Universality Diagram of Phase Transitions in Long-range Statistical Systems

The percolation, Ising, and O($n$) models constitute fundamental systems in statistical and condensed matter physics. For short-range-interacting cases, the nature of their phase transitions is well established by renormalization-group theory. However, the universality of the transitions in these models remains elusive when algebraically decaying long-range interactions $\sim 1/r^{d+\sigma}$ are introduced, where $d$ is the dimensionality and $\sigma$ is the decay exponent. Building upon insights from L\'evy flight, i.e., long-range simple random walk, we propose three universality diagrams in the $(d,\sigma)$ plane for the percolation model, the O($n$) model, and the Fortuin-Kasteleyn Ising model, respectively. The conjectured universality diagrams are consistent with recent high-precision numerical studies and rigorous mathematical results, offering a unified perspective on critical phenomena in systems with long-range interactions.