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We describe the pole-free regions of the one-parameter family of special solutions of P$_\mathrm{II}$, the second Painlev\'e equation, constructed from the Airy functions. This is achieved by exploiting the connection between these solutions and the recurrence coefficients of orthogonal polynomials that appear in the analysis of the ensemble of random matrices corresponding to the cubic potential.

We show that the one-parameter family of special solutions of P$_\mathrm{II}$, the second Painlev\'e equation, constructed from the Airy functions, as well as associated solutions of P$_\mathrm{XXXIV}$ and S$_\mathrm{II}$, can be expressed via the recurrence coefficients of orthogonal polynomials that appear in the analysis of the Hermitian random matrix ensemble with a cubic potential. Exploiting this connection we show that solutions of P$_\mathrm{II}$ that depend only on the first Airy function $ \mathrm{Ai} $ (but not on $ \mathrm{Bi} $) possess a scaling limit in the pole free region, which includes a disk around the origin whose radius grows with the parameter. We then use the scaling limit to show that these solutions are monotone in the parameter on the negative real axis.

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 that in $O_{\mathsf{one-cut}}$ the equilibrium measure is supported by one Jordan arc (cut) and in $O_{\mathsf{two-cut}}$ by two cuts. The regions $O_{\mathsf{one-cut}}$ and $O_{\mathsf{two-cut}}$ are separated by critical curves, which can be calculated in terms of critical trajectories of an auxiliary quadratic differential. In our previous work 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. 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.

We study expectations of powers and correlation functions for characteristic polynomials of $N \times N$ non-Hermitian random matrices. For the $1$-point and $2$-point correlation function, we obtain several characterizations in terms of Painlev\'e transcendents, both at finite-$N$ and asymptotically as $N \to \infty$. In the asymptotic analysis, two regimes of interest are distinguished: boundary asymptotics where parameters of the correlation function can touch the boundary of the limiting eigenvalue support and bulk asymptotics where they are strictly inside the support. For the complex Ginibre ensemble this involves Painlev\'e IV at the boundary as $N \to \infty$. Our approach, together with the results in \cite{HW17} suggests that this should arise in a much broader class of planar models. For the bulk asymptotics, one of our results can be interpreted as the merging of two `planar Fisher-Hartwig singularities' where Painlev\'e V arises in the asymptotics. We also discuss the correspondence of our results with a normal matrix model with $d$-fold rotational symmetries known as the \textit{lemniscate ensemble}, recently studied in \cite{BGM, BGG18}. Our approach is flexible enough to apply to non-Gaussian models such as the truncated unitary ensemble or induced Ginibre ensemble; we show that in the former case Painlev\'e VI arises at finite-$N$. Scaling near the boundary leads to Painlev\'e V, in contrast to the Ginibre ensemble., Comment: Typos corrected, 39 pages, 4 figures, 1 table

We study $n\times n$ Hankel determinants constructed with moments of a Hermite weight with a Fisher-Hartwig singularity on the real line. We consider the case when the singularity is in the bulk and is both of root-type and jump-type. We obtain large $n$ asymptotics for these Hankel determinants, and we observe a critical transition when the size of the jumps varies with $n$. These determinants arise in the thinning of the generalised Gaussian unitary ensembles and in the construction of special function solutions of the Painlev\'e IV equation.

In this paper, we compute the probability that an $N \times N$ matrix from the generalised Gaussian Unitary Ensemble (gGUE) is positive definite, extending a previous result of Dean and Majumdar \cite{DM}. For this purpose, we work out the large degree asymptotics of semi-classical Laguerre polynomials and their recurrence coefficients, using the steepest descent analysis of the corresponding Riemann--Hilbert problem., Comment: 21 pages, 1 figure. Revised version, minor changes and references added

We prove the topological expansion for the cubic log-gas partition function \[ Z_N(t)= \int_\Gamma\cdots\int_\Gamma\prod_{1\leq j

We obtain the double scaling asymptotic behavior of the recurrence coefficients and the partition function at the critical point of the $N\times N$ Hermitian random matrix model with cubic potential. We prove that the recurrence coefficients admit an asymptotic expansion in powers of $N^{-2/5}$, and in the leading order the asymptotic behavior of the recurrence coefficients is given by a Boutroux tronqu\'ee solution to the Painlev\'e I equation. We also obtain the double scaling limit of the partition function, and we prove that the poles of the tronqu\'ee solution are limits of zeros of the partition function. The tools used include the Riemann--Hilbert approach and the Deift--Zhou nonlinear steepest descent method for the corresponding family of complex orthogonal polynomials and their recurrence coefficients, together with the Toda equation in the parameter space., Comment: 49 pages, 12 figures, sections 1 to 3 revised

In this paper we return to the classical work by Br\'ezin, Itzykson, Parisi and Zuber, in which, among other things, the authors explicitly calculated the coefficients of the topological expansion in the cubic random matrix model in genus 0. Our main goal will be to rigorously prove the results of Br\'ezin, Itzykson, Parisi and Zuber and to obtain an explicit formula for the coefficients of the topological expansion in genus 1. We will also prove some formulae and asymptotic results for the coefficients of the topological expansion in higher genera., Comment: 45 pages, 5 figures. New section, appendix and references added