In the present paper we derive a new Hankel determinant representation for the square of the Vorob'ev-Yablonski polynomial Qn(x), x ∈ C. These polynomials are the major ingredients in the construction of rational solutions to the second Painlevé equation uxx = xu + 2u 3 + α. As an application of the new identity, we study the zero distribution of Qn(x) as n → ∞ by asymptotically analyzing a certain collection of (pseudo) orthogonal polynomials connected to the aforementioned Hankel determinant. Our approach reproduces recently obtained results in the same context by Buckingham and Miller [2], which used the Jimbo-Miwa Lax representation of PII equation and the asymptotical analysis thereof.

Abstract. We study the determinant det(I − γKs), 0 < γ < 1, of the integrable Fredholm operator Ks acting on the interval (−1, 1) with kernel Ks(λ, µ) = sin s(λ−µ) π(λ−µ) . This determinant arises in the analysis of a log-gas of interacting particles in the bulk-scaling limit, at inverse temperature β = 2, in the presence of an external potential v = − 1 2 ln(1 − γ) supported on an interval of length 2s π . We evaluate, in particular, the double scaling limit of det(I − γKs) as s → ∞ and γ ↑ 1, in the region 0 ≤ κ = v s = − 1 2s ln(1 − γ) ≤ 1 − δ, for any fixed 0 < δ < 1. This problem was first considered by Dyson in [19].

We analyze the left-tail asymptotics of deformed Tracy-Widom distribution functions describing the fluctuations of the largest eigenvalue in invariant random matrix ensembles after removing each soft edge eigenvalue independently with probability 1−γ ∈ [0, 1]. As γ varies, a transition from Tracy-Widom statistics (γ = 1) to classical Weibull statistics (γ = 0) was observed in the physics literature by Bohigas, de Carvalho, and Pato [12]. We provide a description of this transition by rigorously computing the leading-order left-tail asymptotics of the thinned GOE, GUE and GSE Tracy-Widom distributions. In this paper, we obtain the asymptotic behavior in the non-oscillatory region with γ ∈ [0, 1) fixed (for the GOE, GUE, and GSE distributions) and γ ↑ 1 at a controlled rate (for the GUE distribution). This is the first step in an ongoing program to completely describe the transition between Tracy-Widom and Weibull statistics. As a corollary to our results, we obtain a new total-integral formula involving the Ablowitz-Segur solution to the second Painlevé equation. 4 2 √ π e − 2 3 x 3 2 1 + o(1) , x → +∞

Abstract. The paper contains two main parts: in the first part, we analyze the general case of p ≥ 2 matrices coupled in a chain subject to Cauchy interaction. Similarly to the Itzykson-Zuber interaction model, the eigenvalues of the Cauchy chain form a multi level determinantal point process. We first compute all correlations functions in terms of Cauchy biorthogonal polynomials and locate them as specific entries of a (p + 1) × (p + 1) matrix valued solution of a Riemann-Hilbert problem. In the second part, we fix the external potentials as classical Laguerre weights. We then derive strong asymptotics for the Cauchy biorthogonal polynomials when the support of the equilibrium measures contains the origin. As a result, we obtain a new family of universality classes for multi-level random determinantal point fields which include the Besselν universality for 1-level and the Meijer-G universality for 2-level. Our analysis uses the Deift-Zhou nonlinear steepest descent method and the explicit construction of a (p + 1) × (p + 1) origin parametrix in terms of Meijer G-functions. The solution of the full Riemann-Hilbert problem is derived rigorously only for p = 3 but the general framework of the proof can be extended to the Cauchy chain of arbitrary length p.