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.
Riemannian Geometry of Ising Model in the Bethe Approximation
