The Annals of Applied Probability

The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees

Kenneth S. Alexander

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Abstract

We prove a central limit theorem for the length of the minimal spanning tree of the set of sites of a Poisson process of intensity $\lambda$ in $[0, 1]^2$ as $\lambda \to \infty$. As observed previously by Ramey, the main difficulty is the dependency between the contributions to this length from different regions of $[0, 1]^2$; a percolation-theoretic result on circuits surrounding a fixed site can be used to control this dependency. We prove such a result via a continuum percolation version of the Russo-Seymour-Welsh theorem for occupied crossings of a rectangle. This RSW theorem also yields a variety of results for two-dimensional fixed-radius continuum percolation already well known for lattice models, including a finite-box criterion for percolation and absence of percolation at the critical point.

Article information

Source
Ann. Appl. Probab., Volume 6, Number 2 (1996), 466-494.

Dates
First available in Project Euclid: 18 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1034968140

Digital Object Identifier
doi:10.1214/aoap/1034968140

Mathematical Reviews number (MathSciNet)
MR1398054

Zentralblatt MATH identifier
0855.60009

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 90C27: Combinatorial optimization

Keywords
Central limit theorem occupied crossing continuum percolation minimal spanning tree

Citation

Alexander, Kenneth S. The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees. Ann. Appl. Probab. 6 (1996), no. 2, 466--494. doi:10.1214/aoap/1034968140. https://projecteuclid.org/euclid.aoap/1034968140


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References

  • 1 ALDOUS, D. and STEELE, J. M. 1992. Asy mptotics for Euclidean minimal spanning trees on random points. Probab. Theory Related Fields 92 247 258.
  • 2 ALEXANDER, K. S. 1994. Rates of convergence of means for distance-minimizing subadditive Euclidean functionals. Ann. Appl. Probab. 4 902 922.
  • 3 ALEXANDER, K. S. 1995. Percolation and minimal spanning forests in infinite graphs. Ann. Probab. 23 87 104.
  • 4 AVRAM, F. and BERTSIMAS, D. 1993. On central limit theorems in geometrical probability. Ann. Appl. Probab. 4 1033 1046.
  • 5 BARBOUR, A. D., HOLST, L. and JANSON, S. 1992. Poisson Approximation. Clarendon, Oxford.
  • 6 BEARDWOOD, J., HALTON, J. H. and HAMMERSLEY, J. M. 1959. The shortest path through many points. Proc. Cambridge Philos. Soc. 55 299 327.
  • 7 CHAy ES, J. T. and CHAy ES, L. 1986. Percolation and random media. In Critical PhenomZ. ena, Random Sy stems and Gauge Theories K. Osterwalder and R. Stora, eds. 1001 1142. Elsevier, Amsterdam.
  • 8 CHUNG, K. L. 1974. A Course in Probability Theory. Academic Press, New York.
  • 9 EFRON, B. and STEIN, C. 1981. The jackknife estimate of variance. Ann. Statist. 9 586 596.
  • 10 FEW, L. 1955. The shortest path and shortest road through n points. Mathematika 2 141 144.
  • 11 GRIMMETT, G. R. 1989. Percolation. Springer, New York.
  • 12 HARRIS, T. E. 1960. A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc. 56 13 20.
  • 13 HIGUCHI, Y. 1993. Coexistence of infinite -clusters. II. Ising percolation in two dimensions. Probab. Theory Related Fields 97 1 33.
  • 14 JAILLET, P. 1992. Rates of convergence for quasi-additive smooth Euclidean functionals and application to combinatorial optimization problems. Math. Oper. Res. 17 964 980.
  • 15 KESTEN, H. 1982. Percolation Theory for Mathematicians. Birkhauser, Boston. ¨
  • 16 KESTEN, H. and LEE, S. 1996. The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Probab. 6 495 527.
  • 17 MEESTER, R. and ROY, R. 1996. Continuum Percolation. Cambridge Univ. Press.
  • 18 PREPARATA, F. P. and SHAMOS, M. I. 1985. Computational Geometry: An Introduction. Springer, New York.
  • 19 PRIM, R. C. 1957. Shortest connection networks and some generalizations. Bell Sy stem. Tech. J. 36 1389 1401.
  • 20 RAMEY, D. B. 1982. A non-parametric test of bimodality with applications to cluster analysis. Ph.D. dissertation, Dept. Statistics, Yale Univ.
  • 21 REDMOND, C. and YUKICH, J. 1994. Limit theorems and rates of convergence for Euclidean functionals. Ann. Appl. Probab. 4 1057 1073.
  • 22 RHEE, W. T. 1994. Boundary effects in the traveling salesperson problem. Oper. Res. Lett. 16 19 25.
  • 23 RHEE, W. T. and TALAGRAND, M. 1989. A sharp deviation inequality for the stochastic traveling salesman problem. Ann. Probab. 17 1 8.
  • 24 ROY, R. 1990. The Russo Sey mour Welsh theorem and the equality of critical densities and the ``dual'' critical densities for continuum percolation on 2. Ann. Probab. 18 1563 1575.
  • 25 RUSSO, L. 1978. A note on percolation. Z. Wahrsch. Verw. Gebiete 43 39 48.
  • 26 RUSSO, L. 1981. On the critical percolation probabilities. Z. Wahrsch. Verw. Gebiete 56 229 237.
  • 27 SEy MOUR, P. D. and WELSH, D. J. A. 1978. Percolation probabilities on the square lattice. Z. In Advances in Graph Theory B. Bollobas, ed. 227 245. North-Holland, Amsterdam.
  • 28 STEELE, J. M. 1981. Subadditive Euclidean functionals and nonlinear growth in geometric probability. Ann. Probab. 9 365 376.
  • 29 STEELE, J. M. 1981. Complete convergence of short paths and Karp's algorithm for the TSP. Math. Oper. Res. 6 374 378.
  • 30 STEELE, J. M. 1988. Growth rates of Euclidean minimal spanning trees with power weighted edges. Ann. Probab. 16 1767 1787.
  • 31 ZUEV, S. A. and SIDORENKO, A. F. 1985. Continuous models of percolation theory. I, II. Theoret. and Math. Phy s. 62 72 86, 253 265.
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