arXiv subjects

Ferenc Balogh

Publications and source records attributed to Ferenc Balogh.

9 recordsLinked to original sources

arXiv2023

On longest increasing subsequences in words in which all multiplicities are equal

Gessel's famous Bessel determinant formula gives the generating function of the number of permutations without increasing subsequences of a given length. Ekhad and Zeilberger proposed the challenge of finding a suitable generalization for permutations of multisets in which all multiplicities are equal, that is, to count words of length $rn$ from an alphabet consisting of $n$ letters in which each letter appears exactly $r$ times and which have no increasing subsequences of length $d$. In this paper we present such a generating function expressible as a multiple integral of the product of a Gessel-type Toeplitz determinant with the exponentiated cycle index polynomial of the symmetric group on $r$ elements.

math.CO↗

arXiv2016

Orthogonal polynomials for a class of measures with discrete rotational symmetries in the complex plane

We obtain the strong asymptotics of polynomials $p_n(λ)$, $λ\in\mathbb{C}$, orthogonal with respect to measures in the complex plane of the form $$ e^{-N(|λ|^{2s}-tλ^s-\overline{tλ}^s)}dA(λ), $$ where $s$ is a positive integer, $t$ is a complex parameter and $dA$ stands for the area measure in the plane. Such problem has its origin from normal matrix models. We study the asymptotic behaviour of $p_n(λ)$ in the limit $n,N\to\infty$ in such a way that $n/N\to T$ constant. Such asymptotic behaviour has two distinguished regimes according to the topology of the limiting support of the eigenvalue distribution of the normal matrix model. If $0<|t|^2 T/s$ the eigenvalue distribution support consists of $s$ connected components. Correspondingly the support of the limiting zero distribution of the orthogonal polynomials consists of a closed contour contained in each connected component. Our asymptotic analysis is obtained by reducing the planar orthogonality conditions of the polynomials to an equivalent system of contour integral orthogonality conditions. The strong asymptotics for the orthogonal polynomials is obtained from the corresponding Riemann--Hilbert problem by the Deift--Zhou nonlinear steepest descent method.

math-ph↗

arXiv2015

Hankel Determinant Approach to Generalized Vorob'ev-Yablonski Polynomials and their Roots

Generalized Vorob'ev-Yablonski polynomials have been introduced by Clarkson and Mansfield in their study of rational solutions of the second Painlevé hierarchy. We present new Hankel determinant identities for the squares of these special polynomials in terms of Schur polynomials. As an application of the identities, we analyze the roots of generalized Vorob'ev-Yablonski polynomials and provide formulæ\, for the boundary curves of the highly regular patterns observed numerically in \cite{CM}.

math-ph↗

arXiv2015

Geometric interpretation of Zhou's explicit formula for the Witten-Kontsevich tau function

Based on the work of Itzykson and Zuber on Kontsevich's integrals, we give a geometric interpretation and a simple proof of Zhou's explicit formula for the Witten-Kontsevich tau function. More precisely, we show that the numbers $A_{m,n}^{Zhou}$ defined by Zhou coincide with the affine coordinates for the point of the Sato Grassmannian corresponding to the Witten-Kontsevich tau function. Generating functions and new recursion relations for $A_{m,n}^{Zhou}$ are derived. Our formulation on matrix-valued affine coordinates and on tau functions remains valid for generic Grassmannian solutions of the KdV hierarchy. A by-product of our study indicates an interesting relation between the matrix-valued affine coordinates for the Witten-Kontsevich tau function and the $V$-matrices associated to the $R$-matrix of Witten's $3$-spin structures.

math-ph↗

arXiv2014

Weighted quantile correlation test for the logistic family

We summarize the results of investigating the asymptotic behavior of the weighted quantile correlation tests for the location-scale family associated to the logistic distribution. Explicit representations of the limiting distribution are given in terms of integrals of weighted Brownian bridges or alternatively as infinite series of independent Gaussian random variables. The power of this test and the test for the location logistic family against some alternatives are demonstrated by numerical simulations.

math.ST↗

arXiv2014

The higher spin generalization of the 6-vertex model with domain wall boundary conditions and Macdonald polynomials

The determinantal form of the partition function of the 6-vertex model with domain wall boundary conditions was given by Izergin. It is known that for a special value of the crossing parameter the partition function reduces to a Schur polynomial. Caradoc, Foda and Kitanine computed the partition function of the higher spin generalization of the 6-vertex model. In the present work it is shown that for a special value of the crossing parameter, referred to as the combinatorial point, the partition function reduces to a Macdonald polynomial.

math.CO↗

arXiv2013

Equilibrium measures for a class of potentials with discrete rotational symmetries

In this note the logarithmic energy problem with external potential $|z|^{2n}+tz^d+\bar{t}\bar{z}^d$ is considered in the complex plane, where $n$ and $d$ are positive integers satisfying $d\leq 2n$. Exploiting the discrete rotational invariance of the potential, a simple symmetry reduction procedure is used to calculate the equilibrium measure for all admissible values of $n,d$ and $t$. It is shown that, for fixed $n$ and $d$, there is a critical value $|t|=t_{cr}$ such that the support of the equilibrium measure is simply connected for $|t| t_{cr}$.

math.CV↗

arXiv2013

Strong asymptotics of the orthogonal polynomials with respect to a measure supported on the plane

We consider the orthogonal polynomials $\{P_{n}(z)\}$ with respect to the measure $|z-a|^{2N c} {\rm e}^{-N |z|^2} \,{\rm d} A(z)$ over the whole complex plane. We obtain the strong asymptotic of the orthogonal polynomials in the complex plane and the location of their zeros in a scaling limit where $n$ grows to infinity with $N$. The asymptotics are described in terms of three (probability) measures associated with the problem. The first measure is the limit of the counting measure of zeros of the polynomials, which is captured by the $g$-function much in the spirit of ordinary orthogonal polynomials on the real line. The second measure is the equilibrium measure that minimizes a certain logarithmic potential energy, supported on a region $K$ of the complex plane. The third measure is the harmonic measure of $K^c$ with a pole at $\infty$. This appears as the limit of the probability measure given (up to the normalization constant) by the squared modulus of the $n$-th orthogonal polynomial times the orthogonality measure, i.e. $|P_n(z)|^2 |z-a|^{2N c} {\rm e}^{-N |z|^2} \,{\rm d} A(z)$. The compact region $K$ that is the support of the second measure undergoes a topological transition under the variation of the parameter $t=n/N$; in a double scaling limit near the critical point given by $t_c=a(a+2\sqrt c)$ we observe the Hastings-McLeod solution to Painlevé\ II in the asymptotics of the orthogonal polynomials.

math-ph↗

arXiv2008

Optimal approximation of harmonic growth clusters by orthogonal polynomials

Interface dynamics in two-dimensional systems with a maximal number of conservation laws gives an accurate theoretical model for many physical processes, from the hydrodynamics of immiscible, viscous flows (zero surface-tension limit of Hele-Shaw flows, [1]), to the granular dynamics of hard spheres [2], and even diffusion-limited aggregation [3]. Although a complete solution for the continuum case exists [4, 5], efficient approximations of the boundary evolution are very useful due to their practical applications [6]. In this article, the approximation scheme based on orthogonal polynomials with a deformed Gaussian kernel [7] is discussed, as well as relations to potential theory.

math-ph↗