Asymptotic form of level density distributions for a class of inhomogeneous 1D vertex models

Level density distributions for some Haldane-Shastry like spin chains associated with AN−1 root system have been computed recently by Enciso et al., using the connection of these spin chains with inhomogeneous one-dimensional vertex models. The energy functions of such vertex models are given by specific polynomials of first or second degree. Here we consider a much broader class of one-dimensional vertex models whose energy functions are given by arbitrary polynomials of any possible degree and show that the level densities for this class of vertex models asymptotically follow the Gaussian distribution for large number of vertices.