  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.
