Ising and Potts model coupled to Lorentzian triangulations - Cerda Hernández, José Javier (2014)
The main objective of the present thesis is to investigate: What are the properties of the Ising and Potts model coupled to a CDT emsemble? For that objective, we used two methods: (1) transfer matrix formalism and Krein-Rutman theory. (2) FK representation of the q -state Potts model on CDTs and dual CDTs. Transfer matrix formalism permite us to obtain spectral properties of the transfer matrix using the Krein-Rutman theorem [KR48] on operators preserving the cone of positive func- tions. This yields results on convergence and asymptotic properties of the partition function and the Gibbs measure and allows us to determine regions in the parameter quarter-plane where the free energy converges. Second methods permite us to determine a region in the quadrant of parameters , > 0 where the critical curve for the classical model can be located. We also provide lower and upper bounds for the innite-volume free energy. Finally, using arguments of duality on graph theory and hight-T expansion we study the Potts model coupled to CDTs. This approach permite us to improve the results obtained for Ising model and obtain lower and upper bounds for the critical curve and free energy. Moreover, we obtain an approximation of the maximal eigenvalue of the transfer matrix at lower temperature.