Overlap of two Brownian trajectories: exact results for scaling functions.
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/√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≥4.
Physics Department, Moscow State University, 119991, Moscow, Russia and Department of Applied Mathematics, National Research University Higher School of Economics, 101000, Moscow, Russia.
M V
MV
Tamm
Physics Department, Moscow State University, 119991, Moscow, Russia.
V I
VI
Stadnichuk
Physics Department, Moscow State University, 119991, Moscow, Russia.
A M
AM
Ilyina
Laboratoire de Physique de la Matière Condensée, CNRS-Ecole Polytechnique, F-91128 Palaiseau, France.
D S
DS
Grebenkov