The ising model on a random planar lattice: The structure of the phase transition and the exact critical exponents

Abstract

We investigate the critical properties of a recently proposed exactly soluble Ising model on a planar random dynamical lattice representing a regularization of the zero-dimensional string with internal fermions. The sum over all lattices gives rise to a new quantum degree of freedom - fluctuation of the metric. The whole system of critical exponents is found: α = −1, β = 1 2 , γ = 2, δ = 5, ν· D = 3. To test the universality we have used the planar graphs with the coordination number equal to 4 (θ 4 theory graphs) as well as with the equal to 3 (θ 3 theory graphs or triangulations). The critical exponents coincide for both cases.