1] $, utilising the connection of this Pfaffian point-process with the allied problem in the determinantal point-process of the two-dimensional Cauchy-Laguerre bi-orthogonal polynomial system, now deformed with two variables $s,t$. To this end we develop new general results about Cauchy bi-orthogonal polynomial system for a more general class of weights than the Laguerre densities: in particular a new Christoffel-Darboux formula, reproducing kernels and differential equations for the polynomials and their associated functions. This system is most simply expressed as rank-3 matrix variables and possesses an associated cubic bilinear form. Furthermore under specialisation to truncated Laguerre type densities for the weight, of direct relevance to the Cauchy-Laguerre system, we construct a closed system of constrained, nonlinear differential equations in two deformation variables $s,t$, and observe that the recurrence, spectral and deformation derivative structures form a compatible and integrable triplet of Lax equations.

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arXiv2018

Leading corrections to the scaling function on the diagonal for the two-dimensional Ising model

In the neighbourhood of the critical point, the correlation length of the spin-spin correlation function of the two-dimensional Ising model diverges. The correlation function permits a scaling limit in which the separation $N$ between spins goes to infinity, but the scaling variable $s = N(1-t)/2$ remains fixed, where $t$ is the coupling, and $t=1$ the critical point. Previous work has specified these scaling functions (there is one for the critical point being approached from above, and another if approached from below) in terms of transcendents defined by a particular $σ$-form of the degenerate Painlevé V equation. For the diagonal-diagonal correlation, we characterise the first two leading large $N$ correction terms to the scaling functions --- these occur at orders $N^{-1}$ and $N^{-2}$ --- in terms of solutions of a second order linear differential equation with coefficients given in terms of these transcendents, and show how they can be computed. We show that the order $N^{-1}$ is trivial and can be eliminated through appropriate variables so that the leading non-trivial correction is of order $N^{-2}$. In this respect our result gives precise and full characterisation of claims made in the earlier literature.

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arXiv2016

Singular Values of Products of Ginibre Random Matrices

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 functions ${}_0 F_M$, 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 generalised this formalism to general $M \ge 1$, 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 4-th 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 4-th order equation are discussed, as is a possible relationship of the $M=2$ systems to so-called 4-dimensional Painlevé-type equations.

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arXiv2015

The diagonal two-point correlations of the Ising model on the anisotropic triangular lattice and Garnier systems

The diagonal spin-spin correlations $ \langle σ_{0,0}σ_{N,N} \rangle $ of the Ising model on a triangular lattice with general couplings in the three directions are evaluated in terms of a solution to a three-variable extension of the sixth Painlevé system, namely a Garnier system. This identification, which is accomplished using the theory of bi-orthogonal polynomials on the unit circle with regular semi-classical weights, has an additional consequence whereby the correlations are characterised by a simple system of coupled, nonlinear recurrence relations in the spin separation $ N \in \mathbb{Z}_{\geq 0} $. These later recurrence relations are an example of the discrete Garnier equations which, in turn, are extensions to the "discrete Painlevé V" system.

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arXiv2014

Loop Equation Analysis of the Circular $ β$ Ensembles

We construct a hierarchy of loop equations for invariant circular ensembles. These are valid for general classes of potentials and for arbitrary inverse temperatures $ {\rm Re}\,β>0 $ and number of eigenvalues $ N $. Using matching arguments for the resolvent functions of linear statistics $ f(ζ)=(ζ+z)/(ζ-z) $ in a particular asymptotic regime, the global regime, we systematically develop the corresponding large $ N $ expansion and apply this solution scheme to the Dyson circular ensemble. Currently we can compute the second resolvent function to ten orders in this expansion and also its general Fourier coefficient or moment $ m_{k} $ to an equivalent length. The leading large $ N $, large $ k $, $ k/N $ fixed form of the moments can be related to the small wave-number expansion of the structure function in the bulk, scaled Dyson circular ensemble, known from earlier work. From the moment expansion we conjecture some exact partial fraction forms for the low $ k $ moments. For all of the forgoing results we have made a comparison with the exactly soluble cases of $ β= 1,2,4 $, general $ N $ and even, positive $ β$, $ N=2,3 $.

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arXiv2014

Moments of the Gaussian $β$ Ensembles and the large-$N$ expansion of the densities

The loop equation formalism is used to compute the $1/N$ expansion of the resolvent for the Gaussian $β$ ensemble up to and including the term at $O(N^{-6})$. This allows the moments of the eigenvalue density to be computed up to and including the 12-th power and the smoothed density to be expanded up to and including the term at $O(N^{-6})$. The latter contain non-integrable singularities at the endpoints of the support --- we show how to nonetheless make sense of the average of a sufficiently smooth linear statistic. At the special couplings $β= 1$, $2$ and $4$ there are characterisations of both the resolvent and the moments which allows for the corresponding expansions to be extended, in some recursive form at least, to arbitrary order. In this regard we give fifth order linear differential equations for the density and resolvent at $β= 1$ and $4$, which complements the known third order linear differential equations for these quantities at $β= 2$.

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arXiv2014

On a Family of Integrals that extend the Askey-Wilson Integral

We study a family of integrals parameterised by $ N = 2,3,\dots $ generalising the Askey-Wilson integral $ N=2 $ which has arisen in the theory of $q$-analogs of monodromy preserving deformations of linear differential systems and in theory of the Baxter $Q$ operator for the $ XXZ $ open quantum spin chain. These integrals are particular examples of moments defined by weights generalising the Askey-Wilson weight and we show the integrals are characterised by various $ (N-1) $-th order linear $q$-difference equations which we construct. In addition we demonstrate that these integrals can be evaluated as a finite sum of $ (N-1) $ $ BC_{1} $-type Jackson integrals or $ {}_{2N+2}φ_{2N+1} $ basic hypergeometric functions.

