Limit free energy in the bipartite spin glass model

Determine whether, for the bipartite spin glass with two layers of Ising spins and energy H_N^bip(σ) = (1/√N) ∑_{i,j=1}^N W_{i,j} σ_{1,i} σ_{2,j} with i.i.d. standard Gaussian couplings W_{i,j} and configurations σ=(σ_1,σ_2) ∈ {±1}^N × {±1}^N, the free energy F_N^{bip}(β) = (1/N) E log ∑_{σ} exp(β H_N^bip(σ)) converges as N→∞ for fixed inverse temperature β>0; if it converges, identify its limit.

Background

The bipartite model organizes spins into two layers that interact only across layers, with Hamiltonian H_N^bip(σ) = (1/√N) ∑ W_{i,j} σ{1,i} σ{2,j}. Despite its apparent similarity to the Sherrington–Kirkpatrick model, the bipartite model falls outside the currently understood class for which Parisi-type convexity methods apply, because the relevant covariance structure corresponds to the non-convex mapping (x,y)↦xy.

The authors emphasize that even the existence of the large-N limit of the free energy is unknown, and previously proposed generalizations of the Parisi formula fail for this model, underscoring a fundamental gap in our understanding of its thermodynamic limit.