Nicholas S. Witte
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The organizations/institutes from the scholar's latest article
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Nicholas S. Witte has a total of 4 co-authors. Here are the top co-authors by number of shared publications:
- Peter J. Forrester
School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
7 shared publications Christopher M. Cosgrove
School of Mathematics and Statistics, University of Sydney, Sydney NSW 2006, Australia
2 shared publications- Masahiko Ito
School of Science and Technology for Future Life Tokyo Denki University 120-8551 Tokyo Japan
1 shared publication P. J. Forrester
Department of Mathematics and Statistics University of Melbourne Parkville 3052 Victoria AUSTRALIA
1 shared publication
Latest Publications
- 1 January 2016
- Published by IOP Publishing in Nonlinearity
- Vol. 29 (1) , 131-160
The diagonal two-point correlations of the Ising model on the anisotropic triangular lattice and Garnier systems
The diagonal spin–spin correlations 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 . The later recurrence relations are an example of discrete Garnier equations which, in turn, are extensions to a 'discrete Painlevé V' system.
- 1 January 2016
- Published by IOP Publishing in Nonlinearity
- Vol. 29 (1) , 131-160
The diagonal two-point correlations of the Ising model on the anisotropic triangular lattice and Garnier systems
The diagonal spin–spin correlations 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 . The later recurrence relations are an example of discrete Garnier equations which, in turn, are extensions to a 'discrete Painlevé V' system.
- Open Access
The diagonal two-point correlations of the Ising model on the anisotropic triangular lattice and Garnier systems
The diagonal spin-spin correlations $ \langle \sigma_{0,0}\sigma_{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 Painlev\'e 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 Painlev\'e V" system.Open AccessThe diagonal two-point correlations of the Ising model on the anisotropic triangular lattice and Garnier systems
The diagonal spin-spin correlations $ \langle \sigma_{0,0}\sigma_{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 Painlev\'e 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 Painlev\'e V" system. - Open Access
- 26 February 2015
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2015 (2) , 1-42
Loop equation analysis of the circular β ensembles
- 26 February 2015
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2015 (2) , 1-42
Open AccessLoop equation analysis of the circular β ensembles
- Open AccessLoop Equation Analysis of the Circular $ β$ EnsemblesWe 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}\,\beta>0 $ and number of eigenvalues $ N $. Using matching arguments for the resolvent functions of linear statistics $ f(\zeta)=(\zeta+z)/(\zeta-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 $ \beta = 1,2,4 $, general $ N $ and even, positive $ \beta $, $ N=2,3 $.Open AccessLoop Equation Analysis of the Circular $ β$ EnsemblesWe 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}\,\beta>0 $ and number of eigenvalues $ N $. Using matching arguments for the resolvent functions of linear statistics $ f(\zeta)=(\zeta+z)/(\zeta-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 $ \beta = 1,2,4 $, general $ N $ and even, positive $ \beta $, $ N=2,3 $.
- 27 June 2014
- Published by Springer Nature in Constructive Approximation
- Vol. 41 (3) , 589-613
Painlevé II in Random Matrix Theory and Related Fields
- 27 June 2014
- Published by Springer Nature in Constructive Approximation
- Vol. 41 (3) , 589-613
Painlevé II in Random Matrix Theory and Related Fields
- Open Access
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}\varphi_{2N+1} $ basic hypergeometric functions.Open AccessOn 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}\varphi_{2N+1} $ basic hypergeometric functions. - Open Access
Painleve II in random matrix theory and related fields
We review some occurrences of Painlev\'e II transcendents in the study of two-dimensional Yang-Mills theory, fluctuation formulas for growth models, and as distribution functions within random matrix theory. We first discuss settings in which the parameter $\alpha$ in the Painlev\'e equation is zero, and the boundary condition is that of the Hasting-MacLeod solution. As well as expressions involving the Painlev\'e transcendent itself, one encounters the sigma form of the Painlev\'e II equation, and Lax pair equations in which the Painlev\'e transcendent occurs as coefficients. We then consider settings which give rise to general $\alpha$ Painlev\'e II transcendents. In a particular random matrix setting, new results for the corresponding boundary conditions in the cases $\alpha = \pm 1/2$, 1 and 2 are presented.Open AccessPainleve II in random matrix theory and related fields
