The growth constant of odd cutsets in high dimensions

TL;DR

This paper proves that the number of odd cutsets in high-dimensional integer lattices grows roughly as (2+o(1))^{n/2d}, confirming a conjecture about their exponential growth rate relative to boundary size.

Contribution

It establishes the precise exponential growth rate of odd cutsets in high dimensions, confirming a conjecture and distinguishing their count from general cutsets.

Findings

Number of odd cutsets is approximately (2+o(1))^{n/2d}.

Confirmed the conjectured growth rate of odd cutsets in high dimensions.

Differentiated the growth of odd cutsets from that of general cutsets.

Abstract

A cutset is a non-empty finite subset of $\mathbb{Z}^d$ which is both connected and co-connected. A cutset is odd if its vertex boundary lies in the odd bipartition class of $\mathbb{Z}^d$. Peled suggested that the number of odd cutsets which contain the origin and have $n$ boundary edges may be of order $e^{\Theta(n/d)}$ as $d \to \infty$, much smaller than the number of general cutsets, which was shown by Lebowitz and Mazel to be of order $d^{\Theta(n/d)}$. In this paper, we verify this by showing that the number of such odd cutsets is $(2+o(1))^{n/2d}$.