Intersection Exponents of Simple Random Walks in Two and Three Dimensions

Abstract

The probability that several independent random walks avoid one another decays algebraically with the number of steps $N\to \infty$ in two and three dimensions, $P_N\sim N^{-\xi /2}$, where the intersection exponent $\xi$ depends on the number of random walks. More generally, one may consider several groups of independent random walks, with intersections allowed within each group and forbidden between different groups. We first study intersection exponents in two dimensions, where our results agree with the known exact formulas and test the numerical approach. In three dimensions, where no general exact expression is known, we determine $\xi$ for a range of cases. For two groups containing $k$ and $m$ random walks, respectively, we obtain the exponents $\xi (k,m)$ for $k=1,2$ and $m=1,2,\dots ,7$. We further extend $m$ to a continuous parameter $\lambda$, allowing us to obtain numerical values for $\xi (k,\lambda )$ for $k=1,2,3$ and $0.25\le \lambda \le 3$. Away from the smallest moment orders, these functions increase with $\lambda$ with decreasing slopes, as expected from the strict concavity of Brownian intersection exponents. We also investigate three-group configurations for several representative cases. The resulting integer and continuous exponents supply numerical values for analytical studies of non-intersecting random paths.