Open Access
November 2018 The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane
Christophe Garban, Gábor Pete, Oded Schramm
Ann. Probab. 46(6): 3501-3557 (November 2018). DOI: 10.1214/17-AOP1252


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 $\mathsf{MST}$, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting $\mathsf{MST}$. The topology of convergence is the space of spanning trees introduced by Aizenman et al. [Random Structures Algorithms 15 (1999) 319–365], 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.


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Christophe Garban. Gábor Pete. Oded Schramm. "The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane." Ann. Probab. 46 (6) 3501 - 3557, November 2018.


Received: 1 January 2017; Revised: 1 December 2017; Published: November 2018
First available in Project Euclid: 25 September 2018

zbMATH: 06975492
MathSciNet: MR3857861
Digital Object Identifier: 10.1214/17-AOP1252

Primary: 05C05 , 60K35 , 82B27 , 82B43
Secondary: 60D05 , 81T27 , 81T40

Keywords: conformal invariance , critical and near-critical percolation , Hausdorff dimension , Invasion percolation , Minimal spanning tree , Scaling limit

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 6 • November 2018
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