📝 SummarXiv

Abstract

Statistical mechanical systems at and near their points of phase transition are expected to exhibit rich, fractal-like behaviour that is independent of the small-scale details of the system but depends strongly on the dimension in which the model is defined. Moreover, many models are conjectured to have an upper critical dimension with important quantitative and qualitative differences between critical behaviour at, above, and below the upper critical dimension. For models with long-range interactions, one expects additional transitions between effectively long-range and effectively short-range regimes, with further marginal effects on the boundary of these two regimes, leading to (at least) eight qualitatively distinct forms of critical behaviour in total for each given model. We give a broad overview of these conjectures aimed at a general mathematical audience before surveying the significant recent progress that has been made towards understanding them in the context of long-range percolation.