Multitype Particle Systems

Interacting particle systems with random dynamics are fundamental for modeling phenomena in the physical and social sciences. Such systems can be used to describe chemical reactions, as well as the spread of disease, information, and species through a network. These models often become more meaningful when multiple particle types are incorporated.

For example, the celebrated First Passage Percolation model describes the spread of a single species through an environment; the incorporation of competing species enriches the model. This project seeks to study more realistic variants of well-known models for chemical reactions, epidemic outbreaks, and the spread of information as to deepen our understanding of important phenomena from across the sciences and further develop the mathematics that helps explain them. The project will involve the undergraduate students training.

This project will consider five different stochastic multi-type interacting particle systems: (1) The Diffusion-Limited Annihilation model is an annihilating particle system in which collisions between opposite type particles result in mutual annihilation. The version in which particles have different speeds is difficult to study and conjectured to exhibit anomalous behavior when different particle types are initially balanced. (2) The Ballistic Annihilation model has three particle types and all collisions result in annihilation.

The limiting particle density in different regimes will be characterized. (3) The Chase-Escape model is a competitive stochastic growth model conjectured to have a coexistence phase even when the predatory species is fitter than the prey. The effect of the environment and fitness of the different species on coexistence will be investigated. (4) The A+B->2A model describes a growing system of infected particles.

This process was recently generalized to the continuum, which leads to several new questions at the intersection of discrete and continuous probability, which will be addressed. (5) Lastly, the classical Susceptible-Infected-Removed model is well-understood on random graphs near the critical connectivity threshold. A natural extension is to have susceptible individuals rewire their connections away from infected individuals.

This more realistic modification may change the size of epidemics. These five processes will be studied using a variety of tools from discrete and continuous probability theory such as the mass transport principal, recursion, couplings, large deviation estimates, and stochastic differential equations.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.