Painlevé I asymptotics for orthogonal polynomials with respect to a varying quartic weight

Abstract

We study polynomials that are orthogonal with respect to a varying quartic weight exp(−N (x 2 /2+tx 4 /4)) for t < 0, where the orthogonality takes place on certain contours in the complex plane. Inspired by developments in 2D quantum gravity, Fokas, Its, and Kitaev, showed that there exists a critical value for t around which the asymptotics of the recurrence coefficients are described in terms of exactly specified solutions of the Painlevé I equation. In this paper, we present an alternative and more direct proof of this result by means of the Deift/Zhou steepest descent analysis of the Riemann-Hilbert problem associated with the polynomials. Moreover, we extend the analysis to non-symmetric combinations of contours. Special features in the steepest descent analysis are a modified equililbrium problem and the use of Ψ-functions for the Painlevé I equation in the construction of the local parametrix.

Painlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg—de Vries equation in the small‐dispersion limit

“…An application of particular interest for us is the appearance of the first and second Painlevé equation in the theory of random matrices, see e.g. [21,18,3,6,7]. In random matrix ensembles with certain unitary invariant probability measures, the eigenvalues accumulate on a finite union of intervals when the dimension of the matrices grows [8,9,10].…”

Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models

“…In (1.39) and (1.40) we see that there is no term of order n −1/7 . In the proof of Theorem 1.11 this term will drop out in a similar way as the n −1/5 -term in [16].…”

“…Here, a special solution of the Painlevé I equation occurs instead of a solution of the P 2 I equation and the asymptotics are in powers of n −1/5 . Observe further that in [16] there is no term of order n −1/5 in the asymptotics. In (1.39) and (1.40) we see that there is no term of order n −1/7 .…”

“…Those equilibrium measures will be constructed in Section 2. In order to deal with the deformations V s,t of V 0 , we use modified equilibrium problems where we allow the measures to be negative, which was also done in [5,6,16]. Another modification of the equilibrium problem is that we choose the support of the equilibrium measure fixed, instead of allowing it to choose its own support.…”

“…It is straightforward to check, using the fact that φ s,t,+ + φ s,t,− = 0 on (a, b), that v T has on the interval (a, b) the following factorization, 16) and the opening of the lens is based on this factorization. Observe that, since Re φ s,t,± (x) = 0 and Im φ s,t,± (x) = ∓ b x dν s,t (u) for x ∈ (a, b) (see (3.12)), and since ν s,t is positive on (a+ δ, b− δ) for We deform the RH problem for T into a RH problem for S by opening a lens as shown in Figure 1, so that we obtain a contour Σ.…”