Two geometric lemmas for $S^{N-1}$-valued maps and an application to the homogenization of spin systems

TL;DR

This paper introduces two geometric lemmas for $S^{N-1}$-valued functions that facilitate the homogenization of spin systems by enabling modifications of lattice spin functions to achieve a fixed average, simplifying existing formulas.

Contribution

The paper presents novel geometric lemmas for $S^{N-1}$-valued maps that improve the homogenization process of spin systems by allowing controlled modifications of spin configurations.

Findings

Lemmas enable fixing the average of spin functions with minimal modifications.

Simplified formulas for the homogenization of spin systems.

Applicable to discrete-to-continuum transition in lattice models.

Abstract

We prove two geometric lemmas for $S^{N-1}$-valued functions that allow to modify sequences of lattice spin functions on a small percentage of nodes during a discrete-to-continuum process so as to have a fixed average. This is used to simplify known formulas for the homogenization of spin systems.