The authors study the distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble, where the Laguerre unitary ensemble (\(\text{LUE}_{n,a}\)) of finite rank random matrices is specified by the eigenvalue probability density function
\[ p(\lambda_1,\dots,\lambda_n):=\frac{1}{n!C_{n,n+a}}\prod_{j=1}^{n}e^{-\lambda_j}\lambda^a_j \prod_{1\leq j<k\leq n}(\lambda_k-\lambda_j)^2,\quad \lambda_1,\dots,\lambda_n\in[0,\infty), \] where \(c_{m,n}=\frac{1}{m!}\prod_{j=1}^{m}\Gamma(n-j+1)\Gamma(m-j+2).\) The distribution of the spacing between the smallest and the second smallest eigenvalues has the expression (for \(\text{LUE}_{n+2,a}\))
\[ A_{n,a}:=\int_{0}^{\infty}dx_1p_{(2)}^{\text{LUE}_{n+2,a}}(x_1,x_1+y),\;y\in\mathbb{R}_+. \]
The paper focuses on the study of the distribution \(p_{(2)}^{\text{LUE}_{n+2,a}}\) and its hard edge scaled limit. This study exhibits a presentation of some appropriate results from orthogonal polynomial system theory, including the class of deformed Laguerre orthogonal polynomials (i.e. orthogonal polynomials with deformed Laguerre weight \(w(x;t)=x^2(x+t)^ae^{-x})\). These polynomials are used to characterize the studied distributions by a solution of the fifth Painlevé equation and its associated linear isomonodromic system.
The special case of positive integer values of the parameter \(a\) is analysed. The hard edge limits are studied and it is shown that the corresponding scaled distributions can be characterized by the solution of a certain Painlevé III’ equation and its associated linear isomonodromic system. An analytical and non-formal theory of the solutions of the defining ordinary and partial differential equations is developed and it is used to accurately compute the distribution and its moments for various values of the parameter \(a\). The paper ends with numerical studies at the hard edge.