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Phase transitions in isoperimetric problems on the integers

Joseph Briggs, Chris Wells

TL;DR

This work shows that the natural nesting of extremal sets for vertex- and edge-isoperimetry on Cayley graphs of $\mathbb{Z}$ fails already in dimension one, via a phase-transition mechanism that mimics cylindrical grids. It constructs non-nested extremal families by relating $\mathbb{Z}$-Cayley graphs to cylinder-like embeddings and proves that intervals become eventually optimal for large sizes on any Cayley graph $G=Cay(\mathbb{Z},B)$ generating $\mathbb{Z}$, thereby answering Barber–Erde’s Question 2 positively in dimension one when finitely many sets are ignored. The paper provides precise thresholds: for edge- and vertex-problems on $G= Cay(\mathbb{Z},\pm\{1,b\})$ and $Cay(\mathbb{Z},\pm\{1,b-1,b,b+1\})$, optimizers are intervals for large $n$, but can be translates of $[k]+b[k]$ for certain square sizes $n=k^2$ when $k<(b+1)/2$ or $(b-1)/2$ respectively. It also establishes that intervals are eventually optimal for any Cayley graph on $\mathbb{Z}$, using residue-class and compression arguments, and discusses implications and open problems for higher dimensions.

Abstract

Barber and Erde asked the following question: if $B$ generates $\mathbb Z^d$ as an additive group, then must the extremal sets for the vertex/edge-isoperimetric inequality on the Cayley graph $\operatorname{Cay}(\mathbb Z^d,B)$ form a nested family? We answer this question negatively for both the vertex- and edge-isoperimetric inequalities, specifically in the case of $d=1$. The key is to show that the structure of the cylinder $\mathbb Z\times(\mathbb Z/k\mathbb Z)$ can be mimicked in certain Cayley graphs on $\mathbb Z$, leading to a phase transition. We do, however, show that Barber--Erde's question for Cayley graphs on $\mathbb Z$ has a positive answer if one is allowed to ignore finitely many sets.

Phase transitions in isoperimetric problems on the integers

TL;DR

This work shows that the natural nesting of extremal sets for vertex- and edge-isoperimetry on Cayley graphs of fails already in dimension one, via a phase-transition mechanism that mimics cylindrical grids. It constructs non-nested extremal families by relating -Cayley graphs to cylinder-like embeddings and proves that intervals become eventually optimal for large sizes on any Cayley graph generating , thereby answering Barber–Erde’s Question 2 positively in dimension one when finitely many sets are ignored. The paper provides precise thresholds: for edge- and vertex-problems on and , optimizers are intervals for large , but can be translates of for certain square sizes when or respectively. It also establishes that intervals are eventually optimal for any Cayley graph on , using residue-class and compression arguments, and discusses implications and open problems for higher dimensions.

Abstract

Barber and Erde asked the following question: if generates as an additive group, then must the extremal sets for the vertex/edge-isoperimetric inequality on the Cayley graph form a nested family? We answer this question negatively for both the vertex- and edge-isoperimetric inequalities, specifically in the case of . The key is to show that the structure of the cylinder can be mimicked in certain Cayley graphs on , leading to a phase transition. We do, however, show that Barber--Erde's question for Cayley graphs on has a positive answer if one is allowed to ignore finitely many sets.
Paper Structure (12 sections, 14 theorems, 22 equations, 3 figures)

This paper contains 12 sections, 14 theorems, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Construction of non-nested extremal sets
  4. Mimicking the cylinder
  5. Construction for the edge-isoperimetric problem
  6. Construction for the vertex-isoperimetric problem
  7. Proof of Theorem 1.2
  8. Intervals are eventually optimal
  9. The edge-isoperimetric inequality on $G$.
  10. The vertex-isoperimetric inequality on $G$.
  11. Concluding remarks
  12. The vertex-isoperimetric inequality on the infinity-grid

Key Result

Theorem 1.2

[theorem]not_nested For every integer $N$ and for each $i\in\{\operatorname e,\operatorname v\}$, there is a subset $B\subseteq\mathbb{Z}$ and integers $n_1,n_2$ satisfying

Figures (3)

  • Figure 1: The graphs $\mathop{\mathrm{Cay}}\nolimits(\mathbb{Z},\pm\{1,4\})$ (left) and $C_1(4)$ (right).
  • Figure 2: The graphs $\mathop{\mathrm{Cay}}\nolimits(\mathbb{Z},\pm\{1,3,4,5\})$ (left) and $C_\infty(4)$ (right).
  • Figure 3: The map $\phi'$ from the proof of \ref{['edge_not_nested']} with $b=4$.

Theorems & Definitions (26)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Bollobás--Leader bollobas_edge
  • Theorem 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 16 more