Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The local weak limit of $k$-dimensional hypertrees

András Mészáros

Abstract

Let $\mathcal{C}(n,k)$ be the set of $k$-dimensional simplicial complexes $C$ over a fixed set of $n$ vertices such that: (1) $C$ has a complete $k-1$-skeleton; (2) $C$ has precisely ${{n-1}\choose {k}}$ $k$-faces; (3) the homology group $H_{k-1}(C)$ is finite. Consider the probability measure on $\mathcal{C}(n,k)$ where the probability of a simplicial complex $C$ is proportional to $|H_{k-1}(C)|^2$. For any fixed $k$, we determine the local weak limit of these random simplicial complexes as $n$ tends to infinity. This local weak limit turns out to be the same as the local weak limit of the $1$-out $k$-complexes investigated by Linial and Peled.

The local weak limit of $k$-dimensional hypertrees

Abstract

Let be the set of -dimensional simplicial complexes over a fixed set of vertices such that: (1) has a complete -skeleton; (2) has precisely -faces; (3) the homology group is finite. Consider the probability measure on where the probability of a simplicial complex is proportional to . For any fixed , we determine the local weak limit of these random simplicial complexes as tends to infinity. This local weak limit turns out to be the same as the local weak limit of the -out -complexes investigated by Linial and Peled.

Paper Structure

This paper contains 14 sections, 29 theorems, 117 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local weak convergence
  4. Determinantal probability measures
  5. An equivalent definition of $\nu_{n,k}$
  6. The projection matrix corresponding to $\nu_{n,k}$
  7. Another description of the semi-$k$-ary skeleton tree
  8. The local weak limit of $G_{n,k}$
  9. Formulas for fixed $n$
  10. An asymptotic formula
  11. The proof of Theorem \ref{['thmFO']}
  12. The proof of Theorem \ref{['quanched']}
  13. The proof of Lemma \ref{['generate']}
  14. Open questions

Key Result

Theorem 1.1

For a fixed $k$, the local weak limit of the graphs $G_{n,k}$ is the semi-$k$-ary skeleton tree.

Theorems & Definitions (56)

  • Theorem 1.1
  • Proposition 1.2
  • Remark 1.3
  • Lemma 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Lemma 2.1: lyons2003determinantal
  • Lemma 2.2
  • proof
  • ...and 46 more