Isoperimetric stability in lattices

2020/07/28 by Ben Barber, Barber, Ben, Joshua Erde +5
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods

paper · doi:10.48550/arxiv.2007.14457

openalex publication_date 2020/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We obtain isoperimetric stability theorems for general Cayley digraphs on ℤ^d. For any fixed B that generates ℤ^d over ℤ, we characterise the approximate structure of large sets A that are approximately isoperimetric in the Cayley digraph of B: we show that A must be close to a set of the form kZ ∩ ℤ^d, where for the vertex boundary Z is the conical hull of B, and for the edge boundary Z is the zonotope generated by B.

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