The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane
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 nearcritical percolation ensemble, established in our earlier works.
 Publication:

arXiv eprints
 Pub Date:
 September 2013
 arXiv:
 arXiv:1309.0269
 Bibcode:
 2013arXiv1309.0269G
 Keywords:

 Mathematics  Probability;
 Mathematical Physics
 EPrint:
 56 pages, 21 figures. A thoroughly revised version