The square and add Markov chain

TL;DR

This paper explores a complex random walk generated by squaring and adding over finite fields, revealing new links between Galois theory and probability, with implications for understanding algebraic structures and stochastic processes.

Contribution

It introduces a novel process over finite fields that connects elementary Galois theory with probabilistic behavior, expanding understanding of algebraic and stochastic interactions.

Findings

Identifies a new random walk process over finite fields.

Establishes connections between Galois theory and probability.

Highlights intractability of the process over integers.

Abstract

Squaring and adding $\pm 1$ mod p generates a curiously intractable random walk. A similar process over the finite field $\mathbf{F}_q$ (with $q=2^d$) leads to novel connections between elementary Galois theory and probability.