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arXiv2012

Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles

The density function for the joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles is found in terms of a Painlevé II transcendent and its associated isomonodromic system. As a corollary, the density function for the spacing between these two eigenvalues is similarly characterized.The particular solution of Painlevé II that arises is a double shifted Bäcklund transformation of the Hasting-McLeod solution, which applies in the case of the distribution of the largest eigenvalue at the soft edge. Our deductions are made by employing the hard-to-soft edge transitions to existing results for the joint distribution of the first and second eigenvalue at the hard edge \cite{FW_2007}. In addition recursions under $a \mapsto a+1$ of quantities specifying the latter are obtained. A Fredholm determinant type characterisation is used to provide accurate numerics for the distribution of the spacing between the two largest eigenvalues.

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arXiv2012

Semi-classical Orthogonal Polynomial Systems on Non-uniform Lattices, Deformations of the Askey Table and Analogs of Isomonodromy

A $\mathbb{D}$-semi-classical weight is one which satisfies a particular linear, first order homogeneous equation in a divided-difference operator $\mathbb{D}$. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first order homogeneous matrix equation in the divided-difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the $\mathbb{D}$-semi-classical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the $\mathbb{D}$-semi-classical class it is entirely natural to define a generalisation of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first non-trivial deformation of the Askey-Wilson orthogonal polynomial system defined by the $q$-quadratic divided-difference operator, the Askey-Wilson operator, and derive the coupled first order divided-difference equations characterising its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to the $ E^{(1)}_7 $ $q$-Painlevé system.

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arXiv2011

On the Variance of the Index for the Gaussian Unitary Ensemble

We derive simple linear, inhomogeneous recurrences for the variance of the index by utilising the fact that the generating function for the distribution of the number of positive eigenvalues of a Gaussian unitary ensemble is a $τ$-function of the fourth Painlevé equation. From this we deduce a simple summation formula, several integral representations and finally an exact hypergeometric function evaluation for the variance.

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arXiv2011

Fredholm Determinant evaluations of the Ising Model diagonal correlations and their λ- generalisation

The diagonal spin-spin correlations of the square lattice Ising model, originally expressed as Toeplitz determinants, are given by two distinct Fredholm determinants - one with an integral operator having an Appell function kernel and another with a summation operator having a Gauss hypergeometric function kernel. Either determinant allows for a Neumann expansion possessing a natural λ- parameter generalisation and we prove that both expansions are in fact equal, implying a continuous and a discrete representation of the form factors. Our proof employs an extension of the classic study by Geronimo and Case, applying scattering theory to orthogonal polynomial systems on the unit circle, to the bi-orthogonal situation.

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arXiv2010

Bi-orthogonal systems on the unit circle, regular semi-classical weights and the discrete Garnier equations

We demonstrate that a system of bi-orthogonal polynomials and their associated functions corresponding to a regular semi-classical weight on the unit circle constitute a class of general classical solutions to the Garnier systems by explicitly constructing its Hamiltonian formulation and showing that it coincides with that of a Garnier system. Such systems can also be characterised by recurrence relations of the discrete Painlevé type, for example in the case with one free deformation variable the system was found to be characterised by a solution to the discrete fifth Painlevé equation. Here we derive the canonical forms of the multi-variable generalisation of the discrete fifth Painlevé equation to the Garnier systems, i.e. for arbitrary numbers of deformation variables.

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arXiv2008

Bi-orthogonal systems on the unit circle, Regular Semi-Classical Weights and Integrable Systems - II

We derive the Christoffel-Geronimus-Uvarov transformations of a system of bi-orthogonal polynomials and associated functions on the unit circle, that is to say the modification of the system corresponding to a rational modification of the weight function. In the specialisation of the weight function to the regular semi-classical case with an arbitrary number of regular singularities $ \{z_1, ..., z_M \} $ the bi-orthogonal system is known to be isomonodromy preserving with respect to deformations of the singular points. If the zeros and poles of the Christoffel-Geronimus-Uvarov factors coincide with the singularities then we have the Schlesinger transformations of this isomonodromic system. Compatibility of the Schlesinger transformations with the other structures of the system - the recurrence relations, the spectral derivatives and deformation derivatives is explicitly deduced. Various forms of Hirota-Miwa equations are derived for the $ τ$-functions or equivalently Toeplitz determinants of the system.

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arXiv2007

Boundary Conditions for Scaled Random Matrix Ensembles in the Bulk of the Spectrum

A spectral average which generalises the local spacing distribution of the eigenvalues of random $ N\times N $ hermitian matrices in the bulk of their spectrum as $ N\to\infty $ is known to be a $τ$-function of the fifth Painlevé system. This $τ$-function, $ τ(s) $, has generic parameters and is transcendental but is characterised by particular boundary conditions about the singular point $s=0$, which we determine here. When the average reduces to the local spacing distribution we find that $τ$-function is of the separatrix, or partially truncated type.

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arXiv2007

Isomonodromic deformation theory and the next-to-diagonal correlations of the anisotropic square lattice Ising model

In 1980 Jimbo and Miwa evaluated the diagonal two-point correlation function of the square lattice Ising model as a $τ$-function of the sixth Painlevé system by constructing an associated isomonodromic system within their theory of holonomic quantum fields. More recently an alternative isomonodromy theory was constructed based on bi-orthogonal polynomials on the unit circle with regular semi-classical weights, for which the diagonal Ising correlations arise as the leading coefficient of the polynomials specialised appropriately. Here we demonstrate that the next-to-diagonal correlations of the anisotropic Ising model are evaluated as one of the elements of this isomonodromic system or essentially as the Cauchy-Hilbert transform of one of the bi-orthogonal polynomials.

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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.

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