We use the steepest descents method to study the integral kernel of a family of normal random matrix ensembles with eigenvalue distribution P_{ N}(z_{ 1},...,z_{ N}) = Z_{ N}^{ -1} e^{ -N\Sigma_{ i=1}^{ N}V_{ \alpha}(z_{ i})} \Pi_{ 1\leqi<j\leqN}|z_{ i}-z_{ j}|^{ 2} where V_{ \alpha}(z)=|z|^{ \alpha}, z \in C and \alpha \in ]0,\infty[. Asymptotic analysis with error estimates are obtained. A corollary of this expansion is a scaling limit for the n-point function in terms of the integral kernel for the classical Segal--Bargmann space.
Categories: math-ph math.MP math.PR