Properties of self-avoiding walks and a stress-energy tensor in the O(n) model

Abstract

This thesis is devoted to the study of 2-dimensional models of statistical mechanics. More precisely, we focus on the loop O(n) model and two classical models which can be realized as its particular cases: the Lenz-Ising model (n = 1) and the self-avoiding walk (n = 0). The main goals are to extend our knowledge about these models and to better understand their connection with the Conformal Field Theory which is conjectured to describe the scaling limits. The results known earlier for the self-avoiding walk on the hexagonal lattice are extended to the self-avoiding walk with integrable weights. A discrete stress-energy tensor in the loop O(n) model is constructed and shown to converge to its continuous counterpart for the Ising model. The endpoint of the self-avoiding walk is shown to be delocalized. The main tools used in the thesis are (para)fermionic observable, Yang-Baxter equation and Kesten's pattern lemma.