On Brownian disconnection exponents
The paper starts with a short, intuitive revision of the derivation of the strictly positive constant \(\gamma\), the disconnection exponent for the two-dimensional Brownian motion. The right-hand side of the well-known double inequality \({1\over 2\pi} \leq \gamma \leq {1\over 2}\) is improved by the estimate \(\gamma \leq {1\over 2} - {1\over 2} \biggl( {\log 2\over \pi}\biggr)^2 < 0.47566\). A comprehensive outline of the ideas and methods involved in the proof of this new upper bound for \(\gamma\) is given separately from the technical details of the proof. Furthermore, the arguments of this proof are adapted to verify the following upper bound for the disconnection exponent \(\gamma_n\) (corresponding to the event that the union of \(n\) independent planar Brownian motion paths, stopped at their respective hitting times of a large circle, does not disconnect zero from infinity): \(\gamma \leq {n \over 2} - {(\log 2)^2\over 2\pi^2 n}\) for all \(n > 1\). A well written derivation of \(\gamma_n\) precedes the proof of this second proposition. Using the similarities between disconnection and intersection exponents, some of the previously used arguments for disconnection exponents are modified to also provide some improved upper bounds for intersection exponents of planar Brownian motion. Several interesting conjectures regarding disconnection and intersection exponents are motivated throughout the paper.
scientific article; zbMATH DE number 846993