The hard edge tacnode process and the hard edge Pearcey process with non-intersecting squared Bessel paths
A system of non-intersecting squared Bessel processes is considered which all start from one point and they all return to another point. Under the scaling of the starting and ending points when the macroscopic boundary of the paths touches the hard edge, a limiting critical process is described in the neighbourhood of the touching point which we call the hard edge tacnode process. We derive its correlation kernel in an explicit new form which involves Airy type functions and operators that act on the direct sum of $L^2(\mathbb R_+)$ and a finite dimensional space. As the starting points of the squared Bessel paths are set to 0, a cusp in the boundary appears. The limiting process is described near the cusp and it is called the hard edge Pearcey process. We compute its multi-time correlation kernel which extends the existing formulas for the single-time kernel. Our pre-asymptotic correlation kernel involves the ratio of two Toeplitz determinants which are rewritten using a Borodin-Okounkov type formula.