Discrete Laplace and transition operators over non-Archimedean ordered fields

TL;DR

This paper explores spectral properties of Laplacian and transition operators on finite graphs over non-Archimedean fields, establishing inequalities and analyzing convergence behaviors in this non-traditional setting.

Contribution

It introduces Cheeger's inequality for non-Archimedean graphs and studies the properties of transition operators, extending classical spectral graph theory to non-Archimedean fields.

Findings

Cheeger's inequality is valid for non-Archimedean graphs.

Properties of the transition operator $ ext{P}$ are characterized.

Convergence to equilibrium is analyzed over Levi-Civita fields.

Abstract

We investigate properties of spectrum of normalized Laplacian $\mathcal L$ for finite graphs over non-Archimedean ordered fields. We prove a Cheeger's inequality for first non-zero eigenvalue. Then we describe properties of the operator $\mathcal P=I-\mathcal L$, which is a generalization of transition operator. We show that Cheeger estimate $\alpha_1\preceq \sqrt{1-h^2}$ for the second largest eigenvalue of $\mathcal P$ is crucial for investigation of the convergence of analogue of random walk to equilibrium over a non-Archimedean ordered fields. We consider examples over the Levi-Civita field.