MULTI-SCALE ANALYSIS OF REACTION-DIFFUSION PARTICLE SYSTEMS
Multi-scale analysis of reaction-diffusion particle systems
Abstract: This course focus on the (microscopic) analysis of stochastic processes on particle systems. A few examples include spread of infection among moving particles, the frog model or stochastic combustion model, activated random walks, growth processes with competition, and branching random walk with interactions. A common feature of such models is that correlations do not decay exponentially fast, which brings serious challenges to the application of standard techniques of analysis, such as renormalization techniques and comparison with independent percolation.
Recently, there has been very important progress in this field, with most new results employing a so-called multi-scale analysis to control the dependences in the model. Multi-scale analysis proved to be an extremely powerful technique; it has been employed in areas beyond particle system, one notable example being the study of random interlacements, where it has been playing a fundamental role. Despite its power, a multi-scale analysis can be quite involved to implement and can become very technical. In particular, it usually needs to be developed from scratch for each application as it needs to be tailored to each specific question being analyzed.
The goal of this course is to explain the multi-scale analysis in a didactic and comprehensible way, demystifying this technique, and explaining its main aspects (the ones that are common to most applications) and the most common variations. In order to do this we will concentrate on one problem: the analysis of the spread of an infection among particles that move as independent simple random walks.