We consider the free additive convolution semigroup {μ^t : t ≥ 1} and determine the local behavior of the density of μ^t at the endpoints and at any singular point of its support. We then study the free additive convolution of two multi-cut probability measures and show that its density decays either as a square root or as a cubic root at any endpoint of its support. The probability measures considered in this paper satisfy a power law behavior with exponents strictly between −1 and 1 at the endpoints of their supports.

The Pearcey process is a universal point process in random matrix theory. In this paper, we study the generating function of the Pearcey process on any number m of intervals. We derive an integral representation for it in terms of a Hamiltonian that is related to a system of 6m + 2 coupled nonlinear equations. We also obtain asymptotics for the generating function as the size of the intervals get large, up to and including the constant term. This work generalizes some results of Dai, Xu, and Zhang, which correspond to m = 1.

We consider the free additive convolution semigroup  and determine the local behavior of its density at the endpoints and at any singular pointof its support. We then study the free additive convolution of two multi-cut probability measures and show that its density decays either as a square root or as a cubic root at any endpoints of its support. The probability measures considered in this paper satisfy a power law behavior with exponents strictly between −1 and 1 at the endpoints of their supports.

The Pearcey process is a universal point process in random matrix theory. In this paper, we study the generating function of the Pearcey process on any number m of intervals. We derive an integral representation for it in terms of a Hamiltonian that is related to a system of 6m + 2 coupled nonlinear equations. We also obtain asymptotics for the generating function as the size of the intervals get large, up to and including the constant term. This work generalizes some results of Dai, Xu and Zhang, which correspond to m= 1.

We consider the free additive convolution of two probability measures, supported on respectively n and m disjoint bounded intervals on the real line, and derive a lower bound and an upper bound that is strictly smaller than 2nm, on the number of connected components in its support. We also obtain the corresponding results for the free additive convolution semi-group. Throughout the paper,we consider classes of probability measures with power law behaviors at the endpoints of their supports with exponents ranging from −1 to 1. Our main theorem generalizes a result of Bao, Erdös and Schnelli to the multi-cut setup.