arXiv subjects

Folkmar Bornemann

Publications and source records attributed to Folkmar Bornemann.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotic Expansions of Gaussian and Laguerre Ensembles at the Soft Edge III: Generating Functions

We conclude our work [arXiv:2403.07628, arXiv:2503.12644] on asymptotic expansions at the soft edge for the classical $n$-dimensional Gaussian and Laguerre ensembles, now studying the gap-probability generating functions. We show that the correction terms in the asymptotic expansion are multilinear forms of the higher-order derivatives of the leading-order term, with certain rational polynomial coefficients that are independent of the dummy generating function variable. In this way, the same multilinear structure, with the same polynomial coefficients, is inherited by the asymptotic expansion of any linearly induced quantity such as the distribution of the $k$-th largest level. Whereas the results for the unitary ensembles are presented with proof, the discussion of the orthogonal and symplectic ones is based on some hypotheses. To substantiate the hypotheses, we check the result for the $k$-th largest level in the orthogonal ensembles against simulation data for choices of $n$ and $k$ that require as many as four correction terms to achieve satisfactory accuracy.

math.PR↗

arXiv2025

Asymptotic Expansions of Gaussian and Laguerre Ensembles at the Soft Edge II: Level Densities

We continue our work [arXiv:2403.07628] on asymptotic expansions at the soft edge for the classical $n$-dimensional Gaussian and Laguerre random matrix ensembles. By revisiting the construction of the associated skew-orthogonal polynomials in terms of wave functions, we obtain concise expressions for the level densities that are well suited for proving asymptotic expansions in powers of a certain parameter $h \asymp n^{-2/3}$. In the unitary case, the expansion for the level density can be used to reconstruct the first correction term in an established asymptotic expansion of the associated generating function. In the orthogonal and symplectic cases, we can even reconstruct the conjectured first and second correction terms.

math.PR↗

arXiv2025

Algebraic Independence of an Airy Function, Its Derivative, and Antiderivative

Using tools from the Siegel-Shidlovskii theory of transcendental numbers, we prove that a nontrivial solution of the Airy equation, its derivative, and an antiderivative are algebraically independent over the field of rational functions. Courtesy of Michael Singer, the result is also derived from general considerations in differential Galois theory.

math.CA↗

arXiv2025

Asymptotic Expansions of the Limit Laws of Gaussian and Laguerre (Wishart) Ensembles at the Soft Edge

The large-matrix limit laws of the rescaled largest eigenvalue of the orthogonal, unitary, and symplectic $n$-dimensional Gaussian ensembles -- and of the corresponding Laguerre ensembles (Wishart distributions) for various regimes of the parameter $α$ (degrees of freedom $p$) -- are known to be the Tracy-Widom distributions $F_β$ ($β=1,2,4$). We establish (paying particular attention to large or small ratios $p/n$) that, with careful choices of the rescaling constants and of the expansion parameter $h$, the limit laws embed into asymptotic expansions in powers of $h$, where $h \asymp n^{-2/3}$ resp. $h \asymp (n\,\wedge\,p)^{-2/3}$. We find explicit analytic expressions of the first few expansion terms as linear combinations of higher-order derivatives of the limit law $F_β$ with rational polynomial coefficients. The parametrizations are fine-tuned so that the expansion coefficients in the Gaussian cases are, for given $n$, the limits $p\to\infty$ of those of the Laguerre cases. Whereas the results for $β=2$ are presented with proof, the discussion of the cases $β=1,4$ is based on some hypotheses, focusing on the algebraic aspects of actually computing the polynomial coefficients. For the purposes of illustration and validation, the various results are checked against simulation data with large sample sizes.

math.PR↗

arXiv2024

Asymptotic expansions relating to the lengths of longest monotone subsequences of involutions

