arXiv subjects

Roozbeh Gharakhloo

Publications and source records attributed to Roozbeh Gharakhloo.

13 recordsLinked to original sources

arXiv2026

Universal Asymptotics and Exact Enumeration of Eulerian Maps

We calculate the asymptotics of the number of connected, labeled, genus $g$ Eulerian maps with an arbitrary degree sequence, in the limit as the total number of vertices tends to infinity. This asymptotic is universal, in the sense that the leading order term depends on only finitely many map characteristics. The constant factor in this formula is related to the Painlev\'{e} I equation. Our methods combine the analysis of the recurrence coefficients associated to a particular family of orthogonal polynomials, and the theory of analytic combinatorics of several variables. We also derive an exact formula for the number of connected, labeled, genus $1$ Eulerian maps. These are the first results on this kind of enumeration problem for $g\geq 1$, non-regular (mixed-valence) maps.

math.CO↗

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↗

arXiv2026

Reduction of Multiple Orthogonal Polynomials to Standard Orthogonal Polynomials

In this article, we derive explicit formulae expressing multiple orthogonal polynomials in terms of standard orthogonal polynomials. We treat both the real-line and unit-circle settings: multiple orthogonal polynomials on the real line (MOPRL) are reduced to orthogonal polynomials on the real line (OPRL), while multiple orthogonal polynomials on the unit circle (MOPUC) are reduced to orthogonal polynomials on the unit circle (OPUC). These formulae also yield corresponding reductions of the Christoffel--Darboux kernels, from the MOPRL kernel to the OPRL kernel and from the MOPUC kernel to the OPUC kernel. In particular, they recover Zinn-Justin's kernel for the external-source random matrix model [arXiv:cond-mat/9703033] and Baik's kernel reduction formula in the spiked source model [arXiv:0809.3970]. We also apply our general results to concrete examples: in the real-line setting, we obtain an explicit expression for the multiple Hermite Christoffel--Darboux kernel in terms of classical Hermite polynomials, while in the unit-circle setting, we use arc-indicator weights to exhibit resonance-type zero escape phenomena for type II MOPUC.

math.CA↗

arXiv2025

A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants II

In this article, we continue the development of the Riemann-Hilbert formalism for studying the asymptotics of Toeplitz+Hankel determinants with non-identical symbols, which we initiated in \cite{GI}. In \cite{GI}, we showed that the Riemann-Hilbert problem we formulated admits the Deift-Zhou nonlinear steepest descent analysis, but with a special restriction on the winding numbers of the associated symbols. In particular, the most natural case, namely zero winding numbers, is not allowed. A principal goal of this paper is to develop a framework that extends the asymptotic analysis of Toeplitz+Hankel determinants to a broader range of winding-number configurations. As an application, we consider the case in which the winding numbers of the Szeg\H{o}-type Toeplitz and Hankel symbols are zero and one, respectively, and compute the asymptotics of the norms of the corresponding system of orthogonal polynomials.

math-ph↗

arXiv2025

Combinatorics of Even-Valent Graphs on Riemann Surfaces

In this paper, we derive explicit formulae for the number of regular even-valent graphs, with fixed minimal embedding genus, in which both the valence parameter and the number of vertices are allowed to vary. Our results extend the explicit formulae of Ercolani--McLaughlin--Pierce (2008) for genus $0$ and of Ercolani--Lega--Tippings (2023) for genus $1$. More precisely, we obtain explicit counts $\mathscr{N}_g(2\nu,j)$ -- with $\nu$ and $j$ as variables -- of graphs with $j$ vertices of uniform valence $2\nu$ and minimal embedding genus $g$, for $2\leq g\leq 4$. We also obtain the corresponding formulae for the two-legged counts $\mathcal{N}_g(2\nu,j)$. The method applies to $g\geq 5$, with increasing computational effort as $g$ increases. Finally, we derive leading-order large-valence asymptotics for these counts when $g\leq 4$, and formulate a structural conjecture for higher genus.

math.CO↗

arXiv2024

Bordered and Framed Toeplitz and Hankel Determinants with Applications to Integrable Probability

Bordered and framed Toeplitz/Hankel determinants have the same structure as Toeplitz/Hankel determinants except in small number of matrix rows and/or columns. We review these structured determinants and their connections to orthogonal polynomials, collecting well-known and perhaps less well-known results. We present some applications for these structured determinants to ensembles of non-intersecting paths and the six-vertex model, with an eye towards asymptotic analysis. We also prove some asymptotic formulae for the probability of non-intersection for an ensemble of continuous time random walks for certain choices of starting and ending points as the number of random walkers tends to infinity.

math.CA↗

arXiv2023

Strong Szeg\H{o} Limit Theorems for Multi-Bordered, Framed, and Multi-Framed Toeplitz Determinants

This work provides the general framework for obtaining strong Szeg\H{o} limit theorems for multi-bordered, semi-framed, framed, and multi-framed Toeplitz determinants, extending the results of Basor et al. (2022) beyond the (single) bordered Toeplitz case. For the two-bordered and also the semi-framed Toeplitz determinants, we compute the strong Szeg\H{o} limit theorems associated with certain classes of symbols, and for the $k$-bordered (${k \geq 3}$), framed, and multi-framed Toeplitz determinants we demonstrate the recursive fashion offered by the Dodgson condensation identities via which strong Szeg\H{o} limit theorems can be obtained. One instance of appearance of semi-framed Toeplitz determinants is in calculations related to the entanglement entropy for disjoint subsystems in the XX spin chain (Brightmore et al. (2020) and Jin-Korepin (2011)). In addition, in the recent work Gharakhloo and Liechty (2024) and in an unpublished work of Professor Nicholas Witte, such determinants have found relevance respectively in the study of ensembles of nonintersecting paths and in the study of off-diagonal correlations of the anisotropic square-lattice Ising model. Besides the intrinsic mathematical interest in these structured determinants, the aforementioned applications have further motivated the study of the present work.

