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A fundamental feature of systems modeled with percolation is that these systems undergo phase transitions in response to small changes in external parameters (e.g. density or temperature). The most sensitive regime of the percolation model is the critical regime where it undergoes the transition from a disordered phase to a phase with long-range order. Technically, the transition occurs only in an infinite system. Beyond the transition point, the ordered phase is characterized by the presence of an infinite cluster. The critical regime has been studied extensively by both analytical and numerical methods.
This presentation is a survey of recently developed stochastic geometric methods to study phase transition in percolation. In particular, the speaker will discuss the question of finite-size scaling: namely, how does the critical transition behavior emerge from the behavior of large, finite systems? The new methods rigorously locate the proper window in which to do critical computation and establish features of the phase transition.
Jennifer T. Chayes
Department of Mathematics
University of California, Los Angeles