The relative trace formula for groups with involution |
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Authors: | Nadya Gurevich |
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Institution: | 1. Institut für Mathematik der Universit?t Zürich, Winterthurer Str. 190, 8057, Zürich, Switzerland
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Abstract: | A theorem of Bourgain states that the harmonic measure for a domain in ℝ
d
is supported on a set of Hausdorff dimension strictly less thand 2]. We apply Bourgain’s method to the discrete case, i.e., to the distribution of the first entrance point of a random walk
into a subset of ℤ
d
,d≥2. By refining the argument, we prove that for allβ>0 there existsρ(d,β)<d andN(d,β), such that for anyn>N(d,β), anyx ∈ ℤ
d
, and anyA ⊂ {1,…,n}
d
•{y∈ℤ whereν
A,x
(y) denotes the probability thaty is the first entrance point of the simple random walk starting atx intoA. Furthermore,ρ must converge tod asβ → ∞.
Supported by Swiss NF grant 20-55648.98. |
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Keywords: | |
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