Percolation transition of strongly connected clusters in finite dimensions and on complete graphs
Abstract
We study the percolation of strongly connected clusters (SCCs), in which sites are mutually reachable through directed paths, in systems with randomly oriented bonds by extensive simulations on hypercubic lattices from dimensions d=2 to $7$ and on complete graphs. Below the upper critical dimension du=6, critical SCCs exhibit a nontrivial fractal dimension dSCC with a nonmonotonic dependence on dimension: dSCC reaches a maximum at d=4, in contrast to the fractal dimension of ordinary percolation clusters, which increases monotonically with d. The SCC size distribution follows ∼ s−τSCC, where the Fisher exponent is consistent with the hyperscaling relation τSCC=1+d/dSCC and, unlike dSCC itself, increases monotonically with d. For d≥ du, mean-field behavior is recovered with dSCC/d=1/3, consistent with complete-graph results. However, unlike finite-dimensional lattices, complete graphs exhibit a double-scaling structure in the SCC size distribution: large SCCs are governed by the mean-field exponent τSCC=4, while small SCCs follow a distinct power law with exponent τ'=1. At du=6, logarithmic corrections are also observed. These results show that SCCs remain well-defined fractal objects across dimensions, exhibiting both a nontrivial dimensional dependence and a nontrivial crossover from finite-dimensional to mean-field behavior.
Versions
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BibTeX
@misc{wang2026percolation,
title = {{Percolation transition of strongly connected clusters in finite dimensions and on complete graphs}},
author = {Qi Wang and Ming Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2605.16987},
eprint = {2605.16987},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.16987},
}
RIS
TY - UNPB
TI - Percolation transition of strongly connected clusters in finite dimensions and on complete graphs
AU - Qi Wang
AU - Ming Li
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2605.16987
UR - https://arxiv.org/abs/2605.16987
AB - We study the percolation of strongly connected clusters (SCCs), in which sites are mutually reachable through directed paths, in systems with randomly oriented bonds by extensive simulations on hypercubic lattices from dimensions d=2 to $7$ and on complete graphs. Below the upper critical dimension d_u=6, critical SCCs exhibit a nontrivial fractal dimension d_SCC with a nonmonotonic dependence on dimension: d_SCC reaches a maximum at d=4, in contrast to the fractal dimension of ordinary percolation clusters, which increases monotonically with d. The SCC size distribution follows ∼ s^−τ_SCC, where the Fisher exponent is consistent with the hyperscaling relation τ_SCC=1+d/d_SCC and, unlike d_SCC itself, increases monotonically with d. For d≥ d_u, mean-field behavior is recovered with d_SCC/d=1/3, consistent with complete-graph results. However, unlike finite-dimensional lattices, complete graphs exhibit a double-scaling structure in the SCC size distribution: large SCCs are governed by the mean-field exponent τ_SCC=4, while small SCCs follow a distinct power law with exponent τ'=1. At d_u=6, logarithmic corrections are also observed. These results show that SCCs remain well-defined fractal objects across dimensions, exhibiting both a nontrivial dimensional dependence and a nontrivial crossover from finite-dimensional to mean-field behavior.
ER -