arXiv subjects

Nicholas S. Witte

Publications and source records attributed to Nicholas S. Witte.

13 recordsLinked to original sources

arXiv2026

The $2j-k$ and $j-2k$ Bi-orthogonal Polynomials on the Unit Circle: Further Properties and Riemann-Hilbert Characterizations

In previous work \cite{GW}, we developed a theory of modulated $2j-k$ bi-orthogonal polynomial systems $\{P_n(z;r),Q_n(z;r)\}$ and $j-2k$ bi-orthogonal polynomial systems $\{R_n(z;r),S_n(z;r)\}$, which generalize the classical $j-k$ Toeplitz systems. In the present paper, we further develop this theory in several directions. We derive simplified and unified recurrence relations for both families of polynomials, prove a more transparent Christoffel--Darboux formula, and give Riemann--Hilbert characterizations of the $2j-k$ and $j-2k$ systems.

math.CA↗

arXiv2024

Power spectra of Dyson's circular ensembles

The power spectrum is a Fourier series statistic associated with the covariances of the displacement from average positions of the members of an eigen-sequence. When this eigen-sequence has rotational invariance, as for the eigen-angles of Dyson's circular ensembles, recent work of Riser and Kanzieper has uncovered an exact identity expressing the power spectrum in terms of the generating function for the conditioned gap probability of having $k$ eigenvalues in an interval. These authors moreover showed how for the circular unitary ensemble integrability properties of the generating function, via a particular Painlev\'e VI system, imply a computational scheme for the corresponding power spectrum, and allow for the determination of its large $N$ limit. In the present work, these results are extended to the case of the circular orthogonal ensemble and circular symplectic ensemble, where the integrability is expressed through four particular Painlev\'e VI systems for finite $N$, and two Painlev\'e III$'$ systems for the limit $N\to\infty$, and also via corresponding Fredholm determinants. The relation between the limiting power spectrum $S_\infty(\omega)$, where $\omega$ denotes the Fourier variable, and the limiting generating function for the conditioned gap probabilities is particular direct, involving just a single integration over the gap endpoint in the latter. Interpreting this generating function as the characteristic function of a counting statistic allows for it to be shown that $S_\infty(\omega) \mathop{\sim} \limits_{\omega \to 0} {1 \over \pi \beta | \omega|}$, where $\beta$ is the Dyson index.

math-ph↗

arXiv2023

Integrable Differential Systems for Deformed Laguerre-Hahn Orthogonal Polynomials

Our work studies sequences of orthogonal polynomials $ \{P_{n}(x)\}_{n=0}^{\infty} $ of the Laguerre-Hahn class, whose Stieltjes functions satisfy a Riccati type differential equation with polynomial coefficients, are subject to a deformation parameter $t$. We derive systems of differential equations and give Lax pairs, yielding non-linear differential equations in $t$ for the recurrence relation coefficients and Lax matrices of the orthogonal polynomials. A specialisation to a non semi-classical case obtained via a Möbius transformation of a Stieltjes function related to a modified Jacobi weight is studied in detail, showing this system is governed by a differential equation of the Painlevé type P$_\textrm{VI}$. The particular case of P$_\textrm{VI}$ arising here has the same four parameters as the solution found by Magnus [A.P. Magnus, Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials, J. Comput. Appl. Math., 57:215-237, 1995] but differs in the boundary conditions.

math-ph↗

arXiv2021

Modulated Bi-orthogonal Polynomials on the Unit Circle: The $2j-k$ and $j-2k$ Systems

We construct the systems of bi-orthogonal polynomials on the unit circle where the Toeplitz structure of the moment determinants is replaced by $ \det(w_{2j-k})_{0\leq j,k \leq N-1} $ and the corresponding Vandermonde modulus squared is replaced by $ \prod_{1 \le j < k \le N}(ζ^{2}_k - ζ^{2}_j)(ζ^{-1}_k - ζ^{-1}_j) $. This is the simplest case of a general system of $pj-qk$ with $p,q$ co-prime integers. We derive analogues of the structures well known in the Toeplitz case: third order recurrence relations, determinantal and multiple-integral representations, their reproducing kernel and Christoffel-Darboux sum, and associated (Carath{é}odory) functions. We close by giving full explicit details for the system defined by the simple weight $ w(ζ)=e^ζ$, which is a specialisation of a weight arising from averages of moments of derivatives of characteristic polynomials over $USp(2N)$, $SO(2N)$ and $O^-(2N)$.

math.CA↗

arXiv2017

Large $N$ expansions for the Laguerre and Jacobi $β$ ensembles from the loop equations

The $β$-ensembles of random matrix theory with classical weights have many special properties. One is that the loop equations specifying the resolvent and corresponding multipoint correlators permit a derivation at general order of the correlator via Aomoto's method from the theory of the Selberg integral. We use Aomoto's method to derive the full hierarchy of loop equations for Laguerre and Jacobi $β$ ensembles, and use these to systematically construct the explicit form of the $1/N$ expansion at low orders. This allows us to give the explicit form of corrections to the global density, and allows various moments to be computed, complementing results available in the literature motivated by problems in quantum transport.

