Nucleation scaling in jigsaw percolation

TL;DR

This paper analyzes the conditions under which jigsaw percolation successfully merges all clusters in a puzzle, focusing on the probability thresholds in various graph structures, and confirms several prior conjectures about the model.

Contribution

It provides a detailed analysis of the scaling behavior of jigsaw percolation, including exact thresholds for specific graph types, and resolves existing conjectures in the field.

Findings

Probability of solving depends on pD(log N)

Exact scaling laws for 1D and 2D puzzles

Confirmed several conjectures about the model

Abstract

Jigsaw percolation is a nonlocal process that iteratively merges connected clusters in a deterministic "puzzle graph" by using connectivity properties of a random "people graph" on the same set of vertices. We presume the Erdos--Renyi people graph with edge probability p and investigate the probability that the puzzle is solved, that is, that the process eventually produces a single cluster. In some generality, for puzzle graphs with N vertices of degrees about D (in the appropriate sense), this probability is close to 1 or small depending on whether pD(log N) is large or small. The one dimensional ring and two dimensional torus puzzles are studied in more detail and in many cases the exact scaling of the critical probability is obtained. The paper settles several conjectures posed by Brummitt, Chatterjee, Dey, and Sivakoff who introduced this model.