Universal Limit Theorem for Spectra of iterated inclusion-uniform Subdivisions

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.