Abstract We study four problems in random matrices, The first problem is about the largest eigenvalue distribution of the n × n Laguerre unitary ensemble at the soft edge, in the case of the exponent α =Ο(n) or finite α. We derive the classic Tracy-Widom Painlevé Ⅱ equation. From the ladder operators adapted to the monic orthogonal polynomials associated with this problem, after suitable transformation, we obtain a second order linear differential equation which has power series solutions and thus give another characterization of the orthogonal polynomials. We also propose a method for deducing a Chazy's equation from the general σ-form of Piv and Pv. The second problem is concerning the exceptional solutions of a particular Painlevé Ⅵ equation which characterizes the Hankel determinant for a generalized Jacobi weight. We study the asymptotic expansion of this Painlevé equation near its three regular singular points. For four special cases of the parameters (depending on the dimension of the Hankel determinant) of the hypergeometric weight, we find solutions of that Painlevé Ⅵ equation in terms of algebraic curves. The third problem is on the gap probability (on a symmetric interval) of the Gaussian unitary ensemble (GUE) and the Jacobi unitary ensemble (JUE, where we take the parameters α = β). We show the probability that the interval (-a, a) has no eigenvalues, of an n × n random matrix chosen from GUE, satisfies a large second order differential equation in the variable a. In order to go to the thermodynamic limit, we consider a scenario where n⭢∞ and a⭢0+ such that 2√2n a is finite. After such a double scaling, the large equation “travels down" to the JMMS (M. Jimbo, T. Miwa, Y. Môri and M. Sato, Physica 1D (1980), 80-158) σ form of the Painlevé V equation. We also obtain the asymptotic expansions of the smallest eigenvalue distributions of the Laguerre and shifted Jacobi unitary ensembles after appropriate double scalings, including the constants, With α = ±1/2 in the Gamma density xᵃe⁻ˣ, and the Beta density xᵃ(1-x)ᵝ, we derive the constant term in the asymptotic expansion of the gap probability of GUE (i.e. Widom-Dyson constant) and JUE respectively. The fourth problem is related to the Hankel determinant generated by the weight xᵃe⁻ˣ⁻ᵗ⁄ˣ on [s, ∞). Based on the ladder operators for the associated monic orthogonal polynomials, and from their supplementary conditions and a sum-rule, we show that the sum of the log-derivative of the Hankel determinant over s and t satisfies a large partial differential equation which reduces to a second order second degree equation after suitable double scalings. We derive the asymptotic expansions of the scaled Han-kel determinant for three cases of the scaled variables, obtaining the Widom-Dyson constant.