Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hearing Shapes via p-Adic Laplacians

Patrick Erik Bradley, Ángel Morán Ledezma

TL;DR

The paper develops a non-Archimedean spectral framework for graphs by introducing $p$-adic Laplacians $\Lambda_G^\Delta$ and their spectral curves $P_G^\Delta(X,Y)$. It shows that, under suitable diffusion data, the spectrum organizes into a family $G_r^\Delta$ and that the bivariate polynomial encodes enough graph-structural information (via spanning forests) to reconstruct the graph up to isomorphism; matrix perturbation further demonstrates separation of isospectral graphs. The authors prove a Reconstruction Theorem asserting that there exist diffusion parameters for which $P_G^\Delta(X,Y)$ uniquely determines the graph, and they extend these ideas to applications in $p$-adic geometry by recovering reduction graphs of Mumford curves and products of Tate curves (p-adic tori) from spectral data. This provides a novel inverse-problem toolkit in non-Archimedean spectral geometry with concrete geometric consequences. The work blends $p$-adic analysis (Kozyrev wavelets, Vladimirov operator) with graph theory to build a diffusion-based spectral invariant capable of hearing shapes in both combinatorial and $p$-adic geometric settings.

Abstract

For a finite graph, a spectral curve is constructed as the zero set of a two-variate polynomial with integer coefficients coming from p-adic diffusion on the graph. It is shown that certain spectral curves can distinguish non-isomorphic pairs of isospectral graphs, and can even reconstruct the graph. This allows the graph reconstruction from the spectrum of the associated p-adic Laplacian operator. As an application to p-adic geometry, it is shown that the reduction graph of a Mumford curve and the product reduction graph of a p-adic analytic torus can be recovered from the spectrum of such operators.

Hearing Shapes via p-Adic Laplacians

TL;DR

The paper develops a non-Archimedean spectral framework for graphs by introducing -adic Laplacians and their spectral curves . It shows that, under suitable diffusion data, the spectrum organizes into a family and that the bivariate polynomial encodes enough graph-structural information (via spanning forests) to reconstruct the graph up to isomorphism; matrix perturbation further demonstrates separation of isospectral graphs. The authors prove a Reconstruction Theorem asserting that there exist diffusion parameters for which uniquely determines the graph, and they extend these ideas to applications in -adic geometry by recovering reduction graphs of Mumford curves and products of Tate curves (p-adic tori) from spectral data. This provides a novel inverse-problem toolkit in non-Archimedean spectral geometry with concrete geometric consequences. The work blends -adic analysis (Kozyrev wavelets, Vladimirov operator) with graph theory to build a diffusion-based spectral invariant capable of hearing shapes in both combinatorial and -adic geometric settings.

Abstract

For a finite graph, a spectral curve is constructed as the zero set of a two-variate polynomial with integer coefficients coming from p-adic diffusion on the graph. It is shown that certain spectral curves can distinguish non-isomorphic pairs of isospectral graphs, and can even reconstruct the graph. This allows the graph reconstruction from the spectrum of the associated p-adic Laplacian operator. As an application to p-adic geometry, it is shown that the reduction graph of a Mumford curve and the product reduction graph of a p-adic analytic torus can be recovered from the spectrum of such operators.
Paper Structure (12 sections, 13 theorems, 93 equations, 2 figures)

This paper contains 12 sections, 13 theorems, 93 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and Some Results from $p$-Adic Analysis
  3. Spectral Curves for Diffusion Pairs
  4. $p$-Adic Laplacians for Graphs
  5. The Spectral Curve of a Diffusion Pair
  6. Recovering Spectral Curves
  7. Spectral Curves Which are Separating
  8. Separating Isospectral Pairs via Matrix Perturbation
  9. A Reconstruction Theorem
  10. Hearing Shapes of $p$-adic geometric objects
  11. Hearing the Shape of a Mumford Curve
  12. Hearing the Shape of a $p$-Adic Analytic Torus

Key Result

Theorem 2.1

There exist a complete orthonormal system of eigenfunctions of the operator $\Delta^{\alpha}$ of the form $\psi_{r,n}(x)\in L^2(K)$ , where $r\in \mathbb{Z}$ and $n\in \mathbb{N}$ such that

Figures (2)

  • Figure 1: A pair of non-isomorphic, but isospectral graphs whose first Betti number is $3$.
  • Figure 2: A pair of non-isomorphic bridgeless graphs whose first Betti number is $2$. According to MM2016, it follows that they are not isospectral.

Theorems & Definitions (33)

  • Theorem 2.1: Kozyrev
  • proof
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Definition 3.4
  • Theorem 3.5
  • proof
  • Theorem 3.6
  • ...and 23 more