## The Annals of Applied Probability

### Minimal spanning trees and Stein’s method

#### Abstract

Kesten and Lee [Ann. Appl. Probab. 6 (1996) 495–527] proved that the total length of a minimal spanning tree on certain random point configurations in $\mathbb{R}^{d}$ satisfies a central limit theorem. They also raised the question: how to make these results quantitative? Error estimates in central limit theorems satisfied by many other standard functionals studied in geometric probability are known, but techniques employed to tackle the problem for those functionals do not apply directly to the minimal spanning tree. Thus, the problem of determining the convergence rate in the central limit theorem for Euclidean minimal spanning trees has remained open. In this work, we establish bounds on the convergence rate for the Poissonized version of this problem by using a variation of Stein’s method. We also derive bounds on the convergence rate for the analogous problem in the setup of the lattice $\mathbb{Z}^{d}$.

The contribution of this paper is twofold. First, we develop a general technique to compute convergence rates in central limit theorems satisfied by minimal spanning trees on sequences of weighted graphs, including minimal spanning trees on Poisson points inside a sequence of growing cubes. Second, we present a way of quantifying the Burton–Keane argument for the uniqueness of the infinite open cluster. The latter is interesting in its own right and based on a generalization of our technique, Duminil-Copin, Ioffe and Velenik [Ann. Probab. 44 (2016) 3335–3356] have recently obtained bounds on probability of two-arm events in a broad class of translation-invariant percolation models.

#### Article information

Source
Ann. Appl. Probab., Volume 27, Number 3 (2017), 1588-1645.

Dates
Revised: May 2016
First available in Project Euclid: 19 July 2017

https://projecteuclid.org/euclid.aoap/1500451236

Digital Object Identifier
doi:10.1214/16-AAP1239

Mathematical Reviews number (MathSciNet)
MR3678480

Zentralblatt MATH identifier
1371.60035

#### Citation

Chatterjee, Sourav; Sen, Sanchayan. Minimal spanning trees and Stein’s method. Ann. Appl. Probab. 27 (2017), no. 3, 1588--1645. doi:10.1214/16-AAP1239. https://projecteuclid.org/euclid.aoap/1500451236

