Percolation and Particle Systems: Theory and Simulations

• Grant number: 24/09099-4
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Effective date (Start): August 01, 2024
• Effective date (End): July 31, 2025
• Field of knowledge: Physical Sciences and Mathematics - Probability and Statistics - Probability
• Principal Investigator: Leonardo Trivellato Rolla
• Grantee: Pedro Ricardo Pereira
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 23/13453-5 - Stochastic systems modeling, AP.TEM

Abstract

Percolation theory in mathematics studies the behavior of connected clusters in a random medium to draw crucial conclusions about the connectivity properties of the structure and provide insights into the characteristics of systems. This theory has multiple applications, where networks with some probability parameter can be derived, for example, ferromagnetism, electrical networks, and epidemics or fires. Meanwhile, the theory of Interacting Particle Systems studies the behavior of infinite systems of locally interacting Markov processes.The goal of our project is to study percolation from both theoretical and computational perspectives and, more broadly, Particle Systems through simulations. On the theoretical side, we plan to research the uniqueness of the infinite cluster and (if time permits) the exponential decay as in Mensikov. On the computational side, we plan to conduct simulations for various models of Interacting Particles in Python and Matlab to obtain scientifically significant results and images.