Level density distribution for one-dimensional vertex models related to Haldane-Shastry like spin chains

The energy level density distributions of some Haldane-Shastry like spin chains associated with the $A_{N-1}$ root system have been computed recently by Enciso et al., exploiting the connection of these spin systems with inhomogeneous one-dimensional vertex models whose energy functions depend on the vertices through specific polynomials of first or second degree. Here we consider a much broader class of one-dimensional vertex models whose energy functions depend on the vertices through arbitrary polynomials of any possible degree. We estimate the order of mean and variance for such energy functions and show that the level density distribution of all vertex models belonging to this class asymptotically follow the Gaussian pattern for large number of vertices. We also present some numerical evidence in support of this analytical result.