The Wasserstein Geometry of Real-Space Renormalization

The renormalization group (RG) provides a fundamental description of how physical theories change across scales, often conceptualized as a flow on the space of Hamiltonians. Recent work has formalized momentum-space RG as a gradient flow within the framework of Optimal Transport (OT). However, a corresponding gradient flow understanding of real-space RG, particularly for lattice systems, remains less developed. In this work, we propose a novel framework to study real-space coarse-graining by treating the Ising model as a finite-dimensional parametric manifold of Gibbs measures embedded within the Wasserstein space of all probability distributions. Using tools from the newly-developed theory of "Wasserstein Information Geometry", we interpret RG transformations as paths on this manifold, allowing us to quantify how "optimal" a given RG transformation is and propose novel RG transformations.