The theory of Gibbs measures belongs to the borderlandbetween statistical mechanics and probability theory. In thiscontext, the physical phenomenon of phase transitioncorresponds to the mathematical concept of non-uniqueness for acertain type of probability measures. The most studied model in statistical mechanics is thecelebrated Ising model. The Potts model is a natural extensionof the Ising model, and the beach model, which appears in adifferent mathematical context, is in certain respectsanalogous to the Ising model. The two main parts of this thesisdeal with the Potts model and the beach model,respectively. For theq-state Potts model on an infinite lattice, there areq+1 basic Gibbs measures: one wired-boundary measure foreach state and one free-boundary measure. For infinite trees,we construct "new" invariant Gibbs measures that are not convexcombinations of the basic measures above. To do this, we use anextended version of the random-cluster model together withcoupling techniques. Furthermore, we investigate the rootmagnetization as a function of the inverse temperature.Critical exponents to this function for different parametercombinations are computed. The beach model, which was introduced by Burton and Steif,has many features in common with the Ising model. We generalizesome results for the Ising model to the beach model, such asthe connection between phase transition and a certain agreementpercolation event. We go on to study aq-state variant of the beach model. Using randomclustermodel methods again we obtain some results on where in theparameter space this model exhibits phase transition. Finallywe study the beach model on regular infinite trees as well.Critical values are estimated with iterative numerical methods.In different parameter regions we see indications of both firstand second order phase transition. Keywords and phrases:Potts model, beach model,percolation, randomcluster model, Gibbs measure, coupling,Markov chains on infinite trees, critical exponent.

<p>The theory of Gibbs measures belongs to the borderlandbetween statistical mechanics and probability theory. In thiscontext, the physical phenomenon of phase transitioncorresponds to the mathematical concept of non-uniqueness for acertain type of probability measures.</p><p>The most studied model in statistical mechanics is thecelebrated Ising model. The Potts model is a natural extensionof the Ising model, and the beach model, which appears in adifferent mathematical context, is in certain respectsanalogous to the Ising model. The two main parts of this thesisdeal with the Potts model and the beach model,respectively.</p><p>For the<i>q</i>-state Potts model on an infinite lattice, there are<i>q</i>+1 basic Gibbs measures: one wired-boundary measure foreach state and one free-boundary measure. For infinite trees,we construct "new" invariant Gibbs measures that are not convexcombinations of the basic measures above. To do this, we use anextended version of the random-cluster model together withcoupling techniques. Furthermore, we investigate the rootmagnetization as a function of the inverse temperature.Critical exponents to this function for different parametercombinations are computed.</p><p>The beach model, which was introduced by Burton and Steif,has many features in common with the Ising model. We generalizesome results for the Ising model to the beach model, such asthe connection between phase transition and a certain agreementpercolation event. We go on to study a<i>q</i>-state variant of the beach model. Using randomclustermodel methods again we obtain some results on where in theparameter space this model exhibits phase transition. Finallywe study the beach model on regular infinite trees as well.Critical values are estimated with iterative numerical methods.In different parameter regions we see indications of both firstand second order phase transition.</p><p><b>Keywords and phrases:</b>Potts model, beach model,percolation, randomcluster model, Gibbs measure, coupling,Markov chains on infinite trees, critical exponent.</p>

Wave energy has a great potential in many coastal areas thanks to a number of advantages: the abundant resource, the highest energy density of all renewables, the greater availability factors than e.g. wind or solar energy; and the low environmental and particularly visual impact. In addition, a novel advantage will be investigated in this work: the possibility of a synergetic use for carbon-free energy production and coastal protection. In this context, wave energy can contribute not only to decarbonising the energy supply and reducing greenhouse emissions, but also to mitigating coastal erosion. In effect, wave farms will be deployed nearshore to generate electricity from wave energy, and therefore the leeward coast will be exposed to a milder wave climate, which can potentially mitigate coastal erosion. This thesis aims to determine the effectiveness of wave farms for combating coastal erosion by means of a suite of state-of-the-art process-based numerical models that are applied in several case studies (Perranporth Beach,UK; and Xago Beach, Spain) and at different time scales (from the short-term to the long-term). A wave propagation model, SWAN, is used to establish the effects of the wave farm on the wave conditions. The outcomes of SWAN will be coupled to XBeach, a costal processes model that is applied to analyse the effects of the milder wave conditions on the coast. In addition to these models, empirical classifications and analytical solutions are used as well to characterise the alteration of the beach morphology due to the presence of a wave farm. The analysis of the wave farm impacts on the wave conditions and the beach morphology will be carried out through a set of ad hoc impact indicators. Parameters such as the reduction in the significant wave height, the performance of the wave farm, the effects on the seabed level and the erosion in the beach face area are defined to characterise these impacts. Moreover, the role played by the key design parameters of wave farms, e.g. farm-to-coast distance or layout, is also examined. The results from this analysis demonstrate that wave farms, in addition to their main purpose of generating carbon-free energy, are capable of reducing erosion at the coast. Storm-induced erosion is significantly reduced due to the presence of wave farms in the areas most at risk from this phenomenon. However, the effects of wave farms on the coast do not lend themselves to general statements, for they will depend on the wave farm design (WEC type, layout and farm-to-coast distance) and the characteristics of the area in question, as shown in this document for Perranporth and Xago. In summary, this synergy will improve the economic viability of wave farm projects through savings in conventional coastal defence measures, thereby fostering the development of this nascent renewable, reducing greenhouse gas emission and converging towards a more sustainable energy model. Thus, wave energy contributes to mitigating climate change by two means, one acting on the cause, the other on the effect: (i) by bringing down carbon emissions (cause) through its production of renewable energy, and (ii) by reducing coastal erosion (effect).