  We study uniformly random lozenge tilings of general simply connected
polygons. Under a technical assumption that is presumably generic with respect
to polygon shapes, we show that the local statistics around a cusp point of the
arctic curve converge to the Pearcey process. This verifies the widely
predicted universality of edge statistics in the cusp case. Together with the
smooth and tangent cases proved in Aggarwal-Huang and Aggarwal-Gorin, these are
believed to be the three types of edge statistics that can arise in a generic
polygon. Our proof is via a local coupling of the random tiling with
non-intersecting Bernoulli random walks (NBRW). To leverage this coupling, we
establish an optimal concentration estimate for the tiling height function
around the cusp. As another step and also a result of potential independent
interest, we show that the local statistics of NBRW around a cusp converge to
the Pearcey process when the initial configuration consists of two parts with
proper density growth, via careful asymptotic analysis of the determinantal
formula.
