Tightness of Fluctuations of First Passage Percolation on Some Large Graphs

TL;DR

This paper generalizes a tightness theorem for first passage percolation from binary trees to a broader class of graphs including hyperbolic lattices and lamplighter graphs, expanding understanding of fluctuation behavior.

Contribution

It extends the tightness results of Dekking and Host to new graph classes, including hyperbolic lattices and lamplighter graphs, and explores properties of these graphs.

Findings

Tightness of fluctuations is established for a broad class of graphs.

The class includes all hyperbolic space lattices and lamplighter graphs.

The class is closed under product with bounded degree graphs.

Abstract

The theorem of Dekking and Host regarding tightness around the mean of first passage percolation on the binary tree, from the root to a boundary of a ball, is generalized to a class of graphs which includes all lattices in hyperbolic spaces and the lamplighter graph over N. This class of graphs is closed under product with any bounded degree graph. Few open problems and conjectures are gathered at the end.