In this paper, we address a class of problems in unitary ensembles. Specifically, we study the probability that a gap symmetric about 0, i.e. (−a, a) is found in the Gaussian unitary ensembles (GUE) and the Jacobi unitary ensembles (JUE) (where in the JUE, we take the parameters α = β). By exploiting the even parity of the weight, a doubling of the interval to (a 2 , ∞) for the GUE, and (a 2 , 1), for the (symmetric) JUE, shows that the gap probabilities maybe determined as the product of the smallest eigenvalue distributions of the LUE with parameter α = −1/2, and α = 1/2 and the (shifted) JUE with weights x 1/2 (1 − x) β andThe σ function, namely, the derivative of the log of the smallest eigenvalue distributions of the finite-n LUE or the JUE, satisfies the Jimbo-Miwa-Okamoto σ form of P V and P V I , although in the shift Jacobi case, with the weight x α (1 − x) β , the β parameter does not show up in the equation. We also obtain the asymptotic expansions for the smallest eigenvalue distributions * lvshulin1989@163.com † yangbrookchen@yahoo.co.uk ‡ Corresponding author and faneg@fudan.edu.cn of the Laguerre unitary and Jacobi unitary ensembles after appropriate double scalings, and obtained the constants in the asymptotic expansion of the gap probablities, expressed in term of the Barnes G− function valuated at special point. IntroductionIn the work of Adler and Van Moerbeke [1], the largest eigenvalue distribution of ensembles of n × n random matrices generated by Gaussian, Laguerre and Jacobi weights for general values of the symmetry parameter β, (not to be confused with the β parameter in the Jacobi weight), has been systematically studied, from the perspective of differential operators involving multiple time variables.The gap probabilities that are studied in this paper, the unitary case, denoted by P(a, n), are represented as Fredholm determinant of an integral operator, from the early papers of Mehta, Gaudin and Dyson [19], [20], [22], [31], and [32]. In [28], the gap probability, where a union of disjoint intervals is free of eigenvalues, the integral operator has the sine kernel K(x, y) := sin(x−y) π(x−y) . The (multiple-) gap probability itself was obtained in an expansion in terms of the resolvent of the integral equation. In a tour de force computations, JMMS showed that in the single interval case where (−b, b) is free of eigenvalues, the quantity σ(τ ) := τ d dτ logdet(I − K (−b,b) ), τ = 2b, satisfies a second order non-linear differential equation. Tracy and Widom [36] studied, the finite n version of such problems, namely, the distribution of the smallest eigenvalue in the Laguerre unitary ensembles, and the largest eigenvalue distribution of the Gaussian unitary ensembles starting from the Christoffel-Darboux or Reproducing Kernel, constructed out of the "natural" orthogonal polynomials, namely the Laguerre and Hermite polynomials, respectively. Through a series of differentiation formulas, Tracy and Widom found the finite n version of the differential equations satisfied by the resolvent a...

In this paper, we study the Hankel determinant generated by a singularly perturbed Gaus-By using the ladder operator approach associated with the orthogonal polynomials, we show that the logarithmic derivative of the Hankel determinant satisfies both a non-linear second order difference equation and a non-linear second order differential equation. The Hankel determinant also admits an integral representation involving a Painlevé III ′ . Furthermore, we consider the asymptotics of the Hankel determinant under a double scaling, i.e. n → ∞ and t → 0 such that s = (2n + 1)t is fixed. The asymptotic expansions of the scaled Hankel determinant for large s and small s are established, from which Dyson's constant appears.

We are concerned with the probability that all the eigenvalues of a unitary ensemble with the weight function w(x;t) = x α e −x− t x , x ∈ [0, ∞), α > −1, t ≥ 0, are greater than s. This probability is expressed as the quotient of D n (s,t) and its value at s = 0, where D n (s,t) denotes the determinant of the n dimensional Hankel matrices generated by the moments of w(x;t) on x ∈ [s, ∞). In this paper we focus specifically on the Hankel determinant D n (s,t) and its properties.Based on the ladder operators adapted to the monic polynomials orthogonal with respect to w(x;t), and from the associated supplementary conditions and a sum-rule, we show that the log-derivative of the Hankel determinant, viewed as a function of s and t, satisfies a second order sixth degree partial differential equation, where n appears as a parameter. In order to go to the thermodynamic limit, of infinitely large matrices, we envisage a scenario where n → ∞, s → 0, and t → 0 such that S := 4ns and T := (2n + 1 + α)t are finite. After such a double scaling, the large finite n equation reduces to a second order second degree equation, in the variables S and T , from which we derive the asymptotic expansion of the scaled Hankel determinant in three cases of S and T : S → ∞ with T fixed, S → 0 with T > 0 fixed, and T → ∞ with S > 0 fixed. The constant term in the asymptotic expansion is shown to satisfy a difference equation and one of its solutions is the Tracy-Widom constant.

We study the probability that all the eigenvalues of n × n Hermitian matrices, from the Laguerre unitary ensemble with the weight x γ e −4nx , x ∈ [0, ∞), γ > −1, lie in the interval [0, α]. By using previous results for finite n obtained by the ladder operator approach of orthogonal polynomials, we derive the large n asymptotics of the largest eigenvalue distribution function with α ranging from 0 to the soft edge. In addition, at the soft edge, we compute the constant conjectured by Tracy and Widom [Commun. Math. Phys. 159 (1994), 151-174], later proved by Deift, Its and Krasovsky [Commun. Math. Phys. 278 (2008), 643-678]. Our results are reduced to those of Deift et al. when γ = 0.

We study the Hankel determinant generated by the Laguerre weight with jump discontinuities at t k , k = 1, • • • , m. By employing the ladder operator approach to establish Riccati equations, we show that σ n (t 1 , • • • , t m ), the logarithmic derivative of the n-dimensional Hankel determinant, satisfies a generalization of the σ-from of Painlevé V equation. Through investigating the Riemann-Hilbert problem for the associated orthogonal polynomials and via Lax pair, we express σ n in terms of solutions of a coupled Painlevé V system. We also build relations between the auxiliary quantities introduced in the above two methods, which provides connections between the Riccati equations and Lax pair. In addition, when each t k tends to the hard edge of the spectrum and n goes to ∞, the scaled σ n is shown to satisfy a generalized Painlevé III equation.