arXiv subjects

N. S. Witte

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

At least 37 records · Page 2Linked to original sources

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↗

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↗

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

Gap Probabilities for Double Intervals in Hermitian Random Matrix Ensembles as $τ$-Functions -- Spectrum Singularity case

The probability for the exclusion of eigenvalues from an interval $(-x,x)$ symmetrical about the origin for a scaled ensemble of Hermitian random matrices, where the Fredholm kernel is a type of Bessel kernel with parameter $ a $ (a generalisation of the sine kernel in the bulk scaling case), is considered. It is shown that this probability is the square of a $τ$-function, in the sense of Okamoto, for the Painlevé system \PIII. This then leads to a factorisation of the probability as the product of two $τ$-functions for the Painlevé system \PIIIdash. A previous study has given a formula of this type but involving \PIIIdash systems with different parameters consequently implying an identity between products of $τ$-functions or equivalently sums of Hamiltonians.

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↗

arXiv2003

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

In this work we show that the $ N\times N $ Toeplitz determinants with the symbols $ z^μ\exp(-{1/2}\sqrt{t}(z+1/z)) $ and $ (1+z)^μ(1+1/z)^ν\exp(tz) $ -- known $τ$-functions for the \PIIIa and \PV systems -- are characterised by nonlinear recurrences for the reflection coefficients of the corresponding orthogonal polynomial system on the unit circle. It is shown that these recurrences are entirely equivalent to the discrete Painlevé equations associated with the degenerations of the rational surfaces $ D^{(1)}_{6} \to E^{(1)}_{7} $ (discrete Painlevé {\rm II}) and $ D^{(1)}_{5} \to E^{(1)}_{6} $ (discrete Painlevé {\rm IV}) respectively through the algebraic methodology based upon of the affine Weyl group symmetry of the Painlevé system, originally due to Okamoto. In addition it is shown that the difference equations derived by methods based upon the Toeplitz lattice and Virasoro constraints, when reduced in order by exact summation, are equivalent to our recurrences. Expressions in terms of generalised hypergeometric functions $ {{}^{\vphantom{(1)}}_0}F^{(1)}_1, {{}^{\vphantom{(1)}}_1}F^{(1)}_1 $ are given for the reflection coefficients respectively.

math-ph↗

arXiv2003

Discrete Painlevé equations and random matrix averages

The $τ$-function theory of Painlevé systems is used to derive recurrences in the rank $n$ of certain random matrix averages over U(n). These recurrences involve auxilary quantities which satisfy discrete Painlevé equations. The random matrix averages include cases which can be interpreted as eigenvalue distributions at the hard edge and in the bulk of matrix ensembles with unitary symmetry. The recurrences are illustrated by computing the value of a sequence of these distributions as $n$ varies, and demonstrating convergence to the value of the appropriate limiting distribution.

math-ph↗

arXiv2003

Painlevé transcendent evaluations of finite system density matrices for 1d impenetrable Bosons

The recent experimental realisation of a one-dimensional Bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable Bose gas. Of primary concern is the ground state occupation of effective single particle states in the finite system, and thus the tendency for Bose-Einstein condensation. This requires the computation of the density matrix. For the impenetrable Bose gas on a circle we evaluate the density matrix in terms of a particular Painlevé VI transcendent in $σ$-form, and furthermore show that the density matrix satisfies a recurrence relation in the number of particles. For the impenetrable Bose gas in a harmonic trap, and with Dirichlet or Neumann boundary conditions, we give a determinant form for the density matrix, a form as an average over the eigenvalues of an ensemble of random matrices, and in special cases an evaluation in terms of a transcendent related to Painlevé V and VI. We discuss how our results can be used to compute the ground state occupations.