We study the distribution of the length of longest monotone subsequences in random (fixed-point free) involutions of $n$ integers as $n$ grows large, establishing asymptotic expansions in powers of $n^{-1/6}$ in the general case and in powers of $n^{-1/3}$ in the fixed-point free cases. Whilst the limit laws were shown by Baik and Rains to be one of the Tracy-Widom distributions $F_β$ for $β=1$ or $β=4$, we find explicit analytic expressions of the first few expansion terms as linear combinations of higher order derivatives of $F_β$ with rational polynomial coefficients. Our derivation is based on a concept of generalized analytic de-Poissonization and is subject to the validity of certain hypotheses for which we provide compelling (computational) evidence. In a preparatory step expansions of the hard-to-soft edge transition laws of L$β$E are studied, which are lifted into expansions of the generalized Poissonized length distributions for large intensities. (This paper continues our work arXiv:2301.02022, which established similar results in the case of general permutations and $β=2$.)

math.PR↗

arXiv2024

Asymptotic expansions relating to the distribution of the length of longest increasing subsequences

We study the distribution of the length of longest increasing subsequences in random permutations of $n$ integers as $n$ grows large and establish an asymptotic expansion in powers of $n^{-1/3}$. Whilst the limit law was already shown by Baik, Deift and Johansson to be the GUE Tracy-Widom distribution $F$, we find explicit analytic expressions of the first few finite-size correction terms as linear combinations of higher order derivatives of $F$ with rational polynomial coefficients. Our proof replaces Johansson's de-Poissonization, which is based on monotonicity as a Tauberian condition, by analytic de-Poissonization of Jacquet and Szpankowski, which is based on growth conditions in the complex plane; it is subject to a tameness hypothesis concerning complex zeros of the analytically continued Poissonized length distribution. In a preparatory step an expansion of the hard-to-soft edge transition law of LUE is studied, which is lifted into an expansion of the Poissonized length distribution for large intensities. Finally, expansions of Stirling-type approximations and of the expected value and variance of the length distribution are given.

math.PR↗

arXiv2022

A Stirling-type formula for the distribution of the length of longest increasing subsequences

The discrete distribution of the length of longest increasing subsequences in random permutations of $n$ integers is deeply related to random matrix theory. In a seminal work, Baik, Deift and Johansson provided an asymptotics in terms of the distribution of the scaled largest level of the large matrix limit of GUE. As a numerical approximation, however, this asymptotics is inaccurate for small $n$ and has a slow convergence rate, conjectured to be just of order $n^{-1/3}$. Here, we suggest a different type of approximation, based on Hayman's generalization of Stirling's formula. Such a formula gives already a couple of correct digits of the length distribution for $n$ as small as $20$ but allows numerical evaluations, with a uniform error of apparent order $n^{-2/3}$, for $n$ as large as $10^{12}$; thus closing the gap between a table of exact values (compiled for up to $n=1000$) and the random matrix limit. Being much more efficient and accurate than Monte-Carlo simulations, the Stirling-type formula allows for a precise numerical understanding of the first few finite size correction terms to the random matrix limit. From this we derive expansions of the expected value and variance of the length, exhibiting several more terms than previously put forward.

math.CO↗

arXiv2022

A Jentzsch-Theorem for Kapteyn, Neumann, and General Dirichlet Series

Comparing phase plots of truncated series solutions of Kepler's equation by Lagrange's power series with those by Bessel's Kapteyn series strongly suggest that a Jentzsch-type theorem holds true not only for the former but also for the latter series: each point of the boundary of the domain of convergence in the complex plane is a cluster point of zeros of sections of the series. We prove this result by studying properties of the growth function of a sequence of entire functions. For series, this growth function is computable in terms of the convergence abscissa of an associated general Dirichlet series. The proof then extends, besides including Jentzsch's classical result for power series, to general Dirichlet series, to Kapteyn, and to Neumann series of Bessel functions. Moreover, sections of Kapteyn and Neumann series generally exhibit zeros close to the real axis which can be explained, including their asymptotic linear density, by the theory of the distribution of zeros of entire functions.

math.CV↗

arXiv2022

The Challenge of Sixfold Integrals: The Closed-Form Evaluation of Newton Potentials between Two Cubes

