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One-sided sharp thresholds for homology of random flag complexes

Andrew Newman

TL;DR

It is proved that the random flag complex has a probability regime where the probability of nonvanishing homology is asymptotically bounded away from zero and away from one.

Abstract

We prove that the random flag complex has a probability regime where the probability of nonvanishing homology is asymptotically bounded away from zero and away from one. Related to this main result we also establish new bounds on a sharp threshold for the fundamental group of a random flag complex to be a free group. In doing so we show that there is an intermediate probability regime in which the random flag complex has fundamental group which is neither free nor has Kazhdan's property (T).

One-sided sharp thresholds for homology of random flag complexes

TL;DR

It is proved that the random flag complex has a probability regime where the probability of nonvanishing homology is asymptotically bounded away from zero and away from one.

Abstract

We prove that the random flag complex has a probability regime where the probability of nonvanishing homology is asymptotically bounded away from zero and away from one. Related to this main result we also establish new bounds on a sharp threshold for the fundamental group of a random flag complex to be a free group. In doing so we show that there is an intermediate probability regime in which the random flag complex has fundamental group which is neither free nor has Kazhdan's property (T).

Paper Structure

This paper contains 7 sections, 21 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Overview of the proof
  4. Simplicity of small components
  5. Enumeration of large components
  6. Freeness of the fundamental group
  7. Concluding remarks

Key Result

Theorem 1.1

For $G \sim G(n, c/n)$ with $c \in (0, \infty)$ constant, one has the following:

Theorems & Definitions (44)

  • Theorem 1.1: (Erdős--Rényi ER)
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Remark 2.2
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 34 more