On some aspects of the behaviour of paths and interfaces in discrete and continuous models: random-cluster model, self-repelling polymers and Brownian motion

Abstract

This work is composed of three self-contained parts, where the different models of statistical physics are discussed. In Chapter 1 we discuss the random-cluster model. We present another proof of the well-known fact that for square lattice the critical probability of the random-cluster model $p_{cr}$ is equal to $ rac{sqrt{q}}{1+sqrt{q}}$ for $q in [1,4]$. This proof involves the method of parafermionic observables. In Chapter 3 we study the behaviour of random walks on the square lattice under self-repelling polymers measure. It is a generalisation of a model called self-avoiding walks. We show that, as for self-avoiding walks, self-repelling polymers are sub-ballistic in $Z^d$ with $d ge 2$, i.e that the probability for the walk to go linearly (on the number of steps) far is exponentially small. In the remaining chapter we look at continuous Brownian motion on different three-dimensional spaces. We compare the behaviour of the Brownian motion in the Euclidian space and in the spaces of constant non-zero curvature. Projections of these distributions under certain moment maps corresponds to the Duistermaat-Heckmann measure.