We review some occurrences of Painlev\'e II transcendents in the study of two-dimensional Yang-Mills theory, fluctuation formulas for growth models, and as distribution functions within random matrix theory. We first discuss settings in which the parameter $\alpha$ in the Painlev\'e equation is zero, and the boundary condition is that of the Hasting-MacLeod solution. As well as expressions involving the Painlev\'e transcendent itself, one encounters the sigma form of the Painlev\'e II equation, and Lax pair equations in which the Painlev\'e transcendent occurs as coefficients. We then consider settings which give rise to general $\alpha$ Painlev\'e II transcendents. In a particular random matrix setting, new results for the corresponding boundary conditions in the cases $\alpha = \pm 1/2$, 1 and 2 are presented. - Open AccessAsymptotic forms for hard and soft edge general $β$ conditional gap probabilitiesAn infinite log-gas formalism, due to Dyson, and independently Fogler and Shklovskii, is applied to the computation of conditioned gap probabilities at the hard and soft edges of random matrix $\beta$-ensembles. The conditioning is that there are $n$ eigenvalues in the gap, with $n \ll |t|$, $t$ denoting the end point of the gap. It is found that the entropy term in the formalism must be replaced by a term involving the potential drop to obtain results consistent with known asymptotic expansions in the case $n=0$. With this modification made for general $n$, the derived expansions - which are for the logarithm of the gap probabilities - are conjectured to be correct up to and including terms O$(\log|t|)$. They are shown to satisfy various consistency conditions, including an asymptotic duality formula relating $\beta$ to $4/\beta$.Open AccessAsymptotic forms for hard and soft edge general $β$ conditional gap probabilitiesAn infinite log-gas formalism, due to Dyson, and independently Fogler and Shklovskii, is applied to the computation of conditioned gap probabilities at the hard and soft edges of random matrix $\beta$-ensembles. The conditioning is that there are $n$ eigenvalues in the gap, with $n \ll |t|$, $t$ denoting the end point of the gap. It is found that the entropy term in the formalism must be replaced by a term involving the potential drop to obtain results consistent with known asymptotic expansions in the case $n=0$. With this modification made for general $n$, the derived expansions - which are for the logarithm of the gap probabilities - are conjectured to be correct up to and including terms O$(\log|t|)$. They are shown to satisfy various consistency conditions, including an asymptotic duality formula relating $\beta$ to $4/\beta$.
- 1 July 2002
- Published by Cambridge University Press (CUP) in The ANZIAM Journal
- Vol. 44 (1) , 41-50
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 auxiliary quantities. We summarize and extend earlier work by expressing the probability and some of the auxiliary quantities in terms of Painlevé transcendents.
- 1 July 2002
- Published by Cambridge University Press (CUP) in The ANZIAM Journal
- Vol. 44 (1) , 41-50
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 auxiliary quantities. We summarize and extend earlier work by expressing the probability and some of the auxiliary quantities in terms of Painlevé transcendents.
- 27 March 2002
- Published by Wiley in Communications on Pure and Applied Mathematics
- Vol. 55 (6) , 679-727
Application of the τ‐function theory of Painlevé equations to random matrices: PV, PIII, the LUE, JUE, and CUE
With 〈·〉 denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study is for I = (0, s) and I = (s, ∞), where χ = 1 for λl ∈ I and χ = 0 otherwise. Using Okamoto's development of the theory of the Painlevé V equation, it is shown that ẼN(I; a, μ) is a τ‐function associated with the Hamiltonian therein, and so can be characterized 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, because they correspond to the cumulative distribution and density function, respectively, for the smallest and largest eigenvalue. In the case I = (s, ∞), ẼN(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 ẼN(I; a, μ). In particular, in the hard‐edge scaled limit it is shown that the limiting quantity Ẽhard((O, s); a, μ) can be evaluated as a τ‐function associated with the Hamiltonian in Okamoto's theory of the Painlevé III equation.
- 27 March 2002
- Published by Wiley in Communications on Pure and Applied Mathematics
- Vol. 55 (6) , 679-727
Application of the τ‐function theory of Painlevé equations to random matrices: PV, PIII, the LUE, JUE, and CUE
With 〈·〉 denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study is for I = (0, s) and I = (s, ∞), where χ = 1 for λl ∈ I and χ = 0 otherwise. Using Okamoto's development of the theory of the Painlevé V equation, it is shown that ẼN(I; a, μ) is a τ‐function associated with the Hamiltonian therein, and so can be characterized 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, because they correspond to the cumulative distribution and density function, respectively, for the smallest and largest eigenvalue. In the case I = (s, ∞), ẼN(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 ẼN(I; a, μ). In particular, in the hard‐edge scaled limit it is shown that the limiting quantity Ẽhard((O, s); a, μ) can be evaluated as a τ‐function associated with the Hamiltonian in Okamoto's theory of the Painlevé III equation.