The challenge of explicitly evaluating, in elementary closed form, the weakly singular sixfold integrals for potentials and forces between two cubes has been taken up at various places in the mathematics and physics literature. It created some strikingly specific results, with an aura of arbitrariness, and a single intricate general procedure due to Hackbusch. Those scattered instances were mostly addressing the problem heads on, by successive integration while keeping track of a thicket of primitives generated at intermediate stages. In this paper we present a substantially easier and shorter approach, based on a Laplace transform of the kernel. We clearly exhibit the structure of the results as obtained by an explicit algorithm, just computing with rational polynomials. The method extends, up to the evaluation of single integrals, to higher dimensions. Among other examples, we easily reproduce Fornberg's startling closed form solution of Trefethen's two-cubes problem and Waldvogel's symmetric formula for the Newton potential of a rectangular cuboid.

math.CA↗

arXiv2020

Efficient numerical evaluation of thermodynamic quantities on infinite (semi-)classical chains

This work presents an efficient numerical method to evaluate the free energy density and associated thermodynamic quantities of (quasi) one-dimensional classical systems, by combining the transfer operator approach with a numerical discretization of integral kernels using quadrature rules. For analytic kernels, the technique exhibits exponential convergence in the number of quadrature points. As demonstration, we apply the method to a classical particle chain, to the semiclassical nonlinear Schrödinger equation and to a classical system on a cylindrical lattice.

math.NA↗

arXiv2017

Finite size effects for spacing distributions in random matrix theory: circular ensembles and Riemann zeros

According to Dyson's three fold way, from the viewpoint of global time reversal symmetry there are three circular ensembles of unitary random matrices relevant to the study of chaotic spectra in quantum mechanics. These are the circular orthogonal, unitary and symplectic ensembles, denoted COE, CUE and CSE respectively. For each of these three ensembles and their thinned versions, whereby each eigenvalue is deleted independently with probability $1-ξ$, we take up the problem of calculating the first two terms in the scaled large $N$ expansion of the spacing distributions. It is well known that the leading term admits a characterisation in terms of both Fredholm determinants and Painlevé transcendents. We show that modifications of these characterisations also remain valid for the next to leading term, and that they provide schemes for high precision numerical computations. In the case of the CUE there is an application to the analysis of Odlyzko's data set for the Riemann zeros, and in that case some further statistics are similarly analysed.

math-ph↗

arXiv2016

On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators

By applying an idea of Borodin and Olshanski [J. Algebra 313 (2007), 40-60], we study various scaling limits of determinantal point processes with trace class projection kernels given by spectral projections of selfadjoint Sturm-Liouville operators. Instead of studying the convergence of the kernels as functions, the method directly addresses the strong convergence of the induced integral operators. We show that, for this notion of convergence, the Dyson, Airy, and Bessel kernels are universal in the bulk, soft-edge, and hard-edge scaling limits. This result allows us to give a short and unified derivation of the known formulae for the scaling limits of the classical random matrix ensembles with unitary invariance, that is, the Gaussian unitary ensemble (GUE), the Wishart or Laguerre unitary ensemble (LUE), and the MANOVA (multivariate analysis of variance) or Jacobi unitary ensemble (JUE).

math-ph↗

arXiv2016

A note on the expansion of the smallest eigenvalue distribution of the LUE at the hard edge

In a recent paper, Edelman, Guionnet and Péché conjectured a particular $n^{-1}$ correction term of the smallest eigenvalue distribution of the Laguerre unitary ensemble (LUE) of order $n$ in the hard-edge scaling limit: specifically, the derivative of the limit distribution, that is, the density, shows up in that correction term. We give a short proof by modifying the hard-edge scaling to achieve an optimal $O(n^{-2})$ rate of convergence of the smallest eigenvalue distribution. The appearance of the derivative follows then by a Taylor expansion of the less optimal, standard hard-edge scaling. We relate the $n^{-1}$ correction term further to the logarithmic derivative of the Bessel kernel Fredholm determinant in the work of Tracy and Widom.