math-ph↗

arXiv2002

Finite one dimensional impenetrable Bose systems: Occupation numbers

Bosons in the form of ultra cold alkali atoms can be confined to a one dimensional (1d) domain by the use of harmonic traps. This motivates the study of the ground state occupations $λ_i$ of effective single particle states $ϕ_i$, in the theoretical 1d impenetrable Bose gas. Both the system on a circle and the harmonically trapped system are considered. The $λ_i$ and $ϕ_i$ are the eigenvalues and eigenfunctions respectively of the one body density matrix. We present a detailed numerical and analytic study of this problem. Our main results are the explicit scaled forms of the density matrices, from which it is deduced that for fixed $i$ the occupations $λ_i$ are asymptotically proportional to $\sqrt{N}$ in both the circular and harmonically trapped cases.

cond-mat↗

arXiv2002

New Transformations for Painlevé's Third Transcendent

We present transformations relating the third transcendent of Painlevé with parameter sets located at the corners of the Weyl chamber for the symmetry group of the system, the affine Weyl group of the root system $ B^{(1)}_2 $, to those at the origin. This transformation entails a scaling of the independent variable, and implies additive identities for the canonical Hamiltonians and product identities for the $τ$-functions with these parameter sets.

math.CA↗

arXiv2002

{\bf $τ$-Function Evaluation of Gap Probabilities in Orthogonal and Symplectic Matrix Ensembles}

It has recently been emphasized that all known exact evaluations of gap probabilities for classical unitary matrix ensembles are in fact $τ$-functions for certain Painlevé systems. We show that all exact evaluations of gap probabilities for classical orthogonal matrix ensembles, either known or derivable from the existing literature, are likewise $τ$-functions for certain Painlevé systems. In the case of symplectic matrix ensembles all exact evaluations, either known or derivable from the existing literature, are identified as the mean of two $τ$-functions, both of which correspond to Hamiltonians satisfying the same differential equation, differing only in the boundary condition. Furthermore the product of these two $τ$-functions gives the gap probability in the corresponding unitary symmetry case, while one of those $τ$-functions is the gap probability in the corresponding orthogonal symmetry case.

math-ph↗

arXiv2002

Application of the $τ$-function theory of Painlevé equations to random matrices: \PV, \PIII, the LUE, JUE and CUE

With $<\cdot>$ denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study is $ \tilde{E}_N(I;a,μ) := < \prod_{l=1}^N χ_{(0,\infty)\backslash I}^{(l)} (λ- λ_l)^μ>$ for $I = (0,s)$ and $I = (s,\infty)$, where $χ_I^{(l)} = 1$ for $ λ_l \in I$ and $χ_I^{(l)} = 0$ otherwise. Using Okamoto's development of the theory of the Painlevé V equation, it is shown that $\tilde{E}_N(I;a,μ)$ is a $τ$-function associated with the Hamiltonian therein, and so can be characterised as the solution of a certain second order second degree differential equation, or in terms of the solution of certain difference equations. The cases $μ= 0$ and $μ= 2$ are of particular interest as they correspond to the cumulative distribution and density function respectively for the smallest and largest eigenvalue. In the case $I = (s,\infty)$, $\tilde{E}_N(I;a,μ)$ is simply related to an average in the Jacobi unitary ensemble, and this in turn is simply related to certain averages over the orthogonal group, the unitary symplectic group and the circular unitary ensemble. The latter integrals are of interest for their combinatorial content. Also considered are the hard edge and soft edge scaled limits of $\tilde{E}_N(I;a,μ)$. In particular, in the hard edge scaled limit it is shown that the limiting quantity $E^{\rm hard}((0,s);a,μ)$ can be evaluated as a $τ$-function associated with the Hamiltonian in Okamoto's theory of the Painlevé III equation.

math-ph↗

arXiv2001

Application of the $τ$-Function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE

