The Annals of Probability

The scaling limit of the minimum spanning tree of the complete graph

Louigi Addario-Berry, Nicolas Broutin, Christina Goldschmidt, and Grégory Miermont

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Consider the minimum spanning tree (MST) of the complete graph with $n$ vertices, when edges are assigned independent random weights. Endow this tree with the graph distance renormalized by $n^{1/3}$ and with the uniform measure on its vertices. We show that the resulting space converges in distribution as $n\to\infty$ to a random compact measured metric space in the Gromov–Hausdorff–Prokhorov topology. We additionally show that the limit is a random binary $\mathbb{R}$-tree and has Minkowski dimension $3$ almost surely. In particular, its law is mutually singular with that of the Brownian continuum random tree or any rescaled version thereof. Our approach relies on a coupling between the MST problem and the Erdős–Rényi random graph. We exploit the explicit description of the scaling limit of the Erdős–Rényi random graph in the so-called critical window, established in [Probab. Theory Related Fields 152 (2012) 367–406], and provide a similar description of the scaling limit for a “critical minimum spanning forest” contained within the MST. In order to accomplish this, we introduce the notion of $\mathbb{R}$-graphs, which generalise $\mathbb{R}$-trees, and are of independent interest.

Article information

Ann. Probab., Volume 45, Number 5 (2017), 3075-3144.

Received: January 2015
Revised: April 2016
First available in Project Euclid: 23 September 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems

Minimum spanning tree scaling limit R-tree R-graph Gromov–Hausdorff–Prokhorov distance critical random graphs


Addario-Berry, Louigi; Broutin, Nicolas; Goldschmidt, Christina; Miermont, Grégory. The scaling limit of the minimum spanning tree of the complete graph. Ann. Probab. 45 (2017), no. 5, 3075--3144. doi:10.1214/16-AOP1132.