#### References

• [1] Addario-Berry, L., Broutin, N., Goldschmidt, C. and Miermont, G. (2013). The scaling limit of the minimum spanning tree of the complete graph. Preprint. Available at http://arxiv.org/abs/1301.1664.
• [2] 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.
• [3] Aizenman, M., Kesten, H. and Newman, C. M. (1987). Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Comm. Math. Phys. 111 505–531.
• [4] Aldous, D. (1990). A random tree model associated with random graphs. Random Structures Algorithms 1 383–402.
• [5] Aldous, D. and Steele, J. M. (1992). Asymptotics for Euclidean minimal spanning trees on random points. Probab. Theory Related Fields 92 247–258.
• [6] Alexander, K. S. (1994). Rates of convergence of means for distance-minimizing subadditive Euclidean functionals. Ann. Appl. Probab. 4 902–922.
• [7] Alexander, K. S. (1995). Percolation and minimal spanning forests in infinite graphs. Ann. Probab. 23 87–104.
• [8] Alexander, K. S. (1996). The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees. Ann. Appl. Probab. 6 466–494.
• [9] 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.
• [10] 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.
• [11] Avram, F. and Bertsimas, D. (1993). On central limit theorems in geometrical probability. Ann. Appl. Probab. 3 1033–1046.
• [12] Bai, Z. D., Lee, S. and Penrose, M. D. (2006). Rooted edges of a minimal directed spanning tree on random points. Adv. in Appl. Probab. 38 1–30.
• [13] Baldi, P. and Rinott, Y. (1989). On normal approximations of distributions in terms of dependency graphs. Ann. Probab. 17 1646–1650.
• [14] Baldi, P., Rinott, Y. and Stein, C. (1989). A normal approximation for the number of local maxima of a random function on a graph. In Probability, Statistics, and Mathematics 59–81. Academic Press, Boston, MA.
• [15] Barbour, A. D. (1990). Stein’s method for diffusion approximations. Probab. Theory Related Fields 84 297–322.
• [16] Barbour, A. D., Karoński, M. and Ruciński, A. (1989). A central limit theorem for decomposable random variables with applications to random graphs. J. Combin. Theory Ser. B 47 125–145.
• [17] Beardwood, J., Halton, J. H. and Hammersley, J. M. (1959). The shortest path through many points. Math. Proc. Cambridge Philos. Soc. 55 299–327.
• [18] Bhatt, A. G. and Roy, R. (2004). On a random directed spanning tree. Adv. in Appl. Probab. 36 19–42.
• [19] Bollobás, B. and Riordan, O. (2006). Percolation. Cambridge Univ. Press, Cambridge.
• [20] Bolthausen, E. (1984). An estimate of the remainder in a combinatorial central limit theorem. Probab. Theory Related Fields 66 379–386.
• [21] Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501–505.
• [22] Camia, F., Fontes, L. R. and Newman, C. M. (2006). Two-dimensional scaling limits via marked nonsimple loops. Bull. Braz. Math. Soc. (N.S.) 37 537–559.
• [23] Cerf, R. (2013). A lower bound on the two-arms exponent for critical percolation on the lattice. Ann. Probab. 43 2458–2480.
• [24] Chatterjee, S. (2008). A new method of normal approximation. Ann. Probab. 36 1584–1610.
• [25] Chatterjee, S. (2009). Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields 143 1–40.
• [26] Chatterjee, S. and Soundararajan, K. (2012). Random multiplicative functions in short intervals. Int. Math. Res. Not. IMRN 2012 479–492.
• [27] Chen, L. H. Y. and Shao, Q.-M. (2004). Normal approximation under local dependence. Ann. Probab. 32 1985–2028.
• [28] Duminil-Copin, H., Ioffe, D. and Velenik, Y. (2016). A quantitative Burton–Keane estimate under strong FKG condition. Ann. Probab. 44 3335–3356.
• [29] Frieze, A. M. (1985). On the value of a random minimum spanning tree problem. Discrete Appl. Math. 10 47–56.
• [30] Gandolfi, A., Grimmett, G. and Russo, L. (1988). On the uniqueness of the infinite cluster in the percolation model. Comm. Math. Phys. 114 549–552.
• [31] Goldstein, L. and Reinert, G. (1997). Stein’s method and the zero bias transformation with application to simple random sampling. Ann. Appl. Probab. 7 935–952.
• [32] Goldstein, L. and Rinott, Y. (1996). Multivariate normal approximations by Stein’s method and size bias couplings. J. Appl. Probab. 33 1–17.
• [33] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften 321. Springer, Berlin.
• [34] Häggström, O. (1995). Random-cluster measures and uniform spanning trees. Stochastic Process. Appl. 59 1–75.
• [35] 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.
• [36] Kesten, H. and Lee, S. (1996). The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Probab. 6 495–527.
• [37] Kozma, G. and Nachmias, A. (2010). Arm exponents in high dimensional percolation. J. Amer. Math. Soc. 24 375–409.
• [38] Lachiéze-Rey, R. and Peccati, G. (2015). New Kolmogorov bounds for functionals of binomial point processes. Preprint. Available at http://arxiv.org/abs/1505.04640.
• [39] Last, G., Peccati, G. and Schulte, M. (2014). Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization. Preprint. Available at http://arxiv.org/abs/1401.7568.
• [40] Lee, S. (1997). The central limit theorem for Euclidean minimal spanning trees. I. Ann. Appl. Probab. 7 996–1020.
• [41] Lee, S. (1999). The central limit theorem for Euclidean minimal spanning trees II. Adv. in Appl. Probab. 31 969–984.
• [42] Lyons, R., Peres, Y. and Schramm, O. (2006). Minimal spanning forests. Ann. Probab. 34 1665–1692.
• [43] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge Univ. Press, Cambridge.
• [44] Penrose, M. D. (1996). The random minimal spanning tree in high dimensions. Ann. Probab. 24 1903–1925.
• [45] Penrose, M. D. (1997). The longest edge of the random minimal spanning tree. Ann. Appl. Probab. 7 340–361.
• [46] Penrose, M. D. (1998). Random minimal spanning tree and percolation on the $N$-cube. Random Structures Algorithms 12 63–82.
• [47] Penrose, M. D. (2003). Random Geometric Graphs. Oxford University Press, Oxford.
• [48] Penrose, M. D. and Wade, A. R. (2004). Random minimal directed spanning trees and Dickman-type distributions. Adv. in Appl. Probab. 36 691–714.
• [49] Penrose, M. D. and Yukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 277–303.
• [50] Penrose, M. D. and Yukich, J. E. (2005). Normal approximation in geometric probability. In Stein’s Method and Applications (A. D. Barbour and L. H. Y. Chen, eds.). Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 5 37–58. Singapore Univ. Press, Singapore.
• [51] Pete, G., Garban, C. and Schramm, O. (2013). The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane. Preprint. Available at http://arxiv.org/abs/1309.0269.
• [52] Rinott, Y. and Rotar, V. (1997). On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted $U$-statistics. Ann. Appl. Probab. 7 1080–1105.
• [53] Roy, R. (1990). The Russo–Seymour–Welsh theorem and the equality of critical densities and the “dual” critical densities for continuum percolation on $\mathbb{R}^{2}$. Ann. Probab. 18 1563–1575.
• [54] Smirnov, S. and Werner, W. (2001). Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 729–744.
• [55] Steele, J. M. (1981). Subadditive Euclidean functionals and nonlinear growth in geometric probability. Ann. Probab. 9 365–376.
• [56] Steele, J. M. (1987). On Frieze’s $\zeta(3)$ limit for lengths of minimal spanning trees. Discrete Appl. Math. 18 99–103.
• [57] Steele, J. M. (1988). Growth rates of Euclidean minimal spanning trees with power weighted edges. Ann. Probab. 16 1767–1787.
• [58] Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. of the Sixth Berkeley Sympos. Math. Statist. Probab., Vol. II: Probability Theory 583–602. Univ. California Press, Berkeley, CA.
• [59] Stein, C. (1986). Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes—Monograph Series 7. IMS, Hayward, CA.