A max-plus analogue of the one-dimensional quantum walk is defined, but the proof of the central conservation theorem is invalid and the claimed one-point spectrum is contradicted by an explicit counterexample.
Stationary measure induced by the eigenvalue problem of the one-dimensional Hadamard walk
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abstract
In this paper, we consider the stationary measure of the Hadamard walk on the one-dimensional integer lattice. Here all the stationary measures given by solving the eigenvalue problem are completely determined via the transfer matrix method. Then these stationary measures can be divided into three classes, i.e., quadratic polynomial, bounded, and exponential types. In particular, we present an explicit necessary and sufficient condition for the bounded-type stationary measure to be periodic.
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A walk on max-plus algebra
A max-plus analogue of the one-dimensional quantum walk is defined, but the proof of the central conservation theorem is invalid and the claimed one-point spectrum is contradicted by an explicit counterexample.