Critical random systems. Quasi-randomness and quasi-periodicity

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We study critical phenomena in statistical mechanics. The models (percolation, random-cluster models, random walks) are central objects of modern probability, and are widely applied to the study of physical, biological and social phenomena. One striking feature is the emergence of symmetries in the continuum limits of these discrete random systems, like conformal invariance in two dimensions. (A classical example is Brownian Motion, the limit of random walk.) These are key examples of conformal field theory, and recent mathematical developments have provided not just rigorous proofs, but also greatly improved on the theoretical physics understanding.