math-ph↗

arXiv2012

Construction of a Lax Pair for the $E_6^{(1)}$ $q$-Painlevé System

We construct a Lax pair for the $E^{(1)}_6 $ $q$-Painlevé system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices [arXiv:1204.2328]. Our study treats one special case of such lattices - the $q$-linear lattice - through a natural generalisation of the big $q$-Jacobi weight. As a by-product of our construction we derive the coupled first-order $q$-difference equations for the $E^{(1)}_6 $ $q$-Painlevé system, thus verifying our identification. Finally we establish the correspondences of our result with the Lax pairs given earlier and separately by Sakai and Yamada, through explicit transformations.

math.CA↗

arXiv2012

Painleve II in random matrix theory and related fields

We review some occurrences of Painlevé 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 $α$ in the Painlevé equation is zero, and the boundary condition is that of the Hasting-MacLeod solution. As well as expressions involving the Painlevé transcendent itself, one encounters the sigma form of the Painlevé II equation, and Lax pair equations in which the Painlevé transcendent occurs as coefficients. We then consider settings which give rise to general $α$ Painlevé II transcendents. In a particular random matrix setting, new results for the corresponding boundary conditions in the cases $α= \pm 1/2$, 1 and 2 are presented.

math-ph↗

arXiv2012

Asymptotic forms for hard and soft edge general $β$ conditional gap probabilities

An 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 $β$-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 $β$ to $4/β$.

math-ph↗

arXiv2010

Connection preserving deformations and $q$-semi-classical orthogonal polynomials

We present a framework for the study of $q$-difference equations satisfied by $q$-semi-classical orthogonal systems. As an example, we identify the $q$-difference equation satisfied by a deformed version of the little $q$-Jacobi polynomials as a gauge transformation of a special case of the associated linear problem for $q$-$\mathrm{P}_{\mathrm{VI}}$. We obtain a parameterization of the associated linear problem in terms of orthogonal polynomial variables and find the relation between this parameterization and that of Jimbo and Sakai.

nlin.SI↗

arXiv2009

Physical Combinatorics and Quasiparticles

We consider the physical combinatorics of critical lattice models and their associated conformal field theories arising in the continuum scaling limit. As examples, we consider A-type unitary minimal models and the level-1 sl(2) Wess-Zumino-Witten (WZW) model. The Hamiltonian of the WZW model is the $U_q(sl(2))$ invariant XXX spin chain. For simplicity, we consider these theories only in their vacuum sectors on the strip. Combinatorially, fermionic particles are introduced as certain features of RSOS paths. They are composites of dual-particles and exhibit the properties of quasiparticles. The particles and dual-particles are identified, through an energy preserving bijection, with patterns of zeros of the eigenvalues of the fused transfer matrices in their analyticity strips. The associated (m,n) systems arise as geometric packing constraints on the particles. The analyticity encoded in the patterns of zeros is the key to the analytic calculation of the excitation energies through the Thermodynamic Bethe Ansatz (TBA). As a by-product of our study, in the case of the WZW or XXX model, we find a relation between the location of the Bethe root strings and the location of the transfer matrix 2-strings.

hep-th↗

arXiv2007

The Distribution of the first Eigenvalue Spacing at the Hard Edge of the Laguerre Unitary Ensemble

The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite rank random matrices is found in terms of a Painlevé V system, and the solution of its associated linear isomonodromic system. In particular it is characterised by the polynomial solutions to the isomonodromic equations which are also orthogonal with respect to a deformation of the Laguerre weight. In the scaling to the hard edge regime we find an analogous situation where a certain Painlevé \IIId system and its associated linear isomonodromic system characterise the scaled distribution. We undertake extensive analytical studies of this system and use this knowledge to accurately compute the distribution and its moments for various values of the parameter $ a $. In particular choosing $ a=\pm 1/2 $ allows the first eigenvalue spacing distribution for random real orthogonal matrices to be computed.

math.CA↗

arXiv2006

Hypergeometric Solutions to the q-Painlevé Equation of Type $(A_1+A_1')^{(1)}$

A class of classical solutions to the $q$-Painlevé equation of type $(A_1+A_1')^{(1)}$ (a $q$-difference analog of the Painlevé II equation) is constructed in a determinantal form with basic hypergeometric function elements. The continuous limit of this $q$-Painlevé equation to the Painlevé II equation and its hypergeometric solutions are discussed. The continuous limit of these hypergeometric solutions to the Airy function is obtained through a uniform asymptotic expansion of their integral representation.

nlin.SI↗

arXiv2001

Discriminants and Functional Equations for Polynomials Orthogonal on the Unit Circle

We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and $q$-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients. We also evaluate the discriminants and quantized discriminants of polynomials orthogonal on the unit circle.

math.CA↗