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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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

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

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

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Zentralblatt MATH identifier

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.

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


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.

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  • [1] Addario-Berry, L., Broutin, N. and Goldschmidt, C. (2012). The continuum limit of critical random graphs. Probab. Theory Related Fields 152 367–406.
  • [2] Addario-Berry, L., Broutin, N., Goldschmidt, C. and Miermont, G. (2017). The scaling limit of the minimum spanning tree of the complete graph. Ann. Probab. 45 3075–3144.
  • [3] Aizenman, M. (1997). On the number of incipient spanning clusters. Nuclear Phys. B 485 551–582.
  • [4] Aizenman, M., Burchard, A., Newman, C. M. and Wilson, D. B. (1999). Scaling limits for minimal and random spanning trees in two dimensions. Random Structures Algorithms 15 319–365.
  • [5] Aldous, D. and Steele, J. M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 1–72. Springer, Berlin.
  • [6] Alexander, K. S. (1995). Percolation and minimal spanning forests in infinite graphs. Ann. Probab. 23 87–104.
  • [7] Alexander, K. S. and Molchanov, S. (1994). Percolation of level sets for two-dimensional random fields with lattice symmetry. J. Stat. Phys. 77 627–643.
  • [8] Angel, O., Goodman, J., den Hollander, F. and Slade, G. (2008). Invasion percolation on regular trees. Ann. Probab. 36 420–466.
  • [9] Angel, O., Goodman, J. and Merle, M. (2013). Scaling limit of the invasion percolation cluster on a regular tree. Ann. Probab. 41 229–261.
  • [10] Aumann, S. (2014). Singularity of full scaling limits of planar near-critical percolation. Stochastic Process. Appl. 124 3807–3818.
  • [11] Beffara, V. and Nolin, P. (2009). Numerical estimates for monochromatic percolation exponents. Available at
  • [12] Beffara, V. and Nolin, P. (2011). On monochromatic arm exponents for 2D critical percolation. Ann. Probab. 39 1286–1304.
  • [13] Borgs, C., Chayes, J. T., Kesten, H. and Spencer, J. (2001). The birth of the infinite cluster: Finite-size scaling in percolation. Comm. Math. Phys. 224 153–204.
  • [14] Borůvka, O. (1926). O jistém problému minimálním. Pr. Morav. Přír. Spol. 3 37–58.
  • [15] Broadbent, S. R. and Hammersley, J. M. (1957). Percolation processes I. Crystals and mazes. Proc. Camb. Philos. Soc. 53 629–641.
  • [16] Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501–505.
  • [17] Camia, F., Fontes, L. R. and Newman, C. M. (2006). Two-dimensional scaling limits via marked non-simple loops. Bull. Braz. Math. Soc. (N.S.) 37 537–559.
  • [18] Camia, F. and Newman, C. M. (2006). Two-dimensional critical percolation: The full scaling limit. Comm. Math. Phys. 268 1–38.
  • [19] Chatterjee, S. and Sen, S. (2018). Minimal spanning trees and Stein’s method. Ann. Appl. Probab. To appear. Available at arXiv:1307.1661 [math.PR].
  • [20] Chayes, J. T., Chayes, L. and Newman, C. M. (1985). The stochastic geometry of invasion percolation. Comm. Math. Phys. 101 383–407.
  • [21] Chayes, J. T., Chayes, L. and Newman, C. M. (1987). Bernoulli percolation above threshold: An invasion percolation analysis. Ann. Probab. 15 1272–1287.
  • [22] Damron, M. Recent work on chemical distance in critical percolation. Preprint. Available at arXiv:1602.00775 [math.PR].
  • [23] Damron, M. and Sapozhnikov, A. (2012). Limit theorems for 2D invasion percolation. Ann. Probab. 40 893–920.
  • [24] Damron, M., Sapozhnikov, A. and Vágvölgyi, B. (2009). Relations between invasion percolation and critical percolation in two dimensions. Ann. Probab. 37 2297–2331.
  • [25] Garban, C., Pete, G. and Schramm, O. (2010). The Fourier spectrum of critical percolation. Acta Math. 205 19–104.
  • [26] Garban, C., Pete, G. and Schramm, O. (2010). The scaling limit of the minimal spanning tree—A preliminary report. In XVIth International Congress on Mathematical Physics (P. Exner, ed.) 475–480. World Scientific, Hackensack, NJ.
  • [27] Garban, C., Pete, G. and Schramm, O. (2013). Pivotal, cluster and interface measures for critical planar percolation. J. Amer. Math. Soc. 26 939–1024.