Export citation


  • [1] Abraham, R., Delmas, J.-F. and Hoscheit, P. (2013). A note on the Gromov–Hausdorff–Prokhorov distance between (locally) compact metric measure spaces. Electron. J. Probab. 18 no. 14.
  • [2] Addario-Berry, L. (2013). The local weak limit of the minimum spanning tree of the complete graph. Available at and arXiv:1301.1667 [math.PR].
  • [3] Addario-Berry, L., Broutin, N. and Goldschmidt, C. (2010). Critical random graphs: Limiting constructions and distributional properties. Electron. J. Probab. 15 741–775.
  • [4] Addario-Berry, L., Broutin, N. and Goldschmidt, C. (2012). The continuum limit of critical random graphs. Probab. Theory Related Fields 152 367–406.
  • [5] Addario-Berry, L., Broutin, N. and Reed, B. (2009). Critical random graphs and the structure of a minimum spanning tree. Random Structures Algorithms 35 323–347.
  • [6] Addario-Berry, L., Griffiths, S. and Kang, R. J. (2012). Invasion percolation on the Poisson-weighted infinite tree. Ann. Appl. Probab. 22 931–970.
  • [7] 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 15 319–367.
  • [8] Aldous, D. (1990). A random tree model associated with random graphs. Random Structures Algorithms 1 383–402.
  • [9] Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1–28.
  • [10] Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990). London Mathematical Society Lecture Note Series 167 23–70. Cambridge Univ. Press, Cambridge.
  • [11] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
  • [12] Aldous, D. (1997). Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 812–854.
  • [13] Aldous, D. and Steele, J. M. (1992). Asymptotics for Euclidean minimal spanning trees on random points. Probab. Theory Related Fields 92 247–258.
  • [14] 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.
  • [15] Alexander, K. S. (1995). Percolation and minimal spanning forests in infinite graphs. Ann. Probab. 23 87–104.
  • [16] Alexander, K. S. (1996). The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees. Ann. Appl. Probab. 6 466–494.
  • [17] Alexander, K. S. and Molchanov, S. A. (1994). Percolation of level sets for two-dimensional random fields with lattice symmetry. J. Stat. Phys. 77 627–643.
  • [18] Angel, O., Goodman, J., den Hollander, F. and Slade, G. (2008). Invasion percolation on regular trees. Ann. Probab. 36 420–466.
  • [19] Angel, O., Goodman, J. and Merle, M. (2013). Scaling limit of the invasion percolation cluster on a regular tree. Ann. Probab. 41 229–261.
  • [20] Avram, F. and Bertsimas, D. (1992). The minimum spanning tree constant in geometrical probability and under the independent model: A unified approach. Ann. Appl. Probab. 2 113–130.
  • [21] Beardwood, J., Halton, J. H. and Hammersley, J. M. (1959). The shortest path through many points. Math. Proc. Cambridge Philos. Soc. 55 299–327.
  • [22] Beveridge, A., Frieze, A. and McDiarmid, C. (1998). Random minimum length spanning trees in regular graphs. Combinatorica 18 311–333.
  • [23] Bhamidi, S., Broutin, N., Sen, S. and Wang, X. (2014). Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erdős–Rényi random graph. Available at arXiv:1411.3417 [math.PR].
  • [24] Borůvka, O. (1926). O jistém problému minimálním. Práce Moravské Přírodovědecké Společnosti Brno 3 37–58.
  • [25] Burago, D., Burago, Y. and Ivanov, S. (2001). A Course in Metric Geometry. Graduate Studies in Mathematics 33. Amer. Math. Soc., Providence, RI.
  • [26] Damron, M. and Sapozhnikov, A. (2011). Outlets of 2D invasion percolation and multiple-armed incipient infinite clusters. Probab. Theory Related Fields 150 257–294.
  • [27] Damron, M., Sapozhnikov, A. and Vágvölgyi, B. (2009). Relations between invasion percolation and critical percolation in two dimensions. Ann. Probab. 37 2297–2331.
  • [28] Duquesne, T. and Le Gall, J.-F. (2005). Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 553–603.
  • [29] Erdős, P. and Rényi, A. (1960). On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 17–61.
  • [30] Evans, S. N., Pitman, J. and Winter, A. (2006). Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 81–126.
  • [31] Evans, S. N. and Winter, A. (2006). Subtree prune and regraft: A reversible real tree-valued Markov process. Ann. Probab. 34 918–961.
  • [32] Falconer, K. (1990). Fractal Geometry. Mathematical Foundations and Applications. Wiley, Chichester.
  • [33] Frieze, A., Ruszinkó, M. and Thoma, L. (2000). A note on random minimum length spanning trees. Electron. J. Combin. 7 Research Paper 4, 5.
  • [34] Frieze, A. M. (1985). On the value of a random minimum spanning tree problem. Discrete Appl. Math. 10 47–56.
  • [35] Frieze, A. M. and McDiarmid, C. J. H. (1989). On random minimum length spanning trees. Combinatorica 9 363–374.
  • [36] Garban, C., Pete, G. and Schramm, O. (2013). The scaling limits of the minimal spanning tree and invasion percolation in the plane. Available at arXiv:1309.0269 [math.PR].
  • [37] Graham, R. L. and Hell, P. (1985). On the history of the minimum spanning tree problem. IEEE Ann. Hist. Comput. 7 43–57.
  • [38] Haas, B. and Miermont, G. (2004). The genealogy of self-similar fragmentations with negative index as a continuum random tree. Electron. J. Probab. 9 57–97 (electronic).
  • [39] Janson, S. (1995). The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph. Random Structures Algorithms 7 337–355.
