The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane

TL;DR

This paper establishes the existence and invariance properties of the scaling limits of the Minimal Spanning Tree and Invasion Percolation Tree on a triangular lattice, revealing their geometric features and non-conformal invariance.

Contribution

It proves the unique scaling limits of these trees are invariant under rotations and scalings, and details their geometric properties, extending previous near-critical percolation results.

Findings

Scaling limits are rotation and scale invariant.

The limits are not conformally invariant.

Geometric properties of the limiting MST are characterized.

Abstract

We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of the triangular lattice in the complex plane have unique scaling limits, which are invariant under rotations, scalings, and, in the case of the MST, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting MST. The topology of convergence is the space of spanning trees introduced by Aizenman, Burchard, Newman & Wilson (1999), and the proof relies on the existence and conformal covariance of the scaling limit of the near-critical percolation ensemble, established in our earlier works.