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 $R/\sqrt{t}$. 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 $d\ge 4$.

Figure 1

A realization of two random walks, starting at the origin (a) and at two points separated by distance $R$ (b). The sites visited by both walks are denoted by dark squares.

Figure 2

Scaling functions ${\Phi }_d\left(\xi \right)$ for $d=1,2,3$ (lines) and their asymptotic behaviors in Eqs. (14), (18), and (19) for $0<\xi <1$ (symbols).

Figure 3

Numerical results for the renormalized overlap functions ${\Phi }_d\left(\xi \right)=w_2(0,R,t,t)/w_2(0,0,t,t)$ for $d=1$ (a), $d=2$ (b), $d=3$ (c), and $d=4$ (d). Numerical results were obtained by Monte Carlo simulations of $d$-dimensional random walks on (hyper)cubic lattices with $R$ = 5 (circles), 10 (diamonds), 20 (triangles), and 50 (stars). Each point is an average over 262 144 realizations of random walks up to $t=$ 131 072 steps. The theoretical predictions for the scaling curves in $d=1,2,3$ are shown with solid black lines.