Finitely Dependent Processes on Subshifts

TL;DR

This paper investigates the existence and obstructions of stationary finitely dependent processes on subshifts, using cohomology and height functions, with applications to tilings and graph models.

Contribution

It demonstrates the conditions under which finitely dependent processes exist or are obstructed on subshifts, connecting cohomology and tiling problems.

Findings

Dense set of finitely dependent processes supported on strongly mixing subshifts.

Cohomology can obstruct the existence of finitely dependent processes.

Characterization of finitely dependent processes on tilings using height functions.

Abstract

The existence of stationary finitely dependent processes on combinatorial models like $\mathbb Z^d$ subshifts can be quite mysterious. For instance, Holroyd and Liggett constructed such processes on proper $4$-colorings of $\mathbb Z^d$ for all $d$ while Holroyd, Schramm and Wilson showed that there are no such processes on proper $3$-colorings of $\mathbb Z^d$ for $d>1$. In this paper, we take inspiration from these results and investigate them further. On the positive side, we show that there exists a dense set of stationary finitely dependent processes supported on subshifts with strong mixing properties like the finite extension property. On the negative side, we see that the cohomology of the subshifts can form an obstruction to the existence of such processes. In particular we use Conway-Lagarias-Thurston height functions to characterise when there exists a finitely dependent process on the space of tilings by boxes of $\mathbb Z^2$ answering the tiling problem posed by Gao, Jackson, Krohne and Seward in dimension $2$. The ideas also apply to many other models, such as graph homomorphisms and ribbon tilings. On the way, we also show that continuous cocycles on strongly irreducible subshifts valued in a special class of groups (including torsion free Gromov hyperbolic groups and free product of cyclic groups) are perturbations of group homomorphisms.