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

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1. CMB 2012 (vol 57 pp. 113)

Madras, Neal
A Lower Bound for the End-to-End Distance of Self-Avoiding Walk
For an $N$-step self-avoiding walk on the hypercubic lattice ${\bf Z}^d$, we prove that the mean-square end-to-end distance is at least $N^{4/(3d)}$ times a constant. This implies that the associated critical exponent $\nu$ is at least $2/(3d)$, assuming that $\nu$ exists.

Keywords:self-avoiding walk, critical exponent
Categories:82B41, 60D05, 60K35

2. CMB 2010 (vol 53 pp. 614)

Böröczky, Károly J.; Schneider, Rolf
The Mean Width of Circumscribed Random Polytopes
For a given convex body $K$ in ${\mathbb R}^d$, a random polytope $K^{(n)}$ is defined (essentially) as the intersection of $n$ independent closed halfspaces containing $K$ and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds of optimal orders for the difference of the mean widths of $K^{(n)}$ and $K$ as $n$ tends to infinity. For a simplicial polytope $P$, a precise asymptotic formula for the difference of the mean widths of $P^{(n)}$ and $P$ is obtained.

Keywords:random polytope, mean width, approximation
Categories:52A22, 60D05, 52A27

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