Fractal dimensions of paths on three-dimensional random-walk traces

Source abstract

We study four paths that join the starting site to the final position of a simple random walk of $N=L^2$ steps on the infinite cubic lattice. The paths are selected on the walk trace---the graph of visited sites and traversed bonds---by different rules: the undirected shortest path $S_1$, the directed shortest path $S_2$, the time-respecting shortest path $S_3$, and the depth $D_e$ of the final site in a depth-first-search tree. Monte Carlo simulations for $L$ up to 32768 give mean lengths scaling as $L^{d_O}$, with $d_{S_1}=1.48005(9)$, $d_{S_2}=1.49868(5)$, $d_{S_3}=1.51205(5)$, and $d_{D_e}=1.7099(7)$, so that $d_{S_1}<d_{S_2}<d_{S_3}<d_{\rm LE}<d_{D_e}$, where $d_{\rm LE}=1.62400(5)$ is the known loop-erased-random-walk exponent. Imposing traversal directions increases the shortest-path exponent, and imposing time order increases it further; these constraints therefore change the scaling exponent, not only the amplitude. For every path, the standard deviation scales with the same exponent as the mean, so it is scale-invariant, unlike the visited-site mass; the rescaled length distributions collapse onto a finite-width scaling function. No analytical theory currently predicts these exponents; our results provide precise numerical benchmarks for future studies of path geometry on random-walk traces.