Spin systems and boundary conditions on random planar graphs

Thesis

In this thesis we discuss the multi-matrix integral formulation of spin systems on random planar graphs, as a discretised model of 2D Euclidean quantum gravity coupled to matter.

We begin by extending a recent analysis of the q-states Potts model with p < q allowed, equally weighted spins on a random lattice with a connected boundary. We explore the(q < 4, p < q) parameter space of the model, classify solutions in (q, p) that yield finite-sheeted resolvents, and derive the associated critical exponents. For the particular case of q = 3 we find a new solution with p = 3/2, which we conjecture to be the random lattice analogue of the New boundary condition.

Following this we study the Kramers-Wannier dual of the 3-state Potts model on the random lattice. We explicitly construct the known boundary conditions and show that the mixed boundary condition is dual to the New boundary condition, in accordance with the fixed lattice identification. However, we find that the mixed boundary condition of the dual theory, and the corresponding New boundary condition are not described by conventional resolvents.

Finally, we formalise a combinatorial method for calculating the partition function of the 3-states Potts model on a random planar lattice with various boundary conditions imposed. We determine the p = 1, 2 and 3 boundary conditions via this technique, and discuss other types of boundary condition that may be calculated, before turning our attention to the dilute Ising and dilute Potts models, where we compute the p = 1 boundary condition.

Authors: Kulanthaivelu, A

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Matrix model; Spin systems; Dynamical triangulations; 2D quantum gravity; Liouville quantum gravity; Random planar graphs; Random matrices
• Subjects: Quantum gravity