%0 Journal Article
%T Overlap of two Brownian trajectories: exact results for scaling functions
%A Mikhail V. Tamm
%A Vladimir I. Stadnichuk
%A Alena M. Ilyina
%A Denis S. Grebenkov
%J Physics
%D 2014
%I arXiv
%R 10.1103/PhysRevE.89.042137
%X We consider two random walkers starting at the same time $t=0$ from different points in space separated by a given distance $R$. We compute the average volume of the space visited by both walkers up to time $t$ as a function of $R$ and $t$ and dimensionality of space $d$. For $d<4$, this volume, after proper renormalization, is shown to be expressed through a scaling function of a single variable $R/\sqrt{t}$. We provide general integral formulas for scaling functions for arbitrary dimensionality $d<4$. In contrast, we show that no scaling function exists for higher dimensionalities $d \geq 4$
%U http://arxiv.org/abs/1402.1409v1