Connected allocation to Poisson points in $\mathbb{R}^2$

Maxim Krikun (IECN, Universite Nancy 1)

Abstract


This note answers one question in [1] concerning the connected allocation for the Poisson process in $\mathbb{R}^2$. The proposed solution makes use of the Riemann map from the plane minus the minimal spanning forest of the Poisson point process to the halfplane. A picture of a numerically simulated example is included.

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Pages: 140-145

Publication Date: May 8, 2007

DOI: 10.1214/ECP.v12-1268

References

  1. C. Hoffman, A.E. Holroyd, Y. Peres. A stable marriage of Poisson and Lebesgue. Ann. Probab. 34 (2006), 1241--1272. Math. Review 2257646
  2. K.S. Alexander. Percolation and minimal spanning forests in infinite graphs. Ann. Probab. 23 (1995), 87--104. Math. Review 1330762
  3. S. Chatterjee, R. Peled, Y. Peres, D. Romik. Gravitational allocation to Poisson points (2006), math.PR/0611886
  4. R. Lyons, Y. Peres, O. Schramm. Minimal spanning forests. Ann. Probab. 34 (2006), 1665--1692. Math. Review 2271476
  5. D.E. Marshall, S. Rohde. Convergence of the zipper algorithm for conformal mapping (2006), math.CV/0605532
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