The largest eigenvalue distribution of the laguerre unitary ensemble

We study the probability that all eigenvalues of the Laguerre unitary ensemble of n by n matrices are in (0, t), that is, the largest eigenvalue distribution. Associated with this probability, in the ladder operator approach for orthogonal polynomials, there are recurrence coefficients, namely, an (t) and αn (t) and βn, as well as three auxiliary quantities, denoted by rn (t), Rn (t), and ση (t). We establish the second order differential equations for both βn (t) and rn (t). By investigating the soft edge scaling limit when α = O(n) as n → ∞ or a is finite, we derive a Pu, the σ-form, and the asymptotic solution of the probability. In addition, we develop differential equations for orthogonal polynomials Pn(z) corresponding to the largest eigenvalue distribution of LUE and GUE with n finite or large. For large n, asymptotic formulas are given near the singular points of the ODE. Moreover, we are able to deduce a particular case of Chazy's equation for satisfying the σ-form of PIV or PV.