Ising and Potts model coupled to Lorentzian triangulations

Author(s)

José Javier Cerda Hernández

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2014-08-11

Examining board members

Iouri Mikhailovich Soukhov; Luiz Renato Goncalves Fontes; Domingos Humberto Urbano Marchetti; Rodrigo Bissacot Proença; Stefan Zohren

• Advisor: Anatoli Iambartsev; Iouri Mikhailovich Soukhov

Abstract

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. (AU)

• FAPESP's process: 13/06179-2 - Application of transfer matrix method to causal triangulations models
• Grantee: José Javier Cerda Hernández
• Support Opportunities: Scholarships in Brazil - Doctorate (Direct)