Harmonic-measure distribution functions for a class of multiply connected symmetrical slit domains
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Date
2022
Authors
Green, C.C.
Snipes, M.A.
Ward, L.A.
Crowdy, D.G.
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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2022; 478(2259, article no. 20210832):1-20
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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.
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Copyright 2022 the authors
Access Condition Notes: Accepted manuscript available on open access