A Characteristic Polynomial for The Transition Probability Matrix of A Correlated Random Walk on A Graph

TL;DR

This paper introduces a characteristic polynomial formula for the transition probability matrix of a correlated random walk derived from the Grover walk on a graph, linking spectral properties to zeta functions.

Contribution

It provides a novel determinant-based formula for the characteristic polynomial of the CRW's transition matrix, connecting quantum walk matrices with graph zeta functions.

Findings

Spectrum of transition matrices for CRWs on regular graphs

Spectrum of transition matrices for CRWs on semiregular bipartite graphs

Extension to a different type of correlated random walk

Abstract

We define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph $G$, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW by using a determinant expression for the generalized weighted zeta function of $G$. As applications, we give the spectrum of the transition probability matrices for the CRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs. Furthermore, we consider another type of the CRW on a graph.