arXiv subjects

Geoffrey Grimmett

Publications and source records attributed to Geoffrey Grimmett.

23 records · Page 2Linked to original sources

arXiv2002

On smallest triangles

Pick n points independently at random in R^2, according to a prescribed probability measure mu, and let D^n_1 <= D^n_2 <= ... be the areas of the binomial n choose 3 triangles thus formed, in non-decreasing order. If mu is absolutely continuous with respect to Lebesgue measure, then, under weak conditions, the set {n^3 D^n_i : i >= 1} converges as n --> infinity to a Poisson process with a constant intensity c(mu). This result, and related conclusions, are proved using standard arguments of Poisson approximation, and may be extended to functionals more general than the area of a triangle. It is proved in addition that, if mu is the uniform probability measure on the region S, then c(mu) <= 2/|S|, where |S| denotes the area of S. Equality holds in that c(mu) = 2/|S| if S is convex, and essentially only then. This work generalizes and extends considerably the conclusions of a recent paper of Jiang, Li, and Vitanyi.

math.PR↗

arXiv2002

Stochastic apportionment

The problem of how to allocate to states the seats in the US House of Representatives is the most studied instance of what is termed the `apportionment problem'. We propose a new method of apportionment which is stochastic, which meets the quota condition, and which is fair in the sense of expectations. Two sources of systematic unfairness are identified, firstly the lower bound condition (every state shall receive at least one seat), and secondly the lower quota condition (every state shall receive at least the integer part of its quota).

math.ST↗

arXiv2001

Rigidity of the interface for percolation and random-cluster models

We study conditioned random-cluster measures with edge-parameter p and cluster-weighting factor q satisfying q \ge 1. The conditioning corresponds to mixed boundary conditions for a spin model. Interfaces may be defined in the sense of Dobrushin, and these are proved to be `rigid' in the thermodynamic limit, in three dimensions and for sufficiently large values of p. This implies the existence of non-translation-invariant (conditioned) random-cluster measures in three dimensions. The results are valid in the special case q=1, thus indicating a property of three-dimensional percolation not previously noted.

math.PR↗

arXiv2001

Directed percolation and random walk

Techniques of `dynamic renormalization', developed earlier for undirected percolation and the contact model, are adapted to the setting of directed percolation, thereby obtaining solutions of several problems for directed percolation on $Z^d$ where $d \ge 2$. The first new result is a type of uniqueness theorem: for every pair $x$ and $y$ of vertices which lie in infinite open paths, there exists almost surely a third vertex $z$ which is joined to infinity and which is attainable from $x$ and $y$ along directed open paths. Secondly, it is proved that a random walk on an infinite directed cluster is transient, almost surely, when $d \ge 3$. And finally, the block arguments of the paper may be adapted to systems with infinite range, subject to certain conditions on the edge probabilities.

math.PR↗

arXiv2001

Random Electrical Networks on Complete Graphs II: Proofs

This paper contains the proofs of Theorems 2 and 3 of the article entitled Random Electrical Networks on Complete Graphs, written by the same authors and published in the Journal of the London Mathematical Society, vol. 30 (1984), pp. 171-192. The current paper was written in 1983 but was not published in a journal, although its existence was announced in the LMS paper. This TeX version was created on 9 July 2001. It incorporates minor improvements to formatting and punctuation, but no change has been made to the mathematics. We study the effective electrical resistance of the complete graph $K_{n+2}$ when each edge is allocated a random resistance. These resistances are assumed independent with distribution $P(R=\infty)=1-n^{-1}γ(n)$, $P(R\le x) = n^{-1}γ(n)F(x)$ for $0\le x < \infty$, where $F$ is a fixed distribution function and $γ(n)\toγ\ge 0$ as $n\to\infty$. The asymptotic effective resistance between two chosen vertices is identified in the two cases $γ\le 1$ and $γ>1$, and the case $γ=\infty$ is considered. The analysis proceeds via detailed estimates based on the theory of branching processes.

math.PR↗