Asymptotic behaviour of the Mayer cluster sums for the Ising model in the Bethe approximation

Abstract

The properties of the high-field polynomials L n (u) , where u = exp [ -4 J / ( k B T )] are investigated for the Bethe approximation of the spin 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->∞. In this manner, asymptotic expansions for L n [ u ) are obtained which are uniformly valid in the intervals 0 < u ⩽ u c and u c ⩽ u < 1, where u c = [(σ-1 )/(σ + l)] 2 is the critical value of the variable u , σ ≡ (q-1) and σ > 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 v (σ,n); v = 1, 2, ..., n — 1} which are all located in the real interval u c < u < 1. An asymptotic expansion for u v (σ, n) in powers of n 2/3 is derived from the uniform asymptotic representation for L n ( u ) which is valid in the interval u c ⩽ u < 1. It is also shown that the limiting density of the zeros { u v ( σ, n ); v = 1 ,2 ,..., n -1} as n → ∞ is given by the simple formula ρ ( σ , u ) = n ( 2 π ) − 1 ( σ + 1 ) u − 1 ( u − u c ) 1 / 2 ( 1 − u ) − 1 / 2 where u c < u < 1. Finally, the asymptotic properties of the Bethe polynomial L n ( u ) are determined in the mean-field limit q → ∞ and J → 0 with qJ = J 0 held constant.