Exact log-odds representation and mean-field criticality of a growing social group model

Abstract

We present an exact analytical reformulation of a growing social group model -- a Hamiltonian-free nonequilibrium process in which a group grows by noisy, consensus-driven admission. Cast as a gradient flow on logarithmic time, the fixed-point structure collapses to a single self-consistent equation: $\arctanh(ϕ^*) = m \cdot \arctanh(αϕ^*)$, where $ϕ$ is the polarization, $α=1-2η$ the evaluation reliability, and $m$ the number of evaluators. The equation has a direct log-odds interpretation: each verdict contributes log-likelihood ratio $2\arctanh(αϕ)$; unanimity accumulates $m$ independent evidence pieces. The dynamics thus constitutes an exact mean-field theory of self-consistent inference, ordering when the collective gain $mα$ overcomes the dilution of growth. We develop a systematic three-layer framework: core theory (Landau-like effective potential, comparison with the mean-field Ising model, and features without equilibrium counterpart), mathematical foundations (criticality from correlated verdicts, Pólya-urn martingale convergence, and an RG-like flow with group size as scale), and complementary perspectives on irreversibility and information geometry. A frozen-$N$ Freidlin--Wentzell quasipotential yields Kramers-type escape estimates for metastable states, while Monte Carlo simulations collapse onto a parameter-free deterministic master curve on logarithmic time. Systematic comparison with the mean-field Ising model reveals shared critical exponents but a nested arctanh structure unique to growth. These results provide a detailed analytical characterization of a minimal model of growth-driven collective behavior and map which elements of the equilibrium critical toolbox -- suitably reinterpreted -- survive without a Hamiltonian.