arXiv subjects

Robert Buckingham

Publications and source records attributed to Robert Buckingham.

12 recordsLinked to original sources

arXiv2024

Large-Parameter Asymptotics of Generalized Hasting-McLeod Functions

The generalized Hastings-McLeod solutions to the inhomogeneous Painlevé-II equation arise in multi-critical unitary random matrix ensembles, the chiral two-matrix model for rectangular matrices, non-intersecting squared Bessel paths, and non-intersecting Brownian motions on the circle. We establish the leading-order asymptotic behavior of the generalized Hastings-McLeod functions as the inhomogeneous parameter approaches infinity using the Deift-Zhou nonlinear steepest-descent method for Riemann-Hilbert problems. This analysis is done in both the pole-free region and pole region. The asymptotic formulae show excellent agreement with numerically computed solutions in both regions.

math-ph↗

arXiv2021

Far-Field Asymptotics for Multiple-Pole Solitons in the Large-Order Limit

The integrable focusing nonlinear Schrodinger equation admits soliton solutions whose associated spectral data consist of a single pair of conjugate poles of arbitrary order. We study families of such multiple-pole solitons generated by Darboux transformations as the pole order tends to infinity. We show that in an appropriate scaling, there are four regions in the space-time plane where solutions display qualitatively distinct behaviors: an exponential-decay region, an algebraic-decay region, a non-oscillatory region, and an oscillatory region. Using the nonlinear steepest-descent method for analyzing Riemann-Hilbert problems, we compute the leading-order asymptotic behavior in the algebraic-decay, non-oscillatory, and oscillatory regions.

math.AP↗

arXiv2019

A representation of joint moments of CUE characteristic polynomials in terms of Painleve functions

We establish a representation of the joint moments of the characteristic polynomial of a CUE random matrix and its derivative in terms of a solution of the sigma-Painleve V equation. The derivation involves the analysis of a formula for the joint moments in terms of a determinant of generalised Laguerre polynomials using the Riemann-Hilbert method. We use this connection with the sigma-Painleve V equation to derive explicit formulae for the joint moments and to show that in the large-matrix limit the joint moments are related to a solution of the sigma-Painleve III equation. Using the conformal block expansion of the tau-functions associated with the sigma-Painleve V and the sigma-Painleve III equations leads to general conjectures for the joint moments.

math-ph↗

arXiv2018

Large-order asymptotics for multiple-pole solitons of the focusing nonlinear Schrödinger equation

We analyze the large-$n$ behavior of soliton solutions of the integrable focusing nonlinear Schrödinger equation with associated spectral data consisting of a single pair of conjugate poles of order $2n$. Starting from the zero background, we generate multiple-pole solitons by $n$-fold application of Darboux transformations. The resulting functions are encoded in a Riemann-Hilbert problem using the robust inverse-scattering transform method recently introduced by Bilman and Miller. For moderate values of $n$ we solve the Riemann-Hilbert problem exactly. With appropriate scaling, the resulting plots of exact solutions reveal semiclassical-type behavior, including regions with high-frequency modulated waves and quiescent regions. We compute the boundary of the quiescent regions exactly and use the nonlinear steepest-descent method to prove the asymptotic limit of the solitons is zero in these regions. Finally, we study the behavior of the solitons in a scaled neighborhood of the central peak with amplitude proportional to $n$. We prove that in a local scaling the solitons converge to functions satisfying the second member of the Painlevé-III hierarchy in the sense of Sakka. This function is a generalization of a function recently identified by Suleimanov in the context of geometric optics and by Bilman, Ling, and Miller in the context of rogue wave solutions to the focusing nonlinear Schrödinger equation.

nlin.SI↗

arXiv2017

The k-tacnode process

The tacnode process is a universal behavior arising in nonintersecting particle systems and tiling problems. For Dyson Brownian bridges, the tacnode process describes the grazing collision of two packets of walkers. We consider such a Dyson sea on the unit circle with drift. For any integer k, we show that an appropriate double scaling of the drift and return time leads to a generalization of the tacnode process in which k particles are expected to wrap around the circle. We derive winding number probabilities and an expression for the correlation kernel in terms of functions related to the generalized Hastings-McLeod solutions to the inhomogeneous Painleve-II equation. The method of proof is asymptotic analysis of discrete orthogonal polynomials with a complex weight.

math.PR↗

arXiv2017

Nonintersecting Brownian bridges on the unit circle with drift

Nonintersecting Brownian bridges on the unit circle form a determinantal stochastic process exhibiting random matrix statistics for large numbers of walkers. We investigate the effect of adding a drift term to walkers on the circle conditioned to start and end at the same position. For each return time $T<π^2$ we show that if the absolute value of the drift is less than a critical value then the expected total winding number is asymptotically zero. In addition, we compute the asymptotic distribution of total winding numbers in the double-scaling regime in which the expected total winding is finite. The method of proof is Riemann--Hilbert analysis of a certain family of discrete orthogonal polynomials with varying complex exponential weights. This is the first asymptotic analysis of such a class of polynomials. We determine asymptotic formulas and demonstrate the emergence of a second band of zeros by a mechanism not previously seen for discrete orthogonal polynomials with real weights.

math.PR↗

arXiv2017

Large-degree asymptotics of rational Painleve-IV functions associated to generalized Hermite polynomials

