# Ahmad BarhoumiUniversity of Michigan | U-M · Department of Mathematics

Ahmad Barhoumi

PhD

## About

14

Publications

344

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12

Citations

Introduction

Additional affiliations

August 2020 - present

Education

August 2015 - August 2020

## Publications

Publications (14)

We obtain asymptotics of polynomials satisfying the orthogonality relations where the complex parameter is in the so‐called two‐cut region. As an application, we deduce asymptotic formulas for certain families of solutions of Painlevé‐IV which are indexed by a nonnegative integer and can be written in terms of parabolic cylinder functions. The proo...

The third Painlevé equation in its generic form, often referred to as Painlevé-III($D_6$), is given by $$ \frac{{\rm d}^2u}{{\rm d}x^2} =\frac{1}{u}\left(\frac{{\rm d}u}{{\rm d}x} \right)^2-\frac{1}{x} \frac{{\rm d}u}{{\rm d}x} + \frac{\alpha u^2 + \beta}{x}+4u^3-\frac{4}{u}, \qquad \alpha,\beta \in \mathbb C. $$ Starting from a generic initial sol...

The third Painlev\'e equation in its generic form, often referred to as Painlev\'e-III($D_6$), is given by\[ \dfrac{\mathrm{d}^2u}{\mathrm{d}x^2}=\dfrac{1}{u}\left( \dfrac{\mathrm{d}u}{\mathrm{d}x} \right)^2-\dfrac{1}{x} \dfrac{\mathrm{d}u}{\mathrm{d}x}+\dfrac{\alpha u^2 + \beta}{x}+4u^3-\frac{4}{u}, \quad \alpha,\beta \in \mathbb{C}.\] Starting fr...

We investigate asymptotic behavior of polynomials \( Q_n(z) \) satisfying non-Hermitian orthogonality relations
where \( \Delta \) is a Chebotarëv (minimal capacity) contour connecting three non-collinear points and \( \rho (s) \) is a Jacobi-type weight including a possible power-type singularity at the Chebotarëv center of \( \Delta \).

We investigate asymptotic behavior of polynomials $ Q_n(z) $ satisfying non-Hermitian orthogonality relations $$ \int_\Delta s^kQ_n(s)\rho(s)ds =0, \quad k\in\{0,\ldots,n-1\}, $$ where $ \Delta $ is a Chebotar\"ev (minimal capacity) contour connecting three non-collinear points and $ \rho(s) $ is a Jacobi-type weight including a possible power-type...

We investigate the phase diagram of the complex cubic unitary ensemble of random matrices with the potential [Formula: see text], where t is a complex parameter. As proven in our previous paper [Bleher et al., J. Stat. Phys. 166, 784–827 (2017)], the whole phase space of the model, [Formula: see text], is partitioned into two phase regions, [Formul...

We investigate the phase diagram of the complex cubic unitary ensemble of random matrices with the potential $V(M)=-\frac{1}{3}M^3+tM$ where $t$ is a complex parameter. As proven in our previous paper, the whole phase space of the model, $t\in\mathbb C$, is partitioned into two phase regions, $O_{\mathsf{one-cut}}$ and $O_{\mathsf{two-cut}}$, such...

We study random walks on the vertices of three non-isomorphic halfcubes obtained from a cube by a plane cut through its center. Starting from a particular vertex (called the origin), at each step a particle moves, independently of all previous moves, to one of the vertices adjacent to the current vertex with equal probability. We find the means and...

We investigate asymptotic behaviour of polynomials pnω(z) satisfying varying non-Hermitian orthogonality relations ∫−11xkpnω(x)h(x)eiωxdx=0,k∈{0,…,n−1},where h(x)=h∗(x)(1−x)α(1+x)β, ω=λn, λ≥ 0 and h(x) is holomorphic and non-vanishing in a certain neighbourhood in the plane. These polynomials are an extension of so-called kissing polynomials (α=β=0...

We study a family of monic orthogonal polynomials that are orthogonal with respect to the varying, complex-valued weight function, , over the interval , where is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is, , due to its connection with complex Gaussian quadrature rules...

We investigate asymptotic behavior of polynomials $p^{\omega}_n(z)$ satisfying varying non-Hermitian orthogonality relations $$ \int_{-1}^{1} x^kp^{\omega}_n(x)h(x) e^{\mathrm{i} \omega x}\mathrm{d} x =0, \quad k\in\{0,\ldots,n-1\}, $$ where $h(x) = h^*(x) (1 - x)^{\alpha} (1 + x)^{\beta}, \ \omega = \lambda n, \ \lambda \geq 0 $ and $h(x)$ is holo...

We study a family of monic orthogonal polynomials which are orthogonal with respect to the varying, complex valued weight function, $\exp(nsz)$, over the interval $[-1,1]$, where $s\in\mathbb{C}$ is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is $s\in i \mathbb{R}$, due t...

We investigate asymptotic behavior of polynomials Qn(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ Q_n(z) $$\end{document} satisfying non-Hermitian orthogonality r...

We investigate asymptotic behavior of polynomials $ Q_n(z) $ satisfying non-Hermitian orthogonality relations $$ \int_\Delta s^kQ_n(s)\rho(s)\dd s =0, \quad k\in\{0,\ldots,n-1\}, $$ where $ \Delta := [-a,a]\cup [-\ic b,\ic b] $, $ a,b>0 $, and $ \rho(s) $ is a Jacobi-type weight.