# Asymptotic measures and links in simplicial complexes

## TL;DR

This paper introduces canonical measures on simplicial complexes, analyzes their asymptotic behavior under barycentric subdivisions, and computes face polynomials of links and dual blocks, revealing almost everywhere constant properties.

## Contribution

It presents a novel framework for measures on simplicial complexes and investigates their asymptotic properties under repeated barycentric subdivisions, including face polynomial computations.

## Key findings

- Face polynomial of asymptotic link is almost everywhere constant
- Canonical measures exhibit specific asymptotic behavior under subdivision
- Limit face polynomial studied for finite complexes after multiple subdivisions

## Abstract

We introduce canonical measures on a locally finite simplicial complex $K$ and study their asymptotic behavior under infinitely many barycentric subdivisions. We also compute the face polynomial of the asymptotic link and dual block of a simplex in the $d^{th}$ barycentric subdivision $Sd^d(K)$ of $K$, $d\gg0$. It is almost everywhere constant. When $K$ is finite, we study the limit face polynomial of $Sd^d(K)$ after F.Brenti-V.Welker and E.Delucchi-A.Pixton-L.Sabalka.