Fractal depth-first search paths in statistical physics models

We study the fractal properties of depth-first search (DFS) paths in critical configurations of statistical physics models, including the two-dimensional $O(n)$ loop model for various $n$, and bond percolation in dimensions $d = 2$ to $6$. In the $O(n)$ loop model, across both critical and tricritical Potts regimes, the fractal dimension of the DFS path consistently follows $d_{\rm DFS} = 1 + g/8$, where $g$ is the coupling constant in Coulomb gas theory, related to $n$ via $n^2 = 2 + 2 \cos(\pi g/2)$ with $g \in [8/3, 16/3]$. For bond percolation, the DFS path exhibits nontrivial fractal scaling across all studied dimensions. Interestingly, when DFS is applied to the full lattice without any dilution or criticality, the path is still fractal in two dimensions, with a dimension close to $7/4$, but becomes space-filling in higher dimensions. Our results demonstrate that DFS offers a robust and broadly applicable geometric probe for exploring critical phenomena beyond traditional observables.