Spectral analysis of transition operators, Automata groups and translation in BBS

TL;DR

This paper explores the automata-based description of the box-ball system, its relation to lamplighter groups, and provides spectral analysis of the associated stochastic matrices, revealing spectral coincidences.

Contribution

It introduces automata for the BBS with a carrier, connects it to lamplighter groups, and analyzes the spectral properties of the induced stochastic matrices.

Findings

Spectral coincidence of stochastic matrices derived from automata.

Automata effectively describe BBS time evolution rules.

Spectral analysis links BBS to algebraic structures like lamplighter groups.

Abstract

We give the automata which describe time evolution rules of the box-ball system (BBS) with a carrier. It can be shown by use of tropical geometry, such systems are ultradiscrete analogues of KdV equation. We discuss their relation with the lamplighter group generated by an automaton. We present spectral analysis of the stochastic matrices induced by these automata, and verify their spectral coincidence.