Universal Limit Theorem for Spectra of iterated inclusion-uniform Subdivisions

TL;DR

This paper investigates the spectral behavior of the top-dimensional Laplacian of simplicial complexes under certain subdivisions, revealing a universal limiting distribution depending only on the complex's dimension.

Contribution

It establishes a universal limit theorem for spectra of iterated inclusion-uniform subdivisions, generalizing previous results and connecting spectral limits to graph and fractal analysis.

Findings

Spectral limits depend only on the dimension of the complex.

Universal limiting distributions are identified for inclusion-uniform subdivisions.

Explicit spectral decimation formulas are derived for specific subdivision types.

Abstract

The main object of this work is the top-dimensional Laplacian operator of a simplicial complex $K$. We study its spectral limiting behavior under a given non-trivial subdivision procedure $\text{div}$. It will be shown that in case $\text{div}$ satisfies a property we call inclusion-uniformity its spectrum converges to a universal limiting distribution only depending on the dimension of $K$. This class of subdivisions contains important special cases such as the edgewise subdivision $\text{esd}_r$ for $r\geq 2$ and dimension $d=2$ or the barycentric subdivision $\text{sd}$. This parallels a result of Brenti and Welker showing that the roots of $f$-polynomials of iterated barycentric subdivisions converge to a universal set of roots only depending on the dimension of $K$. Furthermore we determine the family of universal limiting functions for the particular subdivision where the top dimensional faces are replaced by a cone over their boundary. We will show that this choice of $\text{div}$ is the natural generalization of graph subdivision in the spectral sense. These limits are obtained by explicit spectral decimation of the sequence of its dual graphs which is represented as a sequence of Schreier graphs on a rooted regular tree. Finally we will point out that a generic sequence of iterated subdivisions can be realized by a sequence of graphs as in spectral analysis on fractals. We will give a construction of a self-similar sequence of graphs which dualizes the iterated application of subdivision.