Introduction to Random-Cluster Models and Phase Transitions

Statistical mechanics and modern probability theory rely on lattice models to analyze how microscopic interactions produce macroscopic phase transitions. This course provides a structured, accessible entry point into the two-dimensional random-cluster model and the rigorous analysis of critical behavior. You will build a solid theoretical foundation by working through fundamental definitions, exploring correlation inequalities, and understanding how mathematicians determine critical thresholds on planar lattices. What you'll learn: - Understand core probabilistic concepts behind percolation and random-cluster representations - Analyze planar duality, boundary conditions, and spatial correlation inequalities on square lattices - Explore sharp-threshold techniques and differential inequalities used to determine critical points - Connect random-cluster models with classical physical frameworks like the Ising and Potts models - Examine modern Markov chain Monte Carlo approaches for simulating discrete lattice systems The course begins with foundational probability terminology and graph theory basics before progressing systematically through duality properties and critical point derivations. Designed for students, researchers, and math enthusiasts with an introductory background in probability and calculus, this text-based guide requires no prior exposure to advanced statistical physics. Begin reading today to deepen your understanding of phase transitions and percolation theory.