## Abstract

The properties of the high-field polynomials L$_{n}$(u), where u = exp[-4J/(k$_{\text{B}}$T)], are investigated for the Bethe approximation of the spin $\frac{1}{2}$ Ising model on a lattice which has a coordination number q. (The polynomials L$_{n}$(u) are essentially lattice gas analogues of the Mayer cluster integrals b$_{n}$(T) for a continuum gas.) In particular, a contour integral representation for L$_{n}$(u) is established by applying the Lagrange reversion theorem to the implicit equation of state for the Bethe approximation. Various saddle-point methods are then used to analyse the behaviour of the integral representation as n $\rightarrow \infty $. In this manner, asymptotic expansions for L$_{n}$(u) are obtained which are uniformly valid in the intervals 0 < u $\leq $ u$_{\text{c}}$ and u$_{\text{c}}$ $\leq $ u < 1, where u$_{\text{c}}$ = [($\sigma $ - 1)/($\sigma $ + 1)]$^{2}$ is the critical value of the variable u, $\sigma \equiv $ (q - 1) and $\sigma $ > 1. These expansions involve the Airy function Ai (z) and its first derivative. The high-field polynomial L$_{n}$(u) is found to have a trivial zero at u = 0, and n - 1 simple non-trivial zeros {u$_{\nu}$($\sigma $, n); $\nu $ = 1, 2, $\ldots $, n -1} which are all located in the real interval u$_{\text{c}}$ < u < 1. An asymptotic expansion for u$_{\nu}$($\sigma $, n) in powers of n$^{-\frac{2}{3}}$ is derived from the uniform asymptotic representation for L$_{n}$(u) which is valid in the interval u$_{\text{c}}\leq $ u < 1. It is also shown that the limiting density of the zeros {u$_{\nu}$($\sigma $, n); $\nu $ = 1, 2, $\ldots $, n - 1} as n $\rightarrow \infty $ is given by the simple formula $\rho $($\sigma $, u) = n(2$\pi $)$^{-1}$($\sigma $+1)u$^{-1}$(u - u$_{\text{c}}$)$^{\frac{1}{2}}$(1-u)$^{-\frac{1}{2}}$, where u$_{\text{c}}$ < u < 1. Finally, the asymptotic properties of the Bethe polynomial L$_{n}$(u) are determined in the mean-field limit q $\rightarrow \infty $ and J $\rightarrow $ 0 with qJ $\equiv $ J$_{0}$ held constant.