Divergence of the limit free energy beyond the realizability frontier

Determine whether, for the generalized Lotka–Volterra stochastic differential equation with deformed GOE interaction matrix Σ_n = (κ/√n) W_n + α (11^T)/n and associated conditional Gibbs measure with Hamiltonian H_n, the normalized free energy \widetilde F_n = (1/n) log ∫_{R_+^n} exp(H_n(x)) μ_β^{⊗n}(dx) diverges to +∞ as n → ∞ whenever the large‑n limit λ_+(κ,α) of λ_+^{max}(Σ_n) exceeds 1 (i.e., λ_+(κ,α) > 1), equivalently showing that the variational limits given in Theorem \ref{F->P} become infinite in this regime.

Background

The paper studies the invariant distribution of a generalized Lotka–Volterra SDE with symmetric random interactions modeled by an additively deformed GOE matrix Σn = (κ/√n) W_n + α (11^T)/n. Under the condition λ+(κ,α) < 1—where λ+(κ,α) is the almost sure limit of the nonnegative spectral radius λ+^{max}(Σ_n)—the authors derive a Parisi-type variational formula for the large‑n limit of the normalized free energy \widetilde F_n associated with the conditional Gibbs measure.

They note that when λ+(κ,α) > 1, the Gibbs weight is not integrable for large n, suggesting a change of behavior. Immediately after presenting Theorem \ref{F->P}, the authors state a conjecture that, beyond this realizability frontier (λ+ > 1), the limits in their variational expressions are infinite, which would imply divergence of the free energy. Establishing this rigorously would clarify the phase boundary between well-defined and divergent regimes of the model.