Quenching 2D Quantum Gravity

We simulate the Ising model on a set of fixed random φ graphs, which corresponds to a quenched coupling to 2D gravity rather than the annealed coupling that is usually considered. We investigate the critical exponents in such a quenched ensemble and compare them with measurements on dynamical φ graphs, flat lattices and a single fixed φ graph. Submitted to Phys Lett B. 1 Address: Sept. 1993 1994, Permanent Address: Maths Dept, Heriot-Watt University, Edinburgh, Scotland It has recently become possible to calculate the critical exponents for various spin models on a particular class of dynamical connectivity lattices using both the methods of conformal field theory [1] and matrix models [2]. Remarkably, it is even possible to solve the Ising model exactly in the presence of an external field on such a lattice, which has proven impossible on fixed 2D lattices [3]. The key to the calculations, whether in the continuum conformal field theory formalism or the discrete matrix model formalism, lies in the observation that putting the model on a dynamical lattice and allowing it to interact with the lattice is equivalent to coupling the model to 2D quantum gravity. This has the effect of “dressing” the conformal weights ∆0 of the critical continuum theory on a flat 2D lattice, to give new conformal weights ∆ ∆−∆0 = − ξ 2 ∆(∆ − 1), (1) where ξ = − 1 2 √ 3 ( √ 25− c− √ 1− c) (2) and c is the central charge of the theory in question. This, in turn, will effect the critical exponents of theory. These can be calculated from the conformal weights of the energy and spin operators ∆ǫ and ∆σ respectively [4] which give α (the specific heat exponent) and β (the magnetization exponent) α = 1− 2∆ǫ 1−∆ǫ β = ∆σ 1−∆ǫ (3) in the usual fashion. Given α and β from the conformal field theory we can then use the various standard scaling relations to calculate the full array of exponents. For the exactly soluble case of the Ising model on a dynamical lattice the standard scaling relations can be shown to hold, so it is not too great a leap of faith to assume their validity for other models. There is a growing body of numerical evidence confirming the new dynamical exponents in simulations [5, 6, 7], both on dynamical triangulations and their dual dynamical φ graphs with both spherical and toroidal topology lattices . The matrix model approach suggests that any dynamical polygonalization of a surface should give the same exponents for a given model, so there should be universality in this sense. If we think of the numerical simulations purely as an exercise in statistical mechanics it is clear they represent taking an annealed average over the various random graphs on which the spin models live. The partition function of the Ising model on dynamical lattices, for instance, is