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 of 4‐valent connected graphs with
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, [Formula: see text] and [Formula: see text], such that in [Formula: see text] the equilibrium measure is supported by one Jordan arc (cut) and in [Formula: see text] by two cuts. The regions [Formula: see text] and [Formula: see text] 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.
more » « less- NSF-PAR ID:
- 10407716
- Publisher / Repository:
- American Institute of Physics
- Date Published:
- Journal Name:
- Journal of Mathematical Physics
- Volume:
- 63
- Issue:
- 6
- ISSN:
- 0022-2488
- Page Range / eLocation ID:
- Article No. 063303
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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The density matrix formalism is a fundamental tool in studying various problems in quantum information processing. In the space of density matrices, the most well-known measures are the Hilbert–Schmidt and Bures–Hall ensembles. In this work, the averages of quantum purity and von Neumann entropy for an ensemble that interpolates between these two major ensembles are explicitly calculated for finite-dimensional systems. The proposed interpolating ensemble is a specialization of the [Formula: see text]-deformed Cauchy–Laguerre two-matrix model and new results for this latter ensemble are given in full generality, including the recurrence relations satisfied by their associated bi-orthogonal polynomials when [Formula: see text] assumes positive integer values.more » « less
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