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 $n^2 = 2 + 2 \cos(πg/2)$ 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.

💡 Research Summary

This paper presents a novel investigation into the fractal geometry inherent to the paths generated by the Depth-First Search (DFS) algorithm when applied to critical configurations of statistical physics models. The central finding is that these algorithmically defined paths themselves exhibit fractal scaling with dimensions that are intimately connected to universal critical exponents, suggesting DFS as a potent geometric probe for critical phenomena.

The study focuses on three primary systems. First, in the two-dimensional O(n) loop model (which corresponds to critical and tricritical Q-state Potts models with Q=n²), the fractal dimension d_DFS of the DFS path—extracted from scaling of the maximum path length ℓ_max ~ L^{d_DFS}—is measured via extensive Monte Carlo simulations. Strikingly, across both the critical (x-) and tricritical (x+) branches and for all values of n (and thus Q), the numerical estimates for d_DFS show excellent agreement with the formula d_DFS = 1 + g/8. Here, g is the Coulomb-gas coupling constant related to Q by Q = 2 + 2cos(πg/2). This formula was originally derived for the fractal dimension of the external perimeter (d_EP) of Fortuin-Kasteleyn (FK) clusters in the critical Potts model. The paper argues that in sparse (x-) FK clusters, the DFS path tends to trace the outer boundary while skipping fine-scale fjords, explaining its alignment with d_EP. In dense (x+) clusters, the absence of deep fjords causes the hull and external perimeter to coincide, leading to the same agreement.

Second, the research extends to bond percolation on hypercubic lattices from dimensions d=2 to 6. The DFS path remains fractal in all dimensions, displaying non-trivial scaling with d_DFS < d. This finding is significant because the concept of an “external perimeter” is not well-defined in high-dimensional percolation, yet DFS still provides a well-defined geometric observable that captures fractal properties.

Third, a particularly intriguing experiment applies DFS to a full lattice (i.e., a completely connected graph without any dilution or criticality). In two dimensions, the DFS path is still found to be fractal with a dimension close to 7/4 (~1.75), reminiscent of the external perimeter dimension in 2D percolation. However, in dimensions d > 2, the path becomes space-filling, i.e., d_DFS = d. This highlights a fundamental difference in the intrinsic geometric structure generated by the DFS algorithm between two and higher dimensions.

The paper provides an interpretative argument that the DFS path may be related to the dual of the hull of the FK cluster. This perspective offers a consistent framework for understanding why d_DFS matches the external perimeter dimension rather than the hull dimension itself (d_hull = 1 + 2/g).

In conclusion, this work demonstrates that the DFS algorithm is not merely a computational tool for cluster identification but also a source of geometric observables that reflect universal critical properties. It establishes DFS paths as fractal objects whose dimensions serve as robust probes, potentially uncovering new universal features beyond traditional quantities like cluster size or hull geometry. This bridges computer science (graph algorithms) and statistical physics (critical phenomena) in a unique and fruitful way.