The Devil is in the Details: Spectrum and Eigenvalue Distribution of the Discrete Preisach Memory Model

TL;DR

This paper analyzes the spectral properties of the adjacency matrix of the discrete Preisach memory model, providing explicit formulas for eigenvalues, eigenvectors, and the eigenvalue distribution, revealing a scaled Devil's staircase pattern.

Contribution

It offers an explicit solution for the spectrum and eigenvectors of the adjacency matrix, connecting spectral properties to Chebyshev polynomials and the Devil's staircase.

Findings

Eigenvalues are given by Chebyshev polynomial roots.

Eigenvalue distribution approaches a scaled Devil's staircase.

Eigenvectors are expressed analytically.

Abstract

We consider the adjacency matrix associated with a graph that describes transitions between $2^{N}$ states of the discrete Preisach memory model. This matrix can also be associated with the last-in-first-out inventory management rule. We present an explicit solution for the spectrum by showing that the characteristic polynomial is the product of Chebyshev polynomials. The eigenvalue distribution (density of states) is explicitly calculated and is shown to approach a scaled Devil's staircase. The eigenvectors of the adjacency matrix are also expressed analytically.