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The Gaussian Wave for Graphs of Finite Cone Type

Amir Dembo, Theo McKenzie

Abstract

We show that for any infinite tree of finite cone type satisfying a mild expansion condition, the only typical process on its vertices with covariance induced by the Green's function is the Gaussian wave. This generalizes a result of Backhausz and Szegedy, who proved this for the infinite regular tree of degree $d\geq 3$. We do this by giving a reduction to a statement concerning the distribution of the inner product of our process with columns of the Green's function, which in turn are straightforward to calculate. As a consequence, for random bipartite biregular graphs, the distribution of local neighborhoods of eigenvectors must approximate the Gaussian wave. Moreover, for generic configuration models including random lifts, the local distribution of a uniformly chosen eigenvector from any arbitrarily small spectral window likewise converges to the Gaussian wave.

The Gaussian Wave for Graphs of Finite Cone Type

Abstract

We show that for any infinite tree of finite cone type satisfying a mild expansion condition, the only typical process on its vertices with covariance induced by the Green's function is the Gaussian wave. This generalizes a result of Backhausz and Szegedy, who proved this for the infinite regular tree of degree . We do this by giving a reduction to a statement concerning the distribution of the inner product of our process with columns of the Green's function, which in turn are straightforward to calculate. As a consequence, for random bipartite biregular graphs, the distribution of local neighborhoods of eigenvectors must approximate the Gaussian wave. Moreover, for generic configuration models including random lifts, the local distribution of a uniformly chosen eigenvector from any arbitrarily small spectral window likewise converges to the Gaussian wave.
Paper Structure (13 sections, 23 theorems, 76 equations, 2 figures)

This paper contains 13 sections, 23 theorems, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Key entropy inequality
  3. Preliminaries
  4. Notation and conventions
  5. Proof of \ref{['thm:mainthm']}
  6. Entropic identities for typical processes
  7. Entropy of the Gaussian wave
  8. Maximizing the entropy inequality
  9. Moving from the finite to infinite graph
  10. Convergence of mean and covariance
  11. Convergence to the Gaussian wave
  12. Comparing levels
  13. Properties of the Green's Function

Key Result

Theorem 1.4

For any $d\ge 3$, $\epsilon>0$, and $k\in \mathbb{N}$, there exist $N_0\in \mathbb{N}$ and $\delta>0$ such that for all $N\ge N_0$, with probability at least $1-\epsilon$ over the choice of a uniformly random $d$-regular graph $\mathcal{G}$ on $N$ vertices, the following holds: for every vector $\ps

Figures (2)

  • Figure 1: The decomposition of the edge and the star into orthogonal components as given in \ref{['eq:reducedcovar']}.
  • Figure 2: In \ref{['eq:eqforfig']}, we can write the Green's function going into a vertex as a sum over Green's functions leaving it.

Theorems & Definitions (46)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: Theorem 2.2 of backhausz2019almost
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 1.10
  • ...and 36 more