Abstract
We resolve several questions about the harmonic measure distribution function of a planar domain. This function h(r) specifies the harmonic measure of the part of the boundary of the domain which lies within any given distance r of a fixed basepoint in the domain, We focus on the asymptotic behaviour of the function as r decreases towards the distance from the base-point to the boundary of the domain. We show that for each β between zero and one, there is a domain whose distribution function is asymptotically exponential with exponent β, proving our earlier conjecture. We prove that if the base-point in any fixed domain is moved directly towards the closest boundary point, then the value of β cannot decrease. Finally we construct a domain whose distribution function is not asymptotically exponential
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