Regularity theory for lattice based harmonic maps and low temperature Gibbs measures of the spin-$O(N)$ model

Abstract

The purpose of this note is twofold. First, we revisit the classical regularity theory of harmonic maps in the setting of maps defined on a lattice $\mathbb{Z}^d$ instead of $\mathbb{R}^d$ or some other manifold. The target space of our harmonic maps will be spheres $\mathbb{S}^{N-1}, N\geq 2$. A harmonic map $σ: \mathbb{Z}^d \to \mathbb{S}^{N-1}$ is a map which in any finite subgraph $Λ$ minimizes the internal Dirichlet energy $\frac 1 2 \sum_{i \sim j \in Λ} \|σ_i -σ_j\|^2$ given the boundary condition $σ_{|\partial Λ}$. One key difference between our setting and the continuum is that our main Hölder regularity statement in a large domain $Λ\subset \mathbb{Z}^{d=2}$ does not require any regularity assumption on the map $σ$ restricted to the boundary of the domain. For example, when $d=2$ and $N=3$, the Hölder regularity induced in the bulk of the domain holds uniformly in all boundary configurations $σ\in (\mathbb{S}^{2})^{\partial Λ}$ (this is in contrast with the continuum which requires extensions in suitable Sobolev spaces). The fact we obtain results uniformly in the boundary conditions allows us to extend our results for positive temperatures. Namely, as $T \searrow 0$, we obtain a local regularity theory for all infinite volume Spin $O(N)$ Gibbs measures, at least when $d=2$ and $N\geq 3$. We also discuss implications of our analysis in dimensions $d\geq 3$.