A path property of Dyson gaps, Plancherel measures for $Sp(\infty)$, and random surface growth

Abstract

We pursue applications for symplectic Plancherel growth based on a repulsion phenomenon arising in its diffusion limit and on intermediate representation theory underlying its correlation structure. Under diffusive scaling, the dynamics converge to interlaced reflecting Brownian motions with a wall that achieve Dyson non-colliding dynamics. We exhibit non-degeneracy of constraint in this coupled system by deriving a path property that quantifies repulsion between particles coinciding in the limit. We then identify consistent series of Plancherel measures for $Sp(\infty)$ that reflect the odd symplectic groups, despite their non-semisimplicity. As an application, we compute the correlation kernel of the growth model and investigate its local asymptotics: the incomplete beta kernel emerges in the bulk limit, and new variants of the Jacobi and Pearcey kernels arise as edge limits. In particular, we provide further evidence for the universality of the $1/4$-growth exponent and Pearcey point process in the class of anisotropic KPZ with a wall.