Percolation in the loop O(1) model

Specialeforsvar: Boris Bolvig Kjær

Titel: Percolation in the loop O(1) model; Phase transition in the Ising model as seen by uniform even subgraphs

Abstract: The Ising model is the most well known mathematical model of ferromagnetism in condensed matter enabling rigorous mathematical analysis of phase transitions. Elementary phenomena pertinent to phase transition are described below, and proof is given for the classical result of Peierls on the existence of a non-trivial phase in the Ising model. The focus of this thesis is on graphical representations of the Ising model and the relation to percolation theory. Special attention is paid to the so-called loop O(1) model about which several new results are presented. The main result is the statement that the so-called random current model as well as the loop O(1) model corresponding to a super-critical Ising model are always at least critical, meaning that the twopoint correlation functions exhibit decay which is at most polynomial in the graph distance. Uniform even subgraphs play a critical part in the study of the loop O(1) model and are characterised by the Haar measure on the group of even subgraphs. In the infinite setting, this is a key insight when proving percolation on Z d, for d ≥ 3, a result not previously known. This result is extended to proving the existence of a non-trivial phase of the loop O(1) model, also an original contribution.

Vejledere: Albert H. Werner, Frederik Ravn Klausen
Censor:      Poul Hjorth, DTU