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

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.