Discrete Laplace and transition operators over non-Archimedean ordered fields
Abstract
Abstract We investigate the spectrum of the normalized Laplacian L for finite graphs over non-Archimedean ordered fields. We prove a Cheeger’s inequality for the first non-zero eigenvalue. Further, we show that, on the contrary to the real case, for the transition operator P = I − L and for any non-bipartite non-complete graph over a non-Archimedean field there is always a subspace of functions for which an analogue of random walk, i.e. P m f , does not converge. Moreover, we show that the strong Cheeger estimate α 1 ≼ 1 − h 2 for the second largest eigenvalue of P is crucial for the investigation of convergence of P m f to the equilibrium over non-Archimedean ordered fields. We provide examples of convergence and non-convergence for graphs over the Levi-Civita field.
Fields of Study & Subjects
advanced mathematical theories
Mental Health Research Topics
Opinion Dynamics and Social Influence
Publication details
- Journal / source
- Physica Scripta
- Publisher
- IOP Publishing
- Volume
- 101
- Issue
- 9
- Pages
- 095203–095203
- Publication date
- 19 February 2026
- Language
- EN