  • [28] Garban, C., Pete, G. and Schramm, O. (2018). The scaling limits of near-critical and dynamical percolation. J. Eur. Math. Soc. (JEMS) 20 1195–1268.
  • [29] Garban, C. and Steif, J. E. (2012). Noise sensitivity and percolation. In Probability and Statistical Physics in Two and More Dimensions (D. Ellwood, C. Newman, V. Sidoravicius and W. Werner, des.). Clay Math. Proc. 15 49–154. Amer. Math. Soc., Providence, RI.
  • [30] Graham, R. L. and Hell, P. (1985). On the history of the minimum spanning tree problem. Ann. Hist. Comput. 7 43–57.
  • [31] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften 321. Springer, Berlin.
  • [32] Häggström, O., Peres, Y. and Schonmann, R. H. (1999). Percolation on transitive graphs as a coalescent process: Relentless merging followed by simultaneous uniqueness. In Perplexing Problems in Probability, Festschrift in Honor of Harry Kesten (M. Bramson and R. Durrett, eds.) 69–90. Birkhäuser, Boston.
  • [33] Hammond, A., Pete, G. and Schramm, O. (2015). Local time on the exceptional set of dynamical percolation, and the Incipient Infinite Cluster. Ann. Probab. 43 2949–3005.
  • [34] Járai, A. A. (2003). Incipient infinite percolation clusters in 2D. Ann. Probab. 31 444–485.
  • [35] Kesten, H. (1982). Percolation Theory for Mathematicians. Birkhäuser, Boston.
  • [36] Kesten, H. (1986). Subdiffusive behavior of random walk on a random cluster. Ann. Inst. Henri Poincaré Probab. Stat. 22 425–487.
  • [37] Kesten, H. (1987). Scaling relations for 2D-percolation. Comm. Math. Phys. 109 109–156.
  • [38] Kruskal, J. B. (1956). On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Amer. Math. Soc. 7 48–50.
  • [39] Lawler, G. F., Schramm, O. and Werner, W. (2002). One-arm exponent for critical 2D percolation. Electron. J. Probab. 7 Art. ID 2.
  • [40] Lawler, G. F., Schramm, O. and Werner, W. (2004). Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 939–995.
  • [41] Lyons, R. and Peres, Y. (2016). Probability on Trees and Networks. Cambridge Univ. Press, Cambridge.
  • [42] Lyons, R., Peres, Y. and Schramm, O. (2006). Minimal spanning forests. Ann. Probab. 34 1665–1692.
  • [43] Newman, C., Tassion, V. and Wu, W. (2017). Critical percolation and the minimal spanning tree in slabs. Comm. Pure Appl. Math. 70 2084–2120.
  • [44] Nolin, P. (2008). Near-critical percolation in two dimensions. Electron. J. Probab. 13 1562–1623.
  • [45] Nolin, P. and Werner, W. (2009). Asymmetry of near-critical percolation interfaces. J. Amer. Math. Soc. 22 797–819.
  • [46] Penrose, M. D. (1996). The random minimal spanning tree in high dimensions. Ann. Probab. 24 1903–1925.
  • [47] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221–288.
  • [48] Schramm, O. and Smirnov, S. (2011). On the scaling limits of planar percolation. Ann. Probab. 39 1768–1814. With an appendix by Christophe Garban. Available at arXiv:1101.5820 [math.PR].
  • [49] Schramm, O. and Steif, J. E. (2010). Quantitative noise sensitivity and exceptional times for percolation. Ann. of Math. (2) 171 619–672.
  • [50] Smirnov, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 239–244.
  • [51] Smirnov, S. and Werner, W. (2001). Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 729–744.
  • [52] Steif, J. E. (2009). A survey of dynamical percolation. In Fractal Geometry and Stochastics IV 145–174. Birkhäuser, Boston, MA.
  • [53] Sweeney, S. M. and Middleton, A. A. Minimal spanning trees at the percolation threshold: A numerical calculation. Available at arXiv:1307.0043 [cond-mat.dis-nn].
  • [54] Timár, Á. (2006). Ends in free minimal spanning forests. Ann. Probab. 34 865–869.
  • [55] Timár, Á. (2018). Indistinguishability of the components of random spanning forests. Ann. Probab. To appear. Available at arXiv:1506.01370 [math.PR].
  • [56] van den Berg, J. and Conijn, R. (2013). The gaps between the sizes of large clusters in 2D critical percolation. Electron. Commun. Probab. 18 Art. ID 92.
  • [57] Wendelin, W. (2009). Lectures on two-dimensional critical percolation. In Statistical Mechanics. IAS/Park City Math. Ser. 16 297–360. Amer. Math. Soc., Providence, RI.
  • [58] Wieland, B. and Wilson, D. B. (2003). Winding angle variance of Fortuin–Kasteleyn contours. Phys. Rev. E 68 056101.
  • [59] Wilson, D. B. (2004). On the Red–Green–Blue model. Phys. Rev. E 69 037105.
  • [60] Yukich, J. E. (1998). Probability Theory of Classical Euclidean Optimization Problems. Lecture Notes in Math. 1675. Springer, Berlin.