Emergence and complexity in theoretical models of self-organized criticality

TL;DR

This thesis explores theoretical models of self-organized criticality, analyzing pattern growth, emergent quasi-units, and steady states in sandpile models through mathematical and algebraic methods.

Contribution

It provides a mathematical characterization of growth patterns, an exact analysis of the Zhang model, and introduces an operator algebra for stochastic sandpile steady states.

Findings

Patterns exhibit proportionate growth characterized by discrete holomorphic functions.

Quantitative explanation of the emergence of quasi-units in the Zhang model.

Operator algebra determines steady states in stochastic sandpile models.

Abstract

In this thesis we present few theoretical studies of the models of self-organized criticality. Following a brief introduction of self-organized criticality, we discuss three main problems. The first problem is about growing patterns formed in the abelian sandpile model (ASM). The patterns exhibit proportionate growth where different parts of the pattern grow in same rate, keeping the overall shape unchanged. This non-trivial property, often found in biological growth, has received increasing attention in recent years. In this thesis, we present a mathematical characterization of a large class of such patterns in terms of discrete holomorphic functions. In the second problem, we discuss a well known model of self-organized criticality introduced by Zhang in 1989. We present an exact analysis of the model and quantitatively explain an intriguing property known as the emergence of quasi-units. In the third problem, we introduce an operator algebra to determine the steady state of a class of stochastic sandpile models.