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PERCOLATION THEORY: ANALYTICAL SOLUTIONS AND NUMERICAL SOLUTIONS USING MONTE-CARLO AND AI SEARCH METHODS (IN PROGRESS)

Abstract

Percolation theory is one of the simplest models that can accurately describe phase transitions in complex physical systems. Examples of phase transitions range from the macroscopic, such as forest fire propagation, water through porous mediums, and oil dispersion, to microscopic phenomena that include quantum phase transitions and magnetic transitions. While the one-dimensional lattice can be solved explicitly for percolation threshold, mean cluster size, and correlation length, etc., various lattice structures in higher dimensions do not yield explicit solutions due to intense scaling properties which require the use of numerical approximations. Historically, methods such as the Hoshen-Kopelman algorithm were utilized for numerical solutions, but through the devlopment of C++ code that utilizes Monte-Carlo methods, dynamic data structures, and artificial intelligence search methods, best case time and memory complexity is greatly reduced. This presentation will discuss introductory concepts of percolation theory, the development of simulations, and how obtained results are analogous to important physical properties in two and three-dimensional systems.

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