  In recent years, a lot of effort has been put in describing the hydrodynamic
behavior of integrable systems. In this paper, we describe such picture for the
Volterra lattice. Specifically, we are able to explicitly compute the
susceptibility matrix and the current-field correlation matrix in terms of the
density of states of the Volterra lattice endowed with a Generalized Gibbs
ensemble. Furthermore, we apply the theory of linear Generalized Hydrodynamics
to describe the Euler scale behavior of the correlation functions. We
anticipate that the solution to the Generalized Hydrodynamics equations
develops shocks at $\xi_0=\frac{x}{t}$; so this linear approximation does not
fully describe the behavior of correlation functions. Intrigued but this fact,
we performed several numerical investigations which show that, exactly when the
solution to the hydrodynamic equations develops shock, the correlation
functions show an highly oscillatory behavior. In view of this empirical
observation, we believe that at this point $\xi_0$ the diffusive contribution
are not sub-leading corrections to the ballistic transport, but they are of the
same order.
