Non-monotone random growth processes

• Grant number: 25/10177-2
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: September 01, 2025
• End date: August 31, 2027
• Field of knowledge: Physical Sciences and Mathematics - Probability and Statistics - Probability
• Principal Investigator: Luiz Renato Gonçalves Fontes
• Grantee: Leandro Chiarini Medeiros
• 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

In this project, I will investigate phase transitions in stochastic growth processes whose behaviour is non-monotonic.Phase transition refers to the study of systems composed of a large number of particles (or agents) that move spatially and interact with nearby particles, whose macroscopic behaviour changes drastically due to small variations in certain parameters.Among the phase transition models that are well understood from a mathematical perspective, the majority exhibit monotonic behaviour with respect to their parameters. That is, increasing the value of a parameter always favours one of the possible behaviours of the model.For example, in simple population growth models, it is expected that the more frequent the birth of new agents, the more likely it is that the species will survive as a whole.The focus of this project is to develop techniques to study models whose behaviour is non-monotonic in relation to their parameters.This generalisation enables us to understand more complex models than those currently studied.For example, in the study of population growth, this allows for the introduction of a competition mechanism whereby high concentrations of individuals can lead to the elimination of agents. Depending on the nature of the competition mechanism, an increase in the birth rate of new agents may not always be beneficial for the survival of the species. However, to obtain mathematical proofs of such behaviour, it is necessary to develop new theoretical tools. The development of these tools and their application to probabilistic models are the main objectives of this project.The techniques to be developed will be based on the theory of renormalisation (also known as multi-scale analysis). This approach is highly versatile and serves as a fundamental framework for studying various models in probability theory. The central idea is that even in random systems, certain regularities emerge when we observe a large number of events. For example, when a fair coin is tossed one hundred times, we expect approximately fifty heads.The main difference when studying particle systems is that, unlike independent coin tosses, particles interact with one another and do not behave independently. However, when observing large portions of the space where these particles are distributed, we find extensive regions that exhibit expected behaviour. Once this fact is established, we employ an inductive argument to show that even if there exist regions that behave unpredictably, the overall behaviour of the system is dictated by the well-behaved regions.Despite its versatility, multi-scale analysis was originally developed for monotonic systems. Therefore, in order to adapt this methodology to the study of non-monotonic models, it is necessary to develop new theoretical foundations. I have already obtained preliminary results for some non-monotonic models, but further theoretical development is required to enable its application in a broader range of contexts. From a technical perspective, this is the main goal of this project. (AU)