Hamilton–Jacobi limit for the enriched free energy in the bipartite model (Conjecture validity)

Establish whether the enriched free energy F_N(t,q) for the bipartite spin glass with two layers and Hamiltonian H_N^bip(σ) = (1/√N) ∑_{i,j} W_{i,j} σ_{1,i} σ_{2,j} converges, as N→∞, to the unique viscosity solution f of the Hamilton–Jacobi equation ∂_t f(t,q) − ∫_0^1 (∂_{q_1} f(t,q,u))(∂_{q_2} f(t,q,u)) du = 0 on ℝ_+ × Q^2 with initial condition f(0,·) = ψ, where Q^2 denotes pairs of non-decreasing paths q=(q_1,q_2): [0,1]→[0,∞), ∂_{q_d} denotes the L^2-derivative along path q_d, and ψ(q) = F_1(0,q).

Background

To address non-convex covariance structures, the authors introduce an enriched free energy F_N(t,q) depending on a pair of path parameters q=(q_1,q_2) that encode an ultrametric external field, and propose a Hamilton–Jacobi PDE whose solution should describe the large-N limit in the bipartite case.

Because the nonlinearity (x,y)↦xy is neither convex nor concave, standard variational representations are unavailable. The authors prove partial results: any subsequential limit satisfies the PDE almost everywhere, and the viscosity solution furnishes a lower bound. However, the main conjecture that F_N(t,q) converges to this solution remains unresolved.