Large Scale Phenomena in Models of Statistical Mechanics

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

Among the principal tasks of probability theory is to explain why regularity, or determinism and order, arise at large scales in systems where irregularity, or randomness and disorder, dominate at small scales. This is important for theoretical reasons as it helps validate the use of phenomenological theories whenever small-scale phenomena are not expected to be of relevance, but has also practical consequences as one also needs a recipe for computing various material constants that enter the governing phenomenological equations. The present proposal goes some way towards these goals by analyzing specific systems where large scale phenomena give rise to a new degree of regularity. These systems are of interest for material theory and statistical mechanics in general. The specific problems proposed to be studied divide into five specific subareas: (1) Random metric structures on the line associated with long-range percolation, (2) Fluctuation theory and homogenization in random resistor networks, (3) Extreme points of the two-dimensional discrete Gaussian Free Field, (4) Stationary Diffusion-Limited Aggregation, (5) Isoperimetric Wulff problems in random environment. A common feature of these problems is that, in the limit when a natural parameter related to size tends to infinity, a limiting structure emerges. This structure can be deterministic, e.g., a limit shape in question (5), or random, e.g., a Poisson process with random fractal intensity measure in question (3) above. The questions to be studied have bearing on several subject areas of probability and statistical mechanics, e.g., disordered systems, growth processes, extreme-order statistics, random geometry. Methods and tools will be borrowed from other parts of mathematics as well, e.g., harmonic analysis, differential equations, homogenization theory and geometric measure theory.