TY - JOUR
T1 - Overlap of two Brownian trajectories
T2 - Exact results for scaling functions
AU - Tamm, M. V.
AU - Stadnichuk, V. I.
AU - Ilyina, A. M.
AU - Grebenkov, D. S.
PY - 2014/4/21
Y1 - 2014/4/21
N2 - 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.
AB - 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.
U2 - 10.1103/PhysRevE.89.042137
DO - 10.1103/PhysRevE.89.042137
M3 - Article
AN - SCOPUS:84899761756
SN - 1539-3755
VL - 89
JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
IS - 4
M1 - 042137
ER -