Generalized Hydrodynamics for the Volterra lattice: ballistic and non-ballistic behavior of correlation functions

The paper’s Theorem 2.2 expresses the susceptibility matrix C and charge–current matrix B in terms of the Volterra density of states σ_{β,V}, the dressing operator (1−β T_{ρ_{β,V}})^{-1}, and veff = [w^2]_{dr}/[1]_{dr}. The candidate solution reproduces the same dressing-based route: (i) links cumulants to free-energy derivatives, (ii) passes to the large-N variational principle, (iii) derives and linearizes the Euler–Lagrange equation, and (iv) evaluates second derivatives in dressed form to obtain the same bilinear/trilinear identities, including B_{0,n}=−(1/2)C_{n,1}. The only substantive issue is a minor normalization lapse in the model’s notation for q_n and σ (a stray factor κ in q_n and an ambiguous use of σ versus σ_{β,V}); once corrected, it matches the paper’s formulas exactly. The paper’s argument is coherent and complete for Theorem 2.2, relying on established free-energy differentiation identities and a large-deviation/minimization framework, with technical lemmas provided (or cited) for current identities and differentiability (e.g., Lemma 3.2 and Theorem 4.5).