Correlations and Scaling in Disordered and Critical Stochastic Models

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In many physical systems, large and orderly structures are formed without distant parts of the system communicating directly. Understanding these structures is a long-standing challenge in probability theory; this project focuses on the mathematical study of these structures. One class of these, percolation, has arisen to study various random networks: for instance, underground rock formations carrying oil, traffic networks in cities, and polymer chains. Another class of these, Abelian networks, arose to describe situations where energy builds up microscopically until it reaches a certain threshold and then causes a large "avalanche," and has been used to model earthquakes. A major issue the work under this project will touch on is how widely-separated (in space or time) parts of the random structure depend on each other: for instance, how do early topplings in an avalanche influence those occurring much later? These questions have connections to other important physical systems as well as to optimization problems from computer science.The research under this award will study correlations and fluctuations in disordered systems (first-passage percolation and related disordered magnets) and critical systems (percolation, and Bak-Tang-Wiesenfeld models also known as sandpiles). A primary focus in first-passage percolation will be the order of fluctuations of the random metric and its relationship to properties of geodesics. In critical percolation, the work will focus on the scaling of the chemical or graph distance in large finite clusters -- in particular, bounding the chemical distance exponent in two dimensions. A final focus will be structural properties of avalanches in the Abelian sandpile model, including the question of stabilizability of sandpiles in two dimensions.