Limiting object for strongly coupled networks beyond mean-field scaling

Determine the limiting object, if any, for the stochastic network dx^i_t = (f_{p_i}(x^i_t) − ∑_{j=1}^n J^{p_i,p_j}_{ij}(n) b_{p_i,p_j}(x^i_t,x^j_t)) dt + σ_{p_i} dW^i_t as n → ∞ when the interaction scaling satisfies n J^{p,q}_{ij}(n) → ∞ and the fastest divergence is of order γ(n), with g_{p,q} = lim_{n→∞} J^{p,q}_{ij}(n)/γ(n).

Background

Under classical mean-field scaling J ∼ 1/n, propagation of chaos leads to McKean–Vlasov limits. The paper departs from this regime by considering stronger coupling where nJ diverges, introducing a divergence scale γ(n) and associated coefficients g_{p,q}. In this setting, standard mean-field theory does not directly provide a limit.

The authors explicitly state that the limiting object is unclear in this strong-coupling regime. They propose a conjectural description in terms of convergence to a balanced manifold, and they present heuristic and partial results in specific cases, but a general characterization of the limit remains unresolved.