In this thesis we investigate the geometric and algorithmic aspects of the random-cluster model, a correlated bond percolation model of great importance in the field of mathematics and statistical mechanics. We focus on the computational and statistical efficiency of the single-bond or heat-bath Markov chain for the random-cluster model and develop algorithmic techniques that allow for an improvement from a previously known polynomial to a poly-logarithmic runtime scaling of updates for general graphs. The interplay between the (critical) cluster structure of the random-cluster model and algorithmic, as well as statistical, efficiencies is considered, leading to new exact identities. A complementary analysis of certain fragility properties of the Fortuin-Kasteleyn clusters provides new insights into fragmentation phenomena, culminating in a revised scaling relation for a related fragmentation power law exponent, previously only shown for the marginal bond percolation case. By utilising the established structural results, a dynamic fragmentation process is studied that allows for an extraction of characteristics of the equilibrium cluster structure by a careful analysis of the limiting fragments, as well as the entire evolution of the fragmentation process .Besides focussing on structural and computational aspects, in this dissertation we also analyse the efficiency of the coupling from the past perfect sampling algorithm for the random-cluster model via large-scale numerical simulations. Two key results are the particular, close to optimal, efficiency in the off-critical setting and the intriguing observation of its superiority compared to the alternative Chayes-Machta-Swendsen-Wang approach in three dimensions. Governed by a random runtime, the efficiency of the coupling from the past algorithm depends crucially on the fluctuations of the runtime. In this connection a compelling appearance of universal Gumbel fluctuations in the distribution of the runtime of the coupling from the past algorithm is established, both at and off criticality. Fluctuations at a tricritical point and at a discontinuous phase transition are shown to deviate from this Gumbel law. The above findings in two and three dimensions are supported by a rigorous analysis of certain aspects of the algorithm in one dimension, including a proof of the limiting Gumbel law.
|Date of Award||2015|
|Supervisor||Martin Weigel (Supervisor), Nikolaos Fytas (Supervisor) & Christian von Ferber (Supervisor)|