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### Browsing by Author "Barhoumi, Ahmad"

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Item Asymptotics of Polynomials Orthogonal on a Cross with a Jacobi-Type Weight(Springer, 2020) Barhoumi, Ahmad; Yattselev, Maxim L.; Mathematical Sciences, School of ScienceWe investigate asymptotic behavior of polynomials Qn(z) satisfying non-Hermitian orthogonality relations ∫ΔskQn(s)ρ(s)ds=0,k∈{0,…,n−1}, where Δ:=[−a,a]∪[−ib,ib], a,b>0, and ρ(s) is a Jacobi-type weight.Item Investigation of the two-cut phase region in the complex cubic ensemble of random matrices(AIP Publishing, 2022-06) Barhoumi, Ahmad; Bleher, Pavel; Deaño, Alfredo; Yattselev, Maxim; Mathematical Sciences, School of ScienceWe investigate the phase diagram of the complex cubic unitary ensemble of random matrices with the potential V(M) = −1/3(M^3) + tM, 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, t ∈ C, is partitioned into two phase regions, Oone−cut and Otwo−cut, such that in Oone−cut the equilibrium measure is supported by one Jordan arc (cut) and in Otwo−cut by two cuts. The regions Oone−cut and Otwo−cut are separated by critical curves, which can be calculated in terms of critical trajectories of an auxiliary quadratic differential. In Bleher et al. [J. Stat. Phys. 166, 784–827 (2017)], the one-cut phase region was investigated in detail. In the present paper, we investigate the two-cut region. We prove that in the two-cut region, the endpoints of the cuts are analytic functions of the real and imaginary parts of the parameter t, but not of the parameter t itself (so that the Cauchy–Riemann equations are violated for the endpoints). We also obtain the semiclassical asymptotics of the orthogonal polynomials associated with the ensemble of random matrices and their recurrence coefficients. The proofs are based on the Riemann–Hilbert approach to semiclassical asymptotics of the orthogonal polynomials and the theory of S-curves and quadratic differentials.Item Orthogonal Polynomials on S-Curves Associated with Genus One Surfaces(2020-08) Barhoumi, Ahmad; Yattselev, Maxim; Bleher, Pavel; Its, Alexander; Tarasov, VitalyWe consider orthogonal polynomials P_n satisfying orthogonality relations where the measure of orthogonality is, in general, a complex-valued Borel measure supported on subsets of the complex plane. In our consideration we will focus on measures of the form d\mu(z) = \rho(z) dz where the function \rho may depend on other auxiliary parameters. Much of the asymptotic analysis is done via the Riemann-Hilbert problem and the Deift-Zhou nonlinear steepest descent method, and relies heavily on notions from logarithmic potential theory.Item Symmetric Random Walks on Three Half-Cubes(ARF India, 2022-12-29) Barhoumi, Ahmad; Ching Cheung, Chung; Pilla, Michael R.; Sarkar, Jyotirmoy; Mathematical Sciences, School of ScienceWe 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 the standard deviations of the number of steps needed to: (1) return to origin, (2) visit all vertices, and (3) return to origin after visiting all vertices. We also find (4) the probability distribution of the last vertex visited.