# Stationary probability vectors of higher-order two-dimensional transition probability tensors

## TL;DR

This paper characterizes stationary probability vectors of symmetric transition probability tensors in two dimensions, revealing unique or multiple solutions depending on tensor structure, with implications for understanding higher-order Markov processes.

## Contribution

It identifies conditions under which symmetric transition probability tensors have unique or multiple stationary vectors, extending the theory of Markov chains to higher-order tensors.

## Key findings

- Two special tensors have exactly two stationary vectors.
- Most symmetric tensors have a unique stationary vector.
- Irreducible tensors also have a unique stationary vector.

## Abstract

In this paper we investigate stationary probability vectors of higher-order two-dimensional symmetric transition probability tensors. We show that there are two special symmetric transition probability tensors of order $m$ dimension 2, which have and only have two stationary probability vectors; and any other symmetric transition probability tensor of order $m$ dimension 2 has a unique stationary probability vector. As a byproduct, we obtain that any symmetric transition probability tensor of order $m$ dimension 2 has a unique positive stationary probability vector; and that any symmetric irreducible transition probability tensor of order $m$ dimension 2 has a unique stationary probability vector.