Tracy and Widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PIV and PII transcendent respectively. We generalise these results to the evaluation of $\tilde{E}_N(λ;a) := \Big < \prod_{l=1}^N χ_{(-\infty, λ]}^{(l)} (λ- λ_l)^a \Big>$, where $ χ_{(-\infty, λ]}^{(l)} = 1$ for $λ_l \in (-\infty, λ]$ and $ χ_{(-\infty, λ]}^{(l)} = 0$ otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of $F_N(λ;a) := \Big < \prod_{l=1}^N (λ- λ_l)^a \Big >$. Of particular interest are $\tilde{E}_N(λ;2)$ and $F_N(λ;2)$, and their scaled limits, which give the distribution of the largest eigenvalue and the density respectively. Our results are obtained by applying the Okamoto $τ$-function theory of PIV and PII, for which we give a self contained presentation based on the recent work of Noumi and Yamada. We point out that the same approach can be used to study the quantities $\tilde{E}_N(λ;a)$ and $F_N(λ;a)$ for the other classical matrix ensembles.

math-ph↗

arXiv2000

Exact Wigner surmise type evaluation of the spacing distribution in the bulk of the scaled random matrix ensembles

Random matrix ensembles with orthogonal and unitary symmetry correspond to the cases of real symmetric and Hermitian random matrices respectively. We show that the probability density function for the corresponding spacings between consecutive eigenvalues can be written exactly in the Wigner surmise type form $a(s) e^{-b(s)}$ for $a$ simply related to a Painlevé transcendent and $b$ its anti-derivative. A formula consisting of the sum of two such terms is given for the symplectic case (Hermitian matrices with real quaternion elements).

math-ph↗

arXiv2000

Gap probabilities in the finite and scaled Cauchy random matrix ensembles

The probabilities for gaps in the eigenvalue spectrum of finite $ N\times N $ random unitary ensembles on the unit circle with a singular weight, and the related hermitian ensembles on the line with Cauchy weight, are found exactly. The finite cases for exclusion from single and double intervals are given in terms of second order second degree ODEs which are related to certain \mbox{Painlevé-VI} transcendents. The scaled cases in the thermodynamic limit are again second degree and second order, this time related to \mbox{Painlevé-V} transcendents. Using transformations relating the second degree ODE and transcendent we prove an identity for the scaled bulk limit which leads to a simple expression for the spacing p.d.f. We also relate all the variables appearing in the Fredholm determinant formalism to particular \mbox{Painlevé} transcendents, in a simple and transparent way, and exhibit their scaling behaviour.

math-ph↗

arXiv2000

Integrability, Random Matrices and Painlevé Transcendents

The probability that an interval $I$ is free of eigenvalues in a matrix ensemble with unitary symmetry is given by a Fredholm determinant. When the weight function in the matrix ensemble is a classical weight function, and the interval $I$ includes an endpoint of the support, Tracy and Widom have given a formalism which gives coupled differential equations for the required probability and some auxilary quantities. We summarize and extend earlier work by expressing the probability and some of the auxilary quantities in terms of Painlevé transcendents.

math-ph↗

arXiv2000

Gap Probabilities for Edge Intervals in Finite Gaussian and Jacobi Unitary Matrix Ensembles

The probabilities for gaps in the eigenvalue spectrum of the finite dimension $ N \times N $ random matrix Hermite and Jacobi unitary ensembles on some single and disconnected double intervals are found. These are cases where a reflection symmetry exists and the probability factors into two other related probabilities, defined on single intervals. Our investigation uses the system of partial differential equations arising from the Fredholm determinant expression for the gap probability and the differential-recurrence equations satisfied by Hermite and Jacobi orthogonal polynomials. In our study we find second and third order nonlinear ordinary differential equations defining the probabilities in the general $N$ case. For N=1 and N=2 the probabilities and thus the solution of the equations are given explicitly. An asymptotic expansion for large gap size is obtained from the equation in the Hermite case, and also studied is the scaling at the edge of the Hermite spectrum as $ N \to \infty $, and the Jacobi to Hermite limit; these last two studies make correspondence to other cases reported here or known previously. Moreover, the differential equation arising in the Hermite ensemble is solved in terms of an explicit rational function of a {Painlevé-V} transcendent and its derivative, and an analogous solution is provided in the two Jacobi cases but this time involving a {Painlevé-VI} transcendent.

math-ph↗