Correlation Functions in the Multiple Ising Model Coupled to Gravity

Abstract

The model of p Ising spins coupled to 2d gravity, in the form of a sum over planar phi-cubed graphs, is studied and in particular the two-point and spin-spin correlation functions are considered. We first solve a toy model in which only a partial summation over spin configurations is performed and, using a modified geodesic distance, various correlation functions are determined. The two-point function has a diverging length scale associated with it. The critical exponents are calculated and it is shown that all the standard scaling relations apply. Next the full model is studied, in which all spin configurations are included. Many of the considerations for the toy model apply for the full model, which also has a diverging geometric correlation length associated with the transition to a branched polymer phase. Using a transfer function we show that the two-point and spin-spin correlation functions decay exponentially with distance. Finally, by assuming various scaling relations, we make a prediction for the critical exponents at the transition between the magnetized and branched polymer phases in the full model.

29 pages, LaTeX, uses epsf macro, 5 figures

References in corpus (14)

• On the fractal structure of two-dimensional quantum gravity
• Scaling in quantum gravity
• Exact Solution of the O(n) Model on a Random Lattice
• More on the exact solution of the O(n) model on a random lattice and an investigation of the case |n|>2
• Scaling and the Fractal Geometry of Two-Dimensional Quantum Gravity
• Multi-spin systems on a randomly triangulated surface
• A Solvable 2D Quantum Gravity Model with $\GAMMA >0$
• A Universal Fractal Structure of 2D Quantum Gravity for c > 1
• New critical phenomena in 2d quantum gravity
• Two Ising Models Coupled to 2-Dimensional Gravity
• Multiple Ising Spins Coupled to 2d Quantum Gravity
• Grand-Canonical Ensemble of Random Surfaces with Four Species of Ising Spins
• From Trees to Galaxies: The Potts Model on a Random Surface
• The phase diagram of an Ising model on a polymerized random surface

Cited by in corpus (7)

• Quantum geometry of 2d gravity coupled to unitary matter
• A scenario for the c>1 barrier in non-critical bosonic strings
• Geometrical Interpretation of the KPZ Exponents
• A c=1 phase transition in two-dimensional CDT/Horava-Lifshitz gravity?
• Gravity and Random Surfaces on the Lattice - A Review
• Spin-spin correlation functions of spin systems coupled to 2-d quantum gravity for $0<c<1$
• The Hausdorff dimension in polymerized quantum gravity