Discrete Laplace and transition operators over non-Archimedean ordered fields

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 ${\alpha }_1⪯\sqrt{1-h^2}$ 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.