Direct verification of the discretization link between the asymptotic scattering relation and the hydrodynamic effective-velocity equation

Establish a direct, rigorous derivation showing that, under the linear-soliton-trajectory ansatz Q_j(t) ≈ Q_j(0) + t v_eff(λ_j) with v_eff depending only on the Lax eigenvalue λ_j, the Toda lattice asymptotic scattering relation Q_k(t) − Q_k(0) + 2∑_{i: Q_i(t)<Q_k(t)} log|λ_k−λ_i| − 2∑_{i: Q_i(0)<Q_k(0)} log|λ_k−λ_i| ≈ λ_k t is the discretization of the generalized-hydrodynamics integral equation determining the effective velocity v_eff (equation (1.1) in the paper, with scattering shift s(λ,μ)=2 log|λ−μ| for the Toda lattice).

Background

The paper proves a law of large numbers for soliton trajectories in the Toda lattice at thermal equilibrium, using the asymptotic scattering relation (1.4) and concentration bounds. Physically, v_eff is predicted to satisfy an integral equation (1.1) of generalized hydrodynamics.

A standard heuristic suggests that assuming linear trajectories Q_j(t) with velocity v_eff(λ_j) turns the discrete scattering relation into the hydrodynamic integral equation. The authors rely on a different route (regularization and matrix concentration) and explicitly note they do not know how to verify this heuristic directly.

References

In fact, if one were to assume (1.3) for some function veff only dependent on dj, then a concise heuristic (see [1, Appendix B]) would indicate that (1.4) is a discretization of (1.1). Indeed, this intuition is what led to the predicted form of (1.1) in [41, 5, 10]. Unfortunately, we do not know how to verify this hypothesis directly.

— Effective Velocities in the Toda Lattice (2503.11407 - Aggarwal, 14 Mar 2025) in Section 2.1 (Proof Outline)