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Percolation Theory

Discover how local connectivity rules lead to global phase transitions and emergent behavior in networks and materials.

Introduction

Basic concepts of percolation theory

Topics covered:

Site and Bond Percolation
Critical Thresholds
Cluster Statistics
Phase Transitions

Site Percolation

Deep dive into site percolation models

Topics covered:

Occupation Probability
Cluster Size Distribution
Percolation Threshold
Finite Size Effects

Quantities of Interest

Understanding key measurements and critical behavior

Topics covered:

Critical Occupation Probability
Spanning Clusters
Cluster Size Distribution
Fractal Properties

One-Dimensional Percolation

Exploring percolation in one dimension

Topics covered:

Linear Chain Models
Exact Solutions
Critical Behavior
Correlation Length

Phase Transitions

Understanding critical phenomena and phase transitions

Topics covered:

Order Parameters
Critical Exponents
Universality Classes
Scaling Relations

Correlation Functions

Exploring spatial correlations and connectivity

Topics covered:

Two-Point Correlations
Correlation Length
Connected Correlation Functions
Scaling Behavior

Critical Behavior & Bethe Lattices

Analytical solutions and tree-like structures

Topics covered:

Critical Occupation Probability
Bethe Lattice Structure
Exact Solutions
Loop-free Networks

Bethe Lattice Definition

Understanding the structure and properties of Bethe lattices

Topics covered:

Infinite Trees
Coordination Number
Recursive Structure
Boundary Conditions

Bethe Lattice Analysis

Critical behavior and average cluster size in Bethe lattices

Topics covered:

Critical Occupation Probability
Average Cluster Size
Power Law Behavior
Coordination Number Effects

Percolation Transition

Understanding the phase transition between percolating and non-percolating states

Topics covered:

Order Parameter
Critical Point
Finite Size Scaling
Critical Exponents