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arXiv · 2609.33890

FIID Coloring Random Maps

Justin Hsu·Daniel Sium·Riley Thornton
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Abstract

We show that the percolation components on a square grid can be $5$-colored as a factor of iid so that neighboring components get different colors. Above the critical probability, we observe that these regions can be 4-colored. Along the way, we extend the classical correspondence between fiid process and measurable labellings of the Bernoulli shift to this quotient setting, prove a general coloring result about hyperfinite pmp planar graphs, and clarify some foundational issues about how to define planarity for Borel graphs.

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Justin Hsu, Daniel Sium, Riley Thornton. 2026-09-27. FIID Coloring Random Maps. https://arxiv.org/abs/2609.33890

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