Abstract: In long-range percolation on Z^d, each pair of vertices is connected by an edge with probability 1-e^{-beta ||x-y||^{-d-alpha}}, where alpha>0 is fixed and beta is the parameter varied to induce a phase transition. As d and alpha vary, the model exhibits several different forms of critical behaviour, with a transition between effectively high-dimensional and effectively low-dimensional behaviour at the upper critical dimension d_c = min(6,3alpha) and a transition between an effectively long-range and effectively short-range regime at a crossover value alpha_c(d). I will discuss a new series of papers establishing a detailed rigorous theory of the model’s critical behaviour across many of these regimes, including all effectively high-dimensional regimes and all effectively long-range regimes (which may be low or critical dimensional). The key input is a new, more probabilistic style of renormalization group method for analysing critical behaviour in long-range models.
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