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Search: MSC category 60D05 ( Geometric probability and stochastic geometry [See also 52A22, 53C65] )

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1. CJM Online first

Barlow, Martin T.; Járai, Antal A.
Geometry of uniform spanning forest components in high dimensions
We study the geometry of the component of the origin in the uniform spanning forest of $\mathbb{Z}^d$ and give bounds on the size of balls in the intrinsic metric.

Keywords:uniform spanning forest, loop-erased random walk
Categories:60D05, 05C05, 31C20

2. CJM 2009 (vol 61 pp. 1279)

Hoffman, Christopher; Holroyd, Alexander E.; Peres, Yuval
Tail Bounds for the Stable Marriage of Poisson and Lebesgue
Let $\Xi$ be a discrete set in $\rd$. Call the elements of $\Xi$ {\em centers}. The well-known Voronoi tessellation partitions $\rd$ into polyhedral regions (of varying volumes) by allocating each site of $\rd$ to the closest center. Here we study allocations of $\rd$ to $\Xi$ in which each center attempts to claim a region of equal volume $\alpha$. We focus on the case where $\Xi$ arises from a Poisson process of unit intensity. In an earlier paper by the authors it was proved that there is a unique allocation which is {\em stable} in the sense of the Gale--Shapley marriage problem. We study the distance $X$ from a typical site to its allocated center in the stable allocation. The model exhibits a phase transition in the appetite $\alpha$. In the critical case $\alpha=1$ we prove a power law upper bound on $X$ in dimension $d=1$. (Power law lower bounds were proved earlier for all $d$). In the non-critical cases $\alpha<1$ and $\alpha>1$ we prove exponential upper bounds on $X$.

Keywords:stable marriage, point process, phase transition

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