Composite-Operator Scaling on Triadic Hypergraphs: Formation Transitions in Multi-Agent Architectures with Three-Body Coupling

Abstract

We study phase transitions on dynamic triadic hypergraphs, in which a continuous formation field evolves under stochastic Ginzburg--Landau dynamics with a cubic three-body coupling $g_{\ddot{I}}„\ddot{I}{•}_i\ddot{I}{•}_j\ddot{I}{•}_k$, while a discrete opinion variable $s_i\in \{-1,+1\}$ undergoes Kawasaki exchange under a Hamiltonian with pairwise alignment and an irreducible three-body energy $-\hat{I}{»}_{\ddot{I}}„{\prod }_{a\in \ddot{I}„}{s}_a$. Near the formation critical point the cubic coupling is subleading and the transition remains continuous, controlled at leading order by a pairwise Ising baseline with renormalized coupling $J_{\mathrm{eff}}=J+{\hat{I}}^{3}w$. The dominant observable is the triadic formation correlator $\hat{I}{¨}_{\mathrm{form}}\equiv ⟨\ddot{I}{•}_i\ddot{I}{•}_j\ddot{I}{•}_k⟩$, a $k=3$ composite operator built over the underlying ${\mathbb{Z}}_2$-symmetric order parameter. Composite-operator scaling yields the effective exponents ${\hat{I}}_{\mathrm{TF}}^2=3/2$ and ${\hat{I}}_{\mathrm{TF}}^3=-1$. The susceptibility conjugate to $\hat{I}{¨}_{\mathrm{form}}$ vanishes at the critical temperature $T_c$ rather than diverging, in contrast to the divergence characterizing scalar (pairwise) order parameters. The exact partition function of the minimal triad on $\{-1,+1{\}}^3$ identifies a crossover scale $T^{\ast }=4J_{\mathrm{eff}}/\ln 3$. A field-theoretic two-point function argument reproduces the same vanishing susceptibility. Restoring the three-body coupling ($\hat{I}»\ne 0$) makes the transition first-order, with a critical endpoint at $\hat{I}»=0$. The exponent relations ${\hat{I}}_{\mathrm{TF}}^2=3{\hat{I}}_{\mathrm{Ising}}^2$ and ${\hat{I}}_{\mathrm{TF}}^3={\hat{I}}_{\mathrm{Ising}}^3-4{\hat{I}}_{\mathrm{Ising}}^2$ hold exactly on dense hypergraphs via cluster decomposition, and the vanishing-susceptibility signature persists for $d\ge 3$ but fails in $d=2$. A Mori--Zwanzig memory kernel yields a continuously tunable dynamical exponent $z_{\mathrm{TF}}$, completing the composite-operator scaling regime.

v2: Retitled. Scope and framing corrected. Added beyond-mean-field exponent relations, a first-order/critical-endpoint phase diagram, and an appendix with Monte Carlo simulations (with public link to code)