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FIID Coloring Random Maps
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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BibTeX
@misc{hsu2026fiid,
title = {{FIID Coloring Random Maps}},
author = {Justin Hsu and Daniel Sium and Riley Thornton},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33890},
eprint = {2609.33890},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33890},
}
RIS
TY - UNPB
TI - FIID Coloring Random Maps
AU - Justin Hsu
AU - Daniel Sium
AU - Riley Thornton
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33890
UR - https://arxiv.org/abs/2609.33890
AB - 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.
ER -