Fat and thin 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') phi4 random graphs and their dual quadrangulations by matching up the real part of the high- and low-temperature branches of the expression for the free energy. Similar methods work for the mean-field model on generic, 'thin' graphs. Series expansions are very easy to obtain for such random graph Ising models