math.PR↗

arXiv2015

Numerical Methods for the Discrete Map $Z^a$

As a basic example in nonlinear theories of discrete complex analysis, we explore various numerical methods for the accurate evaluation of the discrete map $Z^a$ introduced by Agafonov and Bobenko. The methods are based either on a discrete Painlevé equation or on the Riemann-Hilbert method. In the latter case, the underlying structure of a triangular Riemann-Hilbert problem with a non-triangular solution requires special care in the numerical approach. Complexity and numerical stability are discussed, the results are illustrated by numerical examples

math.NA↗

arXiv2015

The Singular Values of the GOE

As a unifying framework for examining several properties that nominally involve eigenvalues, we present a particular structure of the singular values of the Gaussian orthogonal ensemble (GOE): the even-location singular values are distributed as the positive eigenvalues of a Gaussian ensemble with chiral unitary symmetry (anti-GUE), while the odd-location singular values, conditioned on the even-location ones, can be algebraically transformed into a set of independent $χ$-distributed random variables. We discuss three applications of this structure: first, there is a pair of bidiagonal square matrices, whose singular values are jointly distributed as the even- and odd-location ones of the GOE; second, the magnitude of the determinant of the GOE is distributed as a product of simple independent random variables; third, on symmetric intervals, the gap probabilities of the GOE can be expressed in terms of the Laguerre unitary ensemble (LUE). We work specifically with matrices of finite order, but by passing to a large matrix limit, we also obtain new insight into asymptotic properties such as the central limit theorem of the determinant or the gap probabilities in the bulk-scaling limit. The analysis in this paper avoids much of the technical machinery (e.g. Pfaffians, skew-orthogonal polynomials, martingales, Meijer $G$-function, etc.) that was previously used to analyze some of the applications.

math.PR↗

arXiv2015

Singular Values and Evenness Symmetry in Random Matrix Theory

Complex Hermitian random matrices with a unitary symmetry can be distinguished by a weight function. When this is even, it is a known result that the distribution of the singular values can be decomposed as the superposition of two independent eigenvalue sequences distributed according to particular matrix ensembles with chiral unitary symmetry. We give decompositions of the distribution of singular values, and the decimation of the singular values --- whereby only even, or odd, labels are observed --- for real symmetric random matrices with an orthogonal symmetry, and even weight. This requires further specifying the functional form of the weight to one of three types --- Gauss, symmetric Jacobi or Cauchy. Inter-relations between gap probabilities with orthogonal and unitary symmetry follow as a corollary. The Gauss case has appeared in a recent work of Bornemann and La Croix. The Cauchy case, when appropriately specialised and upon stereographic projection, gives decompositions for the analogue of the singular values for the circular unitary and circular orthogonal ensembles.

math.PR↗

arXiv2013

Automatic Deformation of Riemann-Hilbert Problems with Applications to the Painlevé II Transcendents

The stability and convergence rate of Olver's collocation method for the numerical solution of Riemann-Hilbert problems (RHPs) is known to depend very sensitively on the particular choice of contours used as data of the RHP. By manually performing contour deformations that proved to be successful in the asymptotic analysis of RHPs, such as the method of nonlinear steepest descent, the numerical method can basically be preconditioned, making it asymptotically stable. In this paper, however, we will show that most of these preconditioning deformations, including lensing, can be addressed in an automatic, completely algorithmic fashion that would turn the numerical method into a black-box solver. To this end, the preconditioning of RHPs is recast as a discrete, graph-based optimization problem: the deformed contours are obtained as a system of shortest paths within a planar graph weighted by the relative strength of the jump matrices. The algorithm is illustrated for the RHP representing the Painlevé II transcendents.

math.NA↗

arXiv2012

Optimal Contours for High-Order Derivatives

As a model of more general contour integration problems we consider the numerical calculation of high-order derivatives of holomorphic functions using Cauchy's integral formula. Bornemann (2011) showed that the condition number of the Cauchy integral strongly depends on the chosen contour and solved the problem of minimizing the condition number for circular contours. In this paper we minimize the condition number within the class of grid paths of step size h using Provan's algorithm for finding a shortest enclosing walk in weighted graphs embedded in the plane. Numerical examples show that optimal rectangular paths yield small condition numbers even in those cases where circular contours are known to be of limited use, such as for functions with branch-cut singularities.

math.NA↗