Fractal depth-first search paths in statistical physics models

TL;DR

This work addresses how depth-first search (DFS) traces through critical configurations to reveal fractal geometry beyond traditional observables. By studying DFS paths on FK clusters in the 2D $O(n)$ loop model and on bond percolation clusters in $d=2$–$6$, the authors relate the DFS fractal dimension to the Coulomb-gas coupling via $d_{ m DFS}=1+g/8$ with $n^2=2+2\cos(\pi g/2)$, $g\in[8/3,16/3]$, and show that DFS remains fractal in higher dimensions while full lattices in 2D yield $d_{ m DFS}\approx 7/4$ and higher dims are space-filling. They demonstrate hyperscaling for the DFS-length distribution, establish a duality between DFS paths and hulls of FK clusters, and reveal DFS as a robust geometric probe across models. These results broaden the toolkit for analyzing critical phenomena by linking DFS geometry to established fractal exponents and dual geometric structures, with potential connections to growth processes and random-walk universality.

Abstract

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 $n^2 = 2 + 2 \cos(πg/2)$ 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.

Figures (6)

• Figure 1: (Color online) A schematic illustration of BFS and DFS on a square lattice. (a) A $4\times4$ square lattice with sites labeled from $0$ to $15$. (b) A possible BFS spanning tree starting from site $1$. The tree depth is $5$, representing the shortest path from the initial site $1$ to the farthest site $15$. (c) A possible DFS spanning tree starting from site $1$. The tree depth is $9$. In general, the DFS path is longer and more circuitous than the BFS path.
• Figure 2: (Color online) Finite-size scaling of the maximum DFS path length $\ell_{\rm max}$ in the $O(n)$ loop model for various values of $n$. The solid lines represent the fractal dimensions given by Eq. (\ref{['eq-dep']}). Note that Eq. (\ref{['eq-dep']}) was originally derived for the external perimeter in the critical Potts model; here, the scaling of the DFS path is also consistently described by the same expression for both the critical and tricritical Potts models.
• Figure 3: (Color online) Numerical results for the fractal dimension $d_{\rm DFS}$ of the $O(n)$ loop model as a function of the Coulomb-gas coupling strength $g$. The red and blue regions correspond to the $x_-$ and $x_+$ branches, respectively. The DFS and hull dimensions are given by Eqs. (\ref{['eq-dep']}) and (\ref{['eq-dhull']}) (red and green lines). The external-perimeter dimension coincides with $d_{\rm DFS}$ in the critical Potts model ($x_-$ branch) and with $d_{\rm hull}$ in the tricritical Potts model ($x_+$ branch). The scattered points represent our numerical fit results for $d_{\rm DFS}$, which are consistent well with Eq. (\ref{['eq-dep']}).
• Figure 4: (Color online) The number density $P(\ell,L)$ of the length of the longest DFS path in each cluster for the $O(n)$ loop model with different values of $n$ and linear size $L$. Panels (a) and (b) correspond to the $x_-$ branch, while (c) and (d) correspond to the $x_+$ branch. The dashed lines represent the Fisher exponent predicted by hyperscaling relation, $\tau_{\rm DFS} = 1 + d/d_{\rm DFS}$. The upper insets show the length-resolved count $P_1(\ell,L)$ of DFS paths in the largest cluster, which also exhibits a power-law form, $P_1(\ell,L) \sim \ell^{\tau_1}$, with $\tau_1 = d_f/d_{\rm DFS} - 1$. The lower insets show the scaling collapse plots of $\ell^{\tau_{\rm DFS}} P(\ell,L)$ versus $\ell/L^{d_{\rm DFS}}$, with $d_{\rm DFS}=1+g/8$.
• Figure 5: (Color online) Finite-size scaling of the maximum DFS path length $\ell_{\rm max}$ in the bond percolation model for different spatial dimensions $d$. In all dimensions, a clear power-law scaling $\ell_{\rm max} \sim L^{d_{\rm DFS}}$ is observed, with a $d$-dependent fractal dimension $d_{\rm DFS}$. The solid lines represent the fitting results based on the scaling ansatz given in Eq. (\ref{['eq-fss']}) (Table \ref{['tab2']}). The inset is a rescaled plot of the data for $d=6$, where $\ell_{\rm max}/L^2$ is plotted against $\ln L + d_0$ (with $d_0 = -0.6$) in log-log coordinates. The apparent straight-line behavior with slope $4/21$ demonstrates a multiplicative logarithmic correction of the form $\ell_{\rm max}/L^2 \sim (\ln L + d_0)^{4/21}$.