Damaging 2D Quantum Gravity

We investigate numerically the behaviour of damage spreading in a Kauffman cellular automaton with quenched rules on a dynamical φ graph, which is equivalent to coupling the model to discretized 2D gravity. The model is interesting from the cellular automaton point of view as it lies midway between a fully quenched automaton with fixed rules and fixed connectivity and a (soluble) fully annealed automaton with varying rules and varying connectivity. In addition, we simulate the automaton on a fixed φ graph coming from a 2D gravity simulation as a means of exploring the graph geometry. 1 Address Sept. 1993 1994 2 Permanent Address Cellular automata have found applications in many area of physics and other sciences. One particularly intriguing idea was put foward by Kauffman in the 1960s [1], who suggested that a random Boolean network might serve as a model for cell differentiation. This predated the more recent explosion of papers on the subject that was largely inspired by the work of Wolfram [2]. In the Kauffman model each of i = 1, 2, . . . , N lattice sites contains a spin which can take the values ±1 (or equivalently a Boolean variable that can take the values 0, 1) and the spins evolve according to σi(t+ 1) = fi (σj1 (t), σj2 (t), . . . , σjk(t)) (1) where the functions fi are chosen independently and each site i hasK specified inputs sites j1(i), . . . , jk(i). The original biologically motivated model of Kauffman chose the fi randomly at t = 0 with a probability p = 1/2 for fi = +1 and a probability 1 − p = 1/2 for fi = −1. for each site and also chose the j1(i), j2(i), . . . , jk(i) inputs randomly from the N lattice sites at t = 0. This version of the model thus displays quenched randomness as both the functions fi and the input sites are fixed and has been called “infinite dimensional” because of the non-locality of the inputs. The model must display periodic behaviour for finite N as there are only 2 possible spin configurations, so one can enquire about the length and number of limit cycles as well as quantities such as the Hamming distance between different configurations 1 and 2, D12(t), defined as