Connectivity functions for random walk percolation models

点击翻译按钮获取中文摘要

Historically, it has been of great interest and practical relevance to study spreading phenomena of a fluid (water, pollutants,...) through a medium, and many Mathematical models have been introduced for this purpose. M. Sahimi (1994) assigns two possible categories to such models: diffusion processes and percolation processes. In the first, the assumption is that the source of randomness lies with the fluid, whereas in the second, the motion of the fluid is wholly determined by the structure of the medium, which itself is considered to be randomly generated. The point of view taken by percolation processes was initially studied and contextualised by S. R. Broadbent and J. M. Hammersley (1957), and has since led to further exploration of various random geometries such as the Bernoulli percolation model or Gaussian Free Field (GFF) excursion sets. The latter give rise to new and exciting universality classes, characterized by the presence of long-range correlations between local observables, which decay polynomially with the distance. All these models exhibit what is called a phase transition, presumably of second order and characterized by critical exponents, which takes place as the density changes from a subcritical phase, consisting of small connected islands only, to a supercritical phase comprising an infinite cluster. The goal of this research project is to further develop our mathematical understanding of long-range correlated percolation models. One such model on the integer lattice is called the random interlacements, and was first introduced by A.-S. Sznitman (2010) to deal with probabilistic covering and fragmentation problems attached to random walk traces. Such models were considered e.g. by M.J. Brummelhuis and H.J. Hillhorst (1991) as models of corrosion. By further delving into this area, this project aims to solve several open problems relating to the interlacements and will progress understanding of large deviation properties of the simple random walk. Of particular interest is the behaviour of the truncated two-point function for the vacant set of the walk on the torus at suitable timescales. In determining the precise leading asymptotic behaviour for this observable, both in sub- and supercritical regime, this project aims to tackle a long-lasting and central question in this area. It is likely that the methodology and results obtained for this purpose will be adaptable to other scenarios involving long-range dependence. This project falls within the EPSRC Mathematical Analysis, Probability and Mathematical Physics research areas, in particular relating to random structures, stochastic analysis and critical phenomena.