Limit theorem of the max-plus walk (Mathematical structures of integrable systems, its deepening and expansion)

Abstract

The max-plus algebra is a semiring on Rmax = R⋁{-∞} with addition ⊕ and multiplication ⊗ defined by ⊕ = max and ⊗ = +, respectively. It is known that eigenvalues of max-plus matrices are equivalent to the maximal average weight of the corresponding directed graph. In [9], authors introduced the max-plus walk which is a walk model on one dimensional lattice on Z over max-plus algebra, and discussed its properties such as the conserved quantities and the steady state. In this paper, we will discuss the limit measure of the max-plus walk.