math.CA↗

arXiv2022

Phase Diagram and Topological Expansion in the Complex Quartic Random Matrix Model

We use the Riemann-Hilbert approach, together with string and Toda equations, to study the topological expansion in the quartic random matrix model. The coefficients of the topological expansion are generating functions for the numbers $\mathscr{N}_j(g)$ of $4$-valent connected graphs with $j$ vertices on a compact Riemann surface of genus $g$. We explicitly evaluate these numbers for Riemann surfaces of genus $0,1,2,$ and $3$. Also, for a Riemann surface of an arbitrary genus $g$, we also calculate the leading term in the asymptotics of $\mathscr{N}_j(g)$ as the number of vertices tends to infinity. Using the theory of quadratic differentials, we characterize the critical contours in the complex parameter plane where phase transitions in the quartic model take place, thereby proving a result of David \cite{DAVID}. These phase transitions are of the following four types: a) one-cut to two-cut through the splitting of the cut at the origin, b) two-cut to three-cut through the birth of a new cut at the origin, c) one-cut to three-cut through the splitting of the cut at two symmetric points, and d) one-cut to three-cut through the birth of two symmetric cuts.

math-ph↗

arXiv2022

Openness of Regular Regimes of Complex Random Matrix Models

Consider the general complex polynomial external field $$ V(z)=\frac{z^{k}}{k}+\sum_{j=1}^{k-1} \frac{t_j z^j}{j}, \qquad t_j \in \mathbb{C}, \quad k \in \mathbb{N}. $$ Fix an equivalence class $\mathcal{T}$ of admissible contours whose members approach $\infty$ in two different directions and consider the associated max-min energy problem. When $k=2p$, $p \in \mathbb{N}$, and $\mathcal{T}$ contains the real axis, we show that the set of parameters $t_1, \cdots, t_{2p-1}$ which gives rise to a regular $q$-cut max-min (equilibrium) measure, $1 \leq q \leq 2p-1 $, is an open set in $\mathbb{C}^{2p-1}$. We use the implicit function theorem to prove that the endpoint equations are solvable in a small enough neighborhood of a regular $q$-cut point. We also establish the real-analyticity of the real and imaginary parts of the end-points for all $q$-cut regimes, $1 \leq q \leq 2p-1$, with respect to the real and imaginary parts of the complex parameters in the external field. Our choice of even $k$ and the equivalence class $\mathcal{T} \ni \mathbb{R}$ of admissible contours is only for the simplicity of exposition and our proof extends to all possible choices in an analogous way.

math.CA↗

arXiv2021

Asymptotics of bordered Toeplitz determinants and next-to-diagonal Ising correlations

We prove the analogue of the strong Szeg{\H o} limit theorem for a large class of bordered Toeplitz determinants. In particular, by applying our results to the formula of Au-Yang and Perk \cite{YP} for the next-to-diagonal correlations $\langle σ_{0,0}σ_{N-1,N} \rangle$ in the anisotropic square lattice Ising model, we rigorously justify that the next-to-diagonal long-range order is the same as the diagonal and horizontal ones in the low temperature regime. The anisotropy-dependence of the subleading term in the asymptotics of the next-to-diagonal correlations is also established. We use Riemann-Hilbert and operator theory techniques, independently and in parallel, to prove these results.

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↗

arXiv2021

Asymptotics of Hankel determinants with a Laguerre-type or Jacobi-type potential and Fisher-Hartwig singularities

We obtain large $n$ asymptotics of $n \times n$ Hankel determinants whose weight has a one-cut regular potential and Fisher-Hartwig singularities. We restrict our attention to the case where the associated equilibrium measure possesses either one soft edge and one hard edge (Laguerre-type) or two hard edges (Jacobi-type). We also present some applications in the theory of random matrices. In particular, we can deduce from our results asymptotics for partition functions with singularities, central limit theorems, correlations of the characteristic polynomials, and gap probabilities for (piecewise constant) thinned Laguerre and Jacobi-type ensembles. Finally, we mention some links with the topics of rigidity and Gaussian multiplicative chaos.

math-ph↗

arXiv2020

A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants

In this paper we will formulate $4\times4$ Riemann-Hilbert problems for Toeplitz+Hankel determinants and the associated system of orthogonal polynomials, when the Hankel symbol is supported on the unit circle and also when it is supported on an interval $[a,b]$, $0<a<b<1$. The distinguishing feature of this work is that in the formulation of the Riemann-Hilbert problem no specific relationship is assumed between the Toeplitz and Hankel symbols. We will develop nonlinear steepest descent methods for analysing these problems in the case where the symbols are smooth (i.e., in the absence of Fisher-Hartwig singularities) and admit an analytic continuation in a neighborhood of the unit circle (if the symbol's support is the unit circle). We will finally introduce a model problem and will present its solution requiring certain conditions on the ratio of Hankel and Toeplitz symbols. This in turn will allow us to find the asymptotics of the norms $h_n$ of the corresponding orthogonal polynomials and, in fact, the large $n$ asymptotics of the polynomials themselves. We will explain how this solvable case is related to the recent operator-theoretic approach in [Basor E., Ehrhardt T., in Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics, Oper. Theory Adv. Appl., Vol. 259, Birkhäuser/Springer, Cham, 2017, 125-154, arXiv:1603.00506] to Toeplitz+Hankel determinants. At the end we will discuss the prospects of future work and outline several technical, as well as conceptual, issues which we are going to address next within the $4\times 4$ Riemann-Hilbert framework introduced in this paper.

math-ph↗