Negative dependence in discrete probability :|notions, models and consequences

Abstract

In this thesis we study the negative dependence properties of random cluster measure. First, we introduce the notions of negative correlation and how they relate to each other. We observe that this important notion is difficult to confirm. The uniform spanning tree measure and the uniform spanning forest measure is the measures that we mainly focus as the crucial examples that shows negative dependence properties. Later, we focus on the properties of the random cluster measure which generalizes the uniform spanning tree and uniform spanning forest measures. We prove negative edge dependence of the random cluster measure on the complete graph, giving a partial solution to a well known conjecture of Grimmett, Winkler and Wagner. We than consider a natural problem concerning correlations between collection of connectivity events in graphs with respect to the random cluster measure and relate this natural problem to the Grimmett, Winkler and Wagner conjecture.