A Kertesz line on planar random graphs

The identification of phase transition points, ß_c, with the percolation thresholds of suitably defined clusters of spins has proved immensely fruitful in many areas of statistical mechanics. Some time ago, Kertesz suggested that such percolation thresholds for models defined in field might also have measurable physical consequences for regions of the phase diagram below ß_c,giving rise to a 'Kertesz line' running between ß_c and the bond percolation threshold, ß _p, in the H, ß plane. Although no thermodynamic singularities were associated with this line, it could still be divined by looking for a change in the behaviour of high-field series for quantities such as the free energy or magnetization. Adler and Stauffer did precisely this for the regular square lattice and simple cubic lattice Ising models and did, indeed, find evidence for such a change in high-field series around ß_p. Since there is a general dearth of high-field series, there has been no other work along these lines. In this paper, we use the solution of the Ising model in field on planar random graphs by Boulatov and Kazakov to carry out a similar exercise for the Ising model on random graphs (i.e. coupled to 2D quantum gravity). We generate a high-field series for the Ising model on F ⁴ random graphs and examine its behaviour for evidence of a Kertesz line.