The Painleve-IV equation has three families of rational solutions generated by the generalized Hermite polynomials. Each family is indexed by two positive integers m and n. These functions have applications to nonlinear wave equations, random matrices, fluid dynamics, and quantum mechanics. Numerical studies suggest the zeros and poles form a deformed n by m rectangular grid. Properly scaled, the zeros and poles appear to densely fill certain curvilinear rectangles as m and n tend to infinity with r=m/n fixed. Generalizing a method of Bertola and Bothner used to study rational Painleve-II functions, we express the generalized Hermite rational Painleve-IV functions in terms of certain orthogonal polynomials on the unit circle. Using the Deift-Zhou nonlinear steepest-descent method, we asymptotically analyze the associated Riemann-Hilbert problem in the limit as n tends to infinity with m=r*n for r fixed. We obtain an explicit characterization of the boundary curve and determine the leading-order asymptotic expansion of the functions in the pole-free region.

math-ph↗

arXiv2017

Large deformations of the Tracy-Widom distribution I. Non-oscillatory asymptotics

We analyze the left-tail asymptotics of deformed Tracy-Widom distribution functions describing the fluctuations of the largest eigenvalue in invariant random matrix ensembles after removing each soft edge eigenvalue independently with probability $1-γ\in[0,1]$. As $γ$ varies, a transition from Tracy-Widom statistics ($γ=1$) to classical Weibull statistics ($γ=0$) was observed in the physics literature by Bohigas, de Carvalho, and Pato \cite{BohigasCP:2009}. We provide a description of this transition by rigorously computing the leading-order left-tail asymptotics of the thinned GOE, GUE and GSE Tracy-Widom distributions. In this paper, we obtain the asymptotic behavior in the non-oscillatory region with $γ\in[0,1)$ fixed (for the GOE, GUE, and GSE distributions) and $γ\uparrow 1$ at a controlled rate (for the GUE distribution). This is the first step in an ongoing program to completely describe the transition between Tracy-Widom and Weibull statistics. As a corollary to our results, we obtain a new total-integral formula involving the Ablowitz-Segur solution to the second Painlevé equation.

math-ph↗

arXiv2010

Spectra of Random Hermitian Matrices with a Small-Rank External Source: The critical and near-critical regimes

Random Hermitian matrices are used to model complex systems without time-reversal invariance. Adding an external source to the model can have the effect of shifting some of the matrix eigenvalues, which corresponds to shifting some of the energy levels of the physical system. We consider the case when the $n\times n$ external source matrix has two distinct real eigenvalues: $a$ with multiplicity $r$ and zero with multiplicity $n-r$. For a Gaussian potential, it was shown by Péché \cite{Peche:2006} that when $r$ is fixed or grows sufficiently slowly with $n$ (a small-rank source), $r$ eigenvalues are expected to exit the main bulk for $|a|$ large enough. Furthermore, at the critical value of $a$ when the outliers are at the edge of a band, the eigenvalues at the edge are described by the $r$-Airy kernel. We establish the universality of the $r$-Airy kernel for a general class of analytic potentials for $r=\mathcal{O}(n^γ)$ for $0\leqγ<1/12$.

math-ph↗

arXiv2010

Spectra of random Hermitian matrices with a small-rank external source: supercritical and subcritical regimes

Random Hermitian matrices with a source term arise, for instance, in the study of non-intersecting Brownian walkers \cite{Adler:2009a, Daems:2007} and sample covariance matrices \cite{Baik:2005}. We consider the case when the $n\times n$ external source matrix has two distinct real eigenvalues: $a$ with multiplicity $r$ and zero with multiplicity $n-r$. The source is small in the sense that $r$ is finite or $r=\mathcal O(n^γ)$, for $0< γ<1$. For a Gaussian potential, Péché \cite{Peche:2006} showed that for $|a|$ sufficiently small (the subcritical regime) the external source has no leading-order effect on the eigenvalues, while for $|a|$ sufficiently large (the supercritical regime) $r$ eigenvalues exit the bulk of the spectrum and behave as the eigenvalues of $r\times r$ Gaussian unitary ensemble (GUE). We establish the universality of these results for a general class of analytic potentials in the supercritical and subcritical regimes.

math-ph↗

arXiv2008

Total integrals of global solutions to Painleve II

We evaluate the total integral from negative infinity to positive infinity of all global solutions to the Painleve II equation on the real line. The method is based on the interplay between one of the equations of the associated Lax pair and the corresponding Riemann-Hilbert problem. In addition, we evaluate the total integral of a function related to a special solution to the Painleve V equation. As a corollary, we obtain short proofs of the computation of the constant terms of the limiting gap probabilities in the edge and the bulk of the Gaussian Orthogonal and Gaussian Symplectic Ensembles that were obtained recently in [4] and [18]. We also evaluate the total integrals of certain polynomials of the Painleve functions and their derivatives. These polynomials are the densities of the first integrals of the modified Korteweg-de Vries equation. We discuss the relations of the formulae we have obtained to the classical trace formulae for the Dirac operator on the line.

math.CA↗

arXiv2007

Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function

The Tracy-Widom distribution functions involve integrals of a Painlevé II function starting from positive infinity. In this paper, we express the Tracy-Widom distribution functions in terms of integrals starting from minus infinity. There are two consequences of these new representations. The first is the evaluation of the total integral of the Hastings-McLeod solution of the Painlevé II equation. The second is the evaluation of the constant term of the asymptotic expansions of the Tracy-Widom distribution functions as the distribution parameter approaches minus infinity. For the GUE Tracy-Widom distribution function, this gives an alternative proof of the recent work of Deift, Its, and Krasovsky. The constant terms for the GOE and GSE Tracy-Widom distribution functions are new.

math.FA↗