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Characterizing the fourth-moment phenomenon of monochromatic subgraph counts via influences

Nitya Mani, Dan Mikulincer

Abstract

We investigate the distribution of monochromatic subgraph counts in random vertex $2$-colorings of large graphs. We give sufficient conditions for the asymptotic normality of these counts and demonstrate their essential necessity (particularly for monochromatic triangles). Our approach refines the fourth-moment theorem to establish new, local influence-based conditions for asymptotic normality; these findings more generally provide insight into fourth-moment phenomena for a broader class of Rademacher and Gaussian polynomials.

Characterizing the fourth-moment phenomenon of monochromatic subgraph counts via influences

Abstract

We investigate the distribution of monochromatic subgraph counts in random vertex -colorings of large graphs. We give sufficient conditions for the asymptotic normality of these counts and demonstrate their essential necessity (particularly for monochromatic triangles). Our approach refines the fourth-moment theorem to establish new, local influence-based conditions for asymptotic normality; these findings more generally provide insight into fourth-moment phenomena for a broader class of Rademacher and Gaussian polynomials.
Paper Structure (21 sections, 25 theorems, 119 equations, 1 figure)

This paper contains 21 sections, 25 theorems, 119 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setup and Main Results
  3. Quantitative central limit theorems under a low-influence condition.
  4. Characterizing the asymptotic normality of monochromatic triangles counts with pairwise influences.
  5. Towards a characterization of general monochromatic subgraph counts.
  6. Related Works
  7. Preliminaries
  8. Normal Approximations for Polynomials
  9. Monochromatic Counts as Multi-linear Polynomials
  10. A Fourth Moment Theorem for Monochromatic Counts
  11. Normal Approximation for non-homogeneous polynomials
  12. Graph Polynomials
  13. A Fourth-Moment Theorem for Monochromatic Subgraph Counts
  14. A Closer Look at Monochromatic Triangles
  15. Influential Edges Forbid CLTs
  16. ...and 6 more sections

Key Result

Theorem 1.2

Let $\{G_n\}$ be a sequence of graphs with no influential vertices (asymptotically) and let $H$ be a fixed connected simple graph. Then, Moreover, if $H = \triangle$ is a triangle, we obtain

Figures (1)

  • Figure 1: An example of a graph $G_n$ with $n = 4, b = 3/16, c = 2$ as a disjoint union of graphs $S_4, P_4, B_4$ as pictured

Theorems & Definitions (49)

  • Definition 1.1: Influential vertices and edges
  • Theorem 1.2: Informal
  • Remark
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Theorem 2.1
  • Remark
  • Theorem 2.2
  • Example
  • ...and 39 more