A walk on max-plus algebra

TL;DR

This paper introduces a novel max-plus algebra-based walk model on a one-dimensional lattice, drawing analogies to quantum walks, and analyzes its spectral properties and conserved quantities.

Contribution

It proposes the max-plus walk model as an analogue of quantum walks using max-plus algebra and provides spectral analysis of its evolution operator.

Findings

Eigenvalues of state decision matrices are conserved.

Spectral analysis of the total time evolution operator is performed.

The model offers a new perspective on walks in max-plus algebra.

Abstract

Max-plus algebra is a kind of idempotent semiring over $\mathbb{R}_{\max}:=\mathbb{R}\cup\{-\infty\}$ with two operations $\oplus := \max$ and $\otimes := +$.In this paper, we introduce a new model of a walk on one dimensional lattice on $\mathbb{Z}$, as an analogue of the quantum walk, over the max-plus algebra and we call it max-plus walk. In the conventional quantum walk, the summation of the $\ell^2$-norm of the states over all the positions is a conserved quantity. In contrast, the summation of eigenvalues of state decision matrices is a conserved quantity in the max-plus walk.Moreover, spectral analysis on the total time evolution operator is also given.