# Factor maps and embeddings for random $\mathbb{Z}^d$ shifts of finite type

## TL;DR

This paper investigates the properties of random $
abla^d$ shifts of finite type, establishing conditions under which these shifts can factor onto or embed into given subshifts with high probability as the pattern size grows.

## Contribution

It provides new probabilistic criteria for factorization and embedding of random $
abla^d$ SFTs into specified subshifts, advancing understanding of their structural complexity.

## Key findings

- Conditions for random SFTs to factor onto a given subshift with high probability.
- Conditions for embedding a subshift into a random SFT with high probability.
- Asymptotic probability results as pattern size tends to infinity.

## Abstract

For any $d \geq 1$, random $\mathbb{Z}^d$ shifts of finite type (SFTs) were defined in previous work of the authors. For a parameter $\alpha \in [0,1]$, an alphabet $\mathcal{A}$, and a scale $n \in \mathbb{N}$, one obtains a distribution of random $\mathbb{Z}^d$ SFTs by randomly and independently forbidding each pattern of shape $\{1,\dots,n\}^d$ with probability $1-\alpha$ from the full shift on $\mathcal{A}$. We prove two main results concerning random $\mathbb{Z}^d$ SFTs. First, we establish sufficient conditions on $\alpha$, $\mathcal{A}$, and a $\mathbb{Z}^d$ subshift $Y$ so that a random $\mathbb{Z}^d$ SFT factors onto $Y$ with probability tending to one as $n$ tends to infinity. Second, we provide sufficient conditions on $\alpha$, $\mathcal{A}$ and a $\mathbb{Z}^d$ subshift $X$ so that $X$ embeds into a random $\mathbb{Z}^d$ SFT with probability tending to one as $n$ tends to infinity.