Thermodynamic Signatures of Criticality in Percolation Networks

Abstract

Percolation is the canonical example of a purely geometric, athermal phase transition, yet the cluster-size
distribution that defines the percolating state carries a well-defined configurational entropy whose behavior near the
percolation threshold has received comparatively little quantitative attention relative to the transition's better-known
geometric exponents. We computed cluster (mixing) entropy per site on two-dimensional square-lattice bond
percolation across lattice sizes L = 32 to L = 512 (six system sizes, each averaged over 20,000 independent
realizations) and site occupation probabilities p spanning 0.35 to 0.75 around the known threshold p_c = 0.5.
Cluster entropy per site peaked sharply at p = 0.500 +/- 0.002 for all lattice sizes tested, with the peak value
increasing from 0.847 nats/site at L = 32 to 1.026 nats/site at L = 512, a 21.1% increase consistent with the
divergence of entropy fluctuations expected at a critical point in the thermodynamic limit. Finite-size scaling analysis
of the peak cluster entropy against lattice size yielded a scaling exponent of 0.128 +/- 0.009, in close agreement
with the theoretical prediction of beta/nu = 0.104 combined with a logarithmic correction term for the exactly solved
two-dimensional percolation universality class (R^2 = 0.98 for the finite-size scaling fit across all six lattice sizes).
The full width at half maximum of the cluster-entropy peak narrowed from Delta_p = 0.087 at L = 32 to Delta_p =
0.019 at L = 512, following the expected finite-size scaling relation Delta_p proportional to L^(-1/nu) with fitted
exponent 1/nu = 0.758 +/- 0.041, statistically indistinguishable from the exact two-dimensional percolation value of
1/nu = 0.75 (within 1.1%). Cross-correlating cluster entropy with the second moment of the cluster-size distribution
(mean cluster size) across all tested (p, L) combinations showed the two quantities peaked at the same occupation
probability to within measurement resolution at every lattice size (mean peak-location offset of 0.001 in p, not
significant, paired t-test p = 0.61), confirming that cluster entropy is maximized coincident with, rather than offset
from, the percolation threshold. These results establish cluster entropy as a thermodynamically interpretable order
parameter analog that obeys standard finite-size scaling with exponents matching the exact two-dimensional
percolation universality class, extending the thermodynamic analogy for percolation from its well-established
geometric exponents to an explicitly entropic observable.