Frustrating and Diluting Dynamical Lattice Ising Spins

We investigate what happens to the third order ferromagnetic phase transition displayed by the Ising model on various dynamical planar lattices (ie coupled to 2D quantum gravity) when we introduce annealed bond disorder in the form of either antiferromagnetic couplings or null couplings. We also look at the effect of such disordering for the Ising model on general φ and φ Feynman diagrams. Submitted to Phys Lett B. 1 Address: Sept. 1993 1994, Permanent Address: Maths Dept, Heriot-Watt University, Edinburgh, Scotland In an earlier paper [1] we observed that the Ising antiferromagnet would not undergo a phase transition on dynamical φ or φ planar graphs or on dynamical planar triangulations, essentially because of frustration, but would display a transition on dynamical planar quadrangulations. Here we address a related question, namely: what degree of annealed bond disorder is necessary in order to destroy the ferromagnetic phase transition on a dynamical lattice? For simplicity we consider models in which there are two possible bond factors, +J appearing with probability p and either −J or 0 appearing with probability (1 − p). We consider primarily planar φ and φ graphs, along with planar triangulations and quadrangulations but also look more briefly at general φ and φ graphs of arbitrary topology. We are thus further complicating the annealed connectivity disorder of the Ising model coupled to 2D gravity (ie living on a planar dynamical graph or triangulation) by introducing an additional annealed bond disorder. Such annealed bond disorder has been considered some years ago for the case of fixed regular lattices [2], where it was shown that the critical exponents underwent a Fisher renormalization [3] for models with an initial 1 α > 0. For the two dimensional Ising model it was shown that introducing a finite fraction of either antiferromagnetic or null bonds suppressed the initial phase transition. For the Ising model on the dynamical lattices we consider here with α = −1 the considerations in [2] will not apply, and it is not immediately obvious what behaviour to expect. To investigate the problem we consider the following Ising partition function on an ensemble of planar random graphs G with n vertices.