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 $d \geq 4$

💡 Research Summary

In this paper the authors address the problem of quantifying the overlap between two independent Brownian walkers (or lattice random walks) that start simultaneously at a distance R from each other. The central observable is the average volume (or number of lattice sites) w₂(R,t) that has been visited by both walkers up to a common observation time t. By expressing the probability that a given site x is visited by a particular walker in terms of the propagator g(x,t|x₀) and the persistence probability q(t) (the probability that a walk never returns to its starting point up to time t), they derive an exact representation for w₂ as a double time integral.

For translationally invariant, time‑reversible walks the propagator is Gaussian, g(x,t|x₀) = (4πDt)^{-d/2} exp