A matrix model with a singular weight and Painlevé III

We investigate the matrix model with weight w(x):=exp(−z22x2+tx−x22)and unitary symmetry. In particular we study the double scaling limit as N→∞ and (N−−√t,Nz2)→(u1,u2), where N is the matrix dimension and the parameters (u₁, u₂) remain finite. Using the Deift-Zhou steepest descent method, we compute the asymptotics of the partition function when z and t are of order O(N−1/2). In this regime we discover a phase transition in the (z, N)-plane characterised by the Painlevé III equation. This is the first time that Painlevé III appears in studies of double scaling limits in Random Matrix Theory and is associated to the emergence of an essential singularity in the weighting function. The asymptotics of the partition function is expressed in terms of a particular solution of the Painlevé III equation. We derive explicitly the initial conditions in the limit Nz2→u2 of this solution.