Geoffrey Grimmett

On colouring random graphs

Let ω n denote a random graph with vertex set {1, 2, …, n }, such that each edge is present with a prescribed probability p , independently of the presence or absence of any other edges. We show that the number of vertices in the largest complete subgraph of ω n is, with probability one,

The shortest-path problem for graphs with random arc-lengths

We consider the problem of finding the shortest distance between all pairs of vertices in a complete digraph on n vertices, whose arc-lengths are non-negative random variables. We describe an algorithm which solves this problem in O(n(m+nlogn)) expected time, where m is the expected number of arcs with finite lenght. If m is small enough, this represents a small improvement over the bound in Bloniarz [3]. We consider also the case when the arc-lengths are random variables which are independently distributed with distribution function F, where F(0)=0 and F is differentiable at 0; for this case, we describe an algorithm which runs in O(n2logn) expected time. In our treatment of the shortest-path problem we consider the following problem in combinatorial probability theory. A town contains n people, one of whom knows a rumour. At the first stage he tells someone chosen randomly from the town; at each stage, each person who knows the rumour tells someone else, chosen randomly from the town and indeependently of all other choices. Let Sn be the number of stages before the whole town rnows the rumour. We show that Sn/log2n → 1 + loge2 in probability as n → ∞, and estimate the probabilities of large deviations in Sn.

The supercritical phase of percolation is well behaved

We prove a general result concerning the critical probabilities of subsets of a lattice L. It is a consequence of this result that the critical probability of a percolation process on L equals the limit of the critical probability of a slice of L as the thickness of the slice tends to infinity. This verification of one of the standard hypotheses of the subject settles many questions concerning supercritical percolation.

Weak limits for quantum random walks

We formulate and prove a general weak limit theorem for quantum random walks in one and more dimensions. With X(n) denoting position at time n, we show that X(n)/n converges weakly as n--> infinity to a certain distribution which is absolutely continuous and of bounded support. The proof is rigorous and makes use of Fourier transform methods. This approach simplifies and extends certain preceding derivations valid in one dimension that make use of combinatorial and path integral methods.

First-passage percolation, network flows and electrical resistances

SummaryWe show that the first-passage times of first-passage percolation on ℤ2 are such that P(θ0nn(μ+ɛ)) decay geometrically as n→∞, where θ may represent any of the four usual first-passage-time processes. The former estimate requires no moment condition on the time coordinates, but there exists a geometrically-decaying estimate for the latter quantity if and only if the time coordinate distribution has finite moment generating function near the origin. Here, μ is the time constant and ɛ>0. We study the line-to-line first-passage times and describe applications to the maximum network flow through a randomly-capacitated subsection of ℤ2, and to the asymptotic behaviour of the electrical resistance of a subsection of ℤ2 when the edges of the subsection are wires in an electrical network with random resistances. In the latter case we show, for example, that if each edge-resistance equals 1 or ∞ ohms with probabilities p and 1−p respectively, then the effective resistance Rn across opposite faces of an n by n box satisfies the following:(a)if p<1/2 then P(Rn=∞)→1 as n→∞,(b)if p>1/2 then there exists ν(p)<∞ such that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0Jd9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacI% cacaWGWbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGafyizImQba0ba% daWfqaqaaiGacYgacaGGPbGaaiyBaiGacMgacaGGUbGaaiOzaiaadk% fadaWgaaWcbaGaamOBaaqabaaabaGaamOBaiabgkziUkabg6HiLcqa% baGccuGHKjYOgaqhamaaxababaGaciiBaiaacMgacaGGTbGaci4Cai% aacwhacaGGWbGaamOuamaaBaaaleaacaWGUbaabeaaaeaacaWGUbGa% eyOKH4QaeyOhIukabeaakiqbgsMiJAaaDaGaamODaiaacIcacaWGWb% GaaiykaiaacMcacqGH9aqpcaaIXaaaaa!5DC6!

Random walk on the infinite cluster of the percolation model

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Percolation in half-spaces: equality of critical densities and continuity of the percolation probability

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Probability and Phase Transition

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Directed Percolation and Random Walk

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