Spin-glass phase in the infinite-parameter limit of the syntax network

Determine whether, in the limit as the number of syntactic parameters N tends to infinity, the glassy regime observed in the recurrent binary network model of syntactic evolution—with random pairwise interactions parameterized by an asymmetry angle φ and implication constraints scaled by ζ—constitutes a bona fide spin-glass phase.

Background

The paper models the evolution of syntactic parameters as a recurrent network of binary variables influenced by two types of interactions: (i) explicit, inherently asymmetric implicational constraints scaled by ζ, and (ii) numerous weak, statistically modeled pairwise interactions whose degree of symmetry is controlled by an angle φ. Using a dataset of 94 parameters across 58 languages, the authors observe a glassy regime—metastable states and slow dynamics—when interactions are sufficiently symmetric (φ below roughly 30°).

This behavior is reminiscent of spin-glass models from statistical physics (e.g., the Sherrington–Kirkpatrick model) with asymmetric couplings. While the finite-size system (N=94) exhibits clear glassy dynamics, it remains unresolved whether, in the thermodynamic limit N→∞, this regime corresponds to a true spin-glass phase with associated phase-transition properties.