  • [40] Janson, S. and Luczak, M. J. (2008). Susceptibility in subcritical random graphs. J. Math. Phys. 49 125207, 23.
  • [41] Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley-Interscience, New York.
  • [42] Janson, S. and Wästlund, J. (2006). Addendum to: “The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph” [Random Structures Algorithms 7 (1995), no. 4, 337–355; MR1369071] by Janson. Random Structures Algorithms 28 511–512.
  • [43] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • [44] Kesten, H. and Lee, S. (1996). The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Probab. 6 495–527.
  • [45] Kozma, G., Lotker, Z. and Stupp, G. (2006). The minimal spanning tree and the upper box dimension. Proc. Amer. Math. Soc. 134 1183–1187 (electronic).
  • [46] Lee, S. (1997). The central limit theorem for Euclidean minimal spanning trees. I. Ann. Appl. Probab. 7 996–1020.
  • [47] Lee, S. (1999). The central limit theorem for Euclidean minimal spanning trees. II. Adv. in Appl. Probab. 31 969–984.
  • [48] Le Gall, J.-F. (2005). Random trees and applications. Probab. Surv. 2 245–311.
  • [49] Le Gall, J.-F. (2006). Random real trees. Ann. Fac. Sci. Toulouse Math. (6) 15 35–62.
  • [50] Łuczak, T. (1990). Component behavior near the critical point of the random graph process. Random Structures Algorithms 1 287–310.
  • [51] Łuczak, T. (1998). Random trees and random graphs. Random Structures Algorithms 13 485–500.
  • [52] Łuczak, T., Pittel, B. and Wierman, J. C. (1994). The structure of a random graph at the point of the phase transition. Trans. Amer. Math. Soc. 341 721–748.
  • [53] Lyons, R., Peres, Y. and Schramm, O. (2006). Minimal spanning forests. Ann. Probab. 34 1665–1692.
  • [54] McDiarmid, C., Johnson, T. and Stone, H. S. (1997). On finding a minimum spanning tree in a network with random weights. Random Structures Algorithms 10 187–204.
  • [55] Miermont, G. (2009). Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér. (4) 42 725–781.
  • [56] Nachmias, A. and Peres, Y. (2007). Component sizes of the random graph outside the scaling window. ALEA Lat. Am. J. Probab. Math. Stat. 3 133–142.
  • [57] Newman, C. M. (1997). Topics in Disordered Systems. Birkhäuser, Basel.
  • [58] Papadopoulos, A. (2005). Metric Spaces, Convexity and Nonpositive Curvature. IRMA Lectures in Mathematics and Theoretical Physics 6. European Mathematical Society (EMS), Zürich.
  • [59] Penrose, M. (2003). Random Geometric Graphs. Oxford Studies in Probability 5. Oxford Univ. Press, Oxford.
  • [60] Penrose, M. D. (1996). The random minimal spanning tree in high dimensions. Ann. Probab. 24 1903–1925.
  • [61] Penrose, M. D. (1998). Extremes for the minimal spanning tree on normally distributed points. Adv. in Appl. Probab. 30 628–639.
  • [62] Penrose, M. D. (1998). Random minimal spanning tree and percolation on the $N$-cube. Random Structures Algorithms 12 63–82.
  • [63] Penrose, M. D. (1999). A strong law for the longest edge of the minimal spanning tree. Ann. Probab. 27 246–260.
  • [64] Penrose, M. D. and Yukich, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 1005–1041.
  • [65] Penrose, M. D. and Yukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 277–303.
  • [66] Peres, Y. and Revelle, D. (2005). Scaling limits of the uniform spanning tree and loop-erased random walk on finite graphs. Available at arXiv:math/0410430 [math.PR].
  • [67] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [68] Schriver, A. (2005). On the history of combinatorial optimization (till 1960). In Discrete Optimization (K. Aardal, G. L. Nemhauser and R. Weismantel, eds.). Handbooks in Operations Research and Management 12 1–68. North Holland, Amsterdam.
  • [69] Steele, J. M. (1981). Subadditive Euclidean functionals and nonlinear growth in geometric probability. Ann. Probab. 9 365–376.
  • [70] Steele, J. M. (1987). On Frieze’s $\zeta(3)$ limit for lengths of minimal spanning trees. Discrete Appl. Math. 18 99–103.
  • [71] Steele, J. M. (1988). Growth rates of Euclidean minimal spanning trees with power weighted edges. Ann. Probab. 16 1767–1787.
  • [72] Steele, J. M., Shepp, L. A. and Eddy, W. F. (1987). On the number of leaves of a Euclidean minimal spanning tree. J. Appl. Probab. 24 809–826.
  • [73] Stroock, D. W. (1993). Probability Theory, an Analytic View. Cambridge Univ. Press, Cambridge.
  • [74] Timár, Á. (2006). Ends in free minimal spanning forests. Ann. Probab. 34 865–869.
  • [75] Timofeev, E. A. (1988). On finding the expected length of a random minimal tree. Theory Probab. Appl. 33 361–365.
  • [76] van der Hofstad, R. and Nachmias, A. (2017). Hypercube percolation. J. Eur. Math. Soc. (JEMS) 19 725–814.
  • [77] Varopoulos, N. T. (1985). Long range estimates for Markov chains. Bull. Sci. Math. (2) 109 225–252.
  • [78] Villani, C. (2009). Optimal Transport, Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin.
  • [79] Wilson, D. B. (2004). Red-green-blue model. Phys. Rev. E 69 037105.
  • [80] Yukich, J. E. (1996). Ergodic theorems for some classical problems in combinatorial optimization. Ann. Appl. Probab. 6 1006–1023.
  • [81] Yukich, J. E. (1998). Probability Theory of Classical Euclidean Optimization Problems. Lecture Notes in Math. 1675. Springer, Berlin.
  • [82] Yukich, J. E. (2000). Asymptotics for weighted minimal spanning trees on random points. Stochastic Process. Appl. 85 123–138.