Harmonic-measure distribution functions for a class of multiply connected symmetrical slit domains

Abstract

The harmonic-measure distribution function, or h-function, of a planar domain Ω ⊂ C with respect to a basepoint z0 ∈ Ω is a signature that profiles the behaviour in Ω of a Brownian particle starting from z0. Explicit calculation of h-functions for a wide array of simply connected domains using conformal mapping techniques has allowed many rich connections to be made between the geometry of the domain and the behaviour of its h-function. Until now, almost all h-function computations have been confined to simply connected domains. In this work, we apply the theory of the Schottky-Klein prime function to explicitly compute the h-function of the doubly connected slit domain C \ ([−1/2, −1/6] ∪ [1/6, 1/2]). In view of the connection between the middle-thirds Cantor set and highly multiply connected symmetric slit domains, we then extend our methodology to explicitly construct the h-functions associated with symmetric slit domains of arbitrary even connectivity. To highlight both the versatility and generality of our results, we graph the h-functions associated with quadruply and octuply connected slit domains.