Characterizing phase transitions from statistics of configuration-space distances

This paper systematically reviews a series of works of our research group in the new paradigm of phase transition research, namely “statistical geometry of configuration space”. This method identifies phase transitions from the raw data in an unsupervised manner and extracts universal critical behaviors by analyzing the distance distribution between microscopic configurations sampled by Monte Carlo. We demonstrate how this paradigm has evolved from a numerical exploration of complex models to a universal empirical scaling law, and, finally, to an analytical theoretical framework that connects geometric quantities in configuration space with correlation functions in real space. This work reveals how macroscopic criticality is encoded in the geometry of microscopic configurations, providing a brand-new, “white-box” and analytically tractable data-driven perspective for studying complex phase transition problems with unknown order parameters.