arXiv subjects

Geoffrey R. Grimmett

Publications and source records attributed to Geoffrey R. Grimmett.

46 records · Page 3Linked to original sources

arXiv2013

Inhomogeneous bond percolation on square, triangular and hexagonal lattices

The star-triangle transformation is used to obtain an equivalence extending over the set of all (in)homogeneous bond percolation models on the square, triangular and hexagonal lattices. Among the consequences are box-crossing (RSW) inequalities for such models with parameter-values at which the transformation is valid. This is a step toward proving the universality and conformality of these processes. It implies criticality of such values, thereby providing a new proof of the critical point of inhomogeneous systems. The proofs extend to certain isoradial models to which previous methods do not apply.

math.PR↗

arXiv2013

Bounds on connective constants of regular graphs

Bounds are proved for the connective constant μ of an infinite, connected, Δ-regular graph G. The main result is that μ \ge \sqrt{Δ-1} if G is vertex-transitive and simple. This inequality is proved subject to weaker conditions under which it is sharp.

math.CO↗

arXiv2013

Cluster detection in networks using percolation

We consider the task of detecting a salient cluster in a sensor network, that is, an undirected graph with a random variable attached to each node. Motivated by recent research in environmental statistics and the drive to compete with the reigning scan statistic, we explore alternatives based on the percolative properties of the network. The first method is based on the size of the largest connected component after removing the nodes in the network with a value below a given threshold. The second method is the upper level set scan test introduced by Patil and Taillie [Statist. Sci. 18 (2003) 457-465]. We establish the performance of these methods in an asymptotic decision- theoretic framework in which the network size increases. These tests have two advantages over the more conventional scan statistic: they do not require previous information about cluster shape, and they are computationally more feasible. We make abundant use of percolation theory to derive our theoretical results, and complement our theory with some numerical experiments.

math.ST↗

arXiv2012

Lattice embeddings in percolation

Does there exist a Lipschitz injection of $\mathbb{Z}^d$ into the open set of a site percolation process on $\mathbb{Z}^D$, if the percolation parameter p is sufficiently close to 1? We prove a negative answer when d=D and also when $d\geq2$ if the Lipschitz constant M is required to be 1. Earlier work of Dirr, Dondl, Grimmett, Holroyd and Scheutzow yields a positive answer for d<D and M=2. As a result, the above question is answered for all d, D and M. Our proof in the case d=D uses Tucker's lemma from topological combinatorics, together with the aforementioned result for d<D. One application is an affirmative answer to a question of Peled concerning embeddings of random patterns in two and more dimensions.

math.PR↗

arXiv2012

Self-avoiding walks and the Fisher transformation

The Fisher transformation acts on cubic graphs by replacing each vertex by a triangle. We explore the action of the Fisher transformation on the set of self-avoiding walks of a cubic graph. Iteration of the transformation yields a sequence of graphs with common critical exponents, and with connective constants converging geometrically to the golden mean. We consider the application of the Fisher transformation to one of the two classes of vertices of a bipartite cubic graph. The connective constant of the ensuing graph may be expressed in terms of that of the initial graph. When applied to the hexagonal lattice, this identifies a further lattice whose connective constant may be computed rigorously.

math.CO↗

arXiv2012

Percolation since Saint-Flour

This is a short survey of work on percolation and first-passage percolation since the publication (in 1996 and 1984, respectively) of the two authors' Saint-Flour notes on these topics.

math.PR↗

arXiv2011

European Apportionment via the Cambridge Compromise

Seven mathematicians and one political scientist met at the Cambridge Apportionment Meeting in January 2011. They agreed a unanimous recommendation to the European Parliament for its future apportionments between the EU Member States. This is a short factual account of the reasons that led to the Meeting, of its debates and report, and of some of the ensuing Parliamentary debate.

math.HO↗

arXiv2010

Plaquettes, Spheres, and Entanglement

The high-density plaquette percolation model in d dimensions contains a surface that is homeomorphic to the (d-1)-sphere and encloses the origin. This is proved by a path-counting argument in a dual model. When d=3, this permits an improved lower bound on the critical point p_e of entanglement percolation, namely p_e >= μ^-2 where μis the connective constant for self-avoiding walks on Z^3. Furthermore, when the edge density p is below this bound, the radius of the entanglement cluster containing the origin has an exponentially decaying tail.

math.PR↗

arXiv2010

Geometry of Lipschitz percolation

We prove several facts concerning Lipschitz percolation, including the following. The critical probability p_L for the existence of an open Lipschitz surface in site percolation on Z^d with d\ge 2 satisfies the improved bound p_L \le 1-1/[8(d-1)]. Whenever p > p_L, the height of the lowest Lipschitz surface above the origin has an exponentially decaying tail. The lowest surface is dominated stochastically by the boundary of a union of certain independent, identically distributed random subsets of Z^d. As a consequence, for p sufficiently close to 1, the connected regions of Z^{d-1} above which the surface has height 2 or more exhibit stretched-exponential tail behaviour.

math.PR↗

arXiv2008

Percolation of arbitrary words in one dimension

We consider a type of long-range percolation problem on the positive integers, motivated by earlier work of others on the appearance of (in)finite words within a site percolation model. The main issue is whether a given infinite binary word appears within an iid Bernoulli sequence at locations that satisfy certain constraints. We settle the issue in some cases, and provide partial results in others.

math.PR↗