CRITICAL PHENOMENA IN RANDOM DIMER MODELS

This thesis introduces the concept of criticality and universal behaviour for vastly different systems. The assumption of conformal invariance for systems at critical temperature is shown to yield incredible predictions with the use of Conformal Field Theory. By focusing on the properties of geometrical shapes at criticality, we introduce Schramm-Loewner Evolution (SLE), a powerful tool to extract information about conformality. Rigorously proving that a discrete system exhibits critical behaviour and conformal invariance in the thermodynamic limit is no easy task. When the system is imbued with quenched disorder, this task becomes even harder. As of today, there is no rigorous proof that a single disordered system exhibits SLE-compatible excitations. This thesis, albeit not providing any such proof, analyses numerically and extensively shows under which conditions such behaviour can be observed in a system with quenched disorder, namely the Random Dimer Model. The Random Dimer Model consists of a dimer cover of a weighted graph. Its ground state corresponds to the minimum-weight perfect matching. Dimers are often used in Statistical Mechanic, providing a fruitful mapping from a model to a combinatorial problem. By topological perturbations, we obtain excitations in the form of fractal curves. Observables based on these geometrical objects, such as their length, are shown to behave as power laws compatible with other well-known systems at critical point. Moreover, the extensive analysis of winding angle and Left Passage Probability at the thermodynamic limit, strongly suggests that excitation curves in the Random Dimer Model are conformal and compatible with Schramm-Loewner Evolution. The latter casts light on the yet vastly unexplored question of conformal behaviour for disordered models at critical temperature in the thermodynamic limit.