The Annals of Probability

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

Christophe Garban, Gábor Pete, and Oded Schramm

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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 $\mathsf{MST}$, 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.

Article information

Source
Ann. Probab., Volume 46, Number 6 (2018), 3501-3557.

Dates
Received: January 2017
Revised: December 2017
First available in Project Euclid: 25 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1537862439

Digital Object Identifier
doi:10.1214/17-AOP1252

Mathematical Reviews number (MathSciNet)
MR3857861

Zentralblatt MATH identifier
06975492

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B27: Critical phenomena 82B43: Percolation [See also 60K35] 05C05: Trees
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 81T27: Continuum limits 81T40: Two-dimensional field theories, conformal field theories, etc.

Keywords
Minimal spanning tree invasion percolation critical and near-critical percolation scaling limit conformal invariance Hausdorff dimension

Citation

Garban, Christophe; Pete, Gábor; Schramm, Oded. The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane. Ann. Probab. 46 (2018), no. 6, 3501--3557. doi:10.1214/17-AOP1252. https://projecteuclid.org/euclid.aop/1537862439


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