Fat Fisher zeroes

We show that it is possible to determine the locus of Fisher zeroes in the thermodynamic limit for the Ising model on planar ("fat") phi (4) random gr aphs and their dual quadrangulations by matching up the real part of the hi gh and low temperature branches of the expression for the free energy. The form of this expression for the free energy also means that series expansio n results for the zeroes may be obtained with rather less effort than might appear necessary at first sight by simply reverting the series expansion o f a function g(z) which appears in the solution and taking a logarithm. Unlike regular 2D lattices where numerous unphysical critical points exist with non-standard exponents, the Ising model on planar phi (4) graphs displ ays only the physical transition at c exp(-2 beta) 1/4 and a mirror transit ion at c = -1/4 both with KPZ/DDK exponents (alpha = -1, beta= 1/2, gamma = 2). The relation between the phi (4) locus and that of the dual quadrangul ations is akin to that between the (regular) triangular and honeycomb latti ces since there is no self-duality. (C) 2001 Elsevier Science B.V. All righ ts reserved.