Dominic Welsh

Percolation Probabilities on the Square Lattice

Publisher Summary This chapter focuses on bond percolation on the square lattice and briefly describes percolation model; this model is a special but perhaps the most interesting case of the general theory of percolation. It introduces the FKG inequality of Fortuin, Kasteleyn, and Ginibre; it proves a remarkable inequality showing that nondecreasing functions on a finite distributive lattice are positively correlated by all positive measures, which have a certain convexity property. The problem of percolation through an n × n sponge is introduced. The chapter also examines two of the possible critical probabilities pT and pH and uses the theory developed for the sponge problem to prove the result pT + pH = 1. By the percolation model on G, one mean the assignment of open or closed to each edge of G with probabilities p and q = 1 - p respectively, the assignments to be independent for each edge.

Random planar graphs

We study various properties of the random planar graph Rn, drawn uniformly at random from the class Pn of all simple planar graphs on n labelled vertices. In particular, we show that the probability that Rn is connected is bounded away from 0 and from 1. We also show for example that each positive integer k, with high probability Rn has linearly many vertices of a given degree, in each embedding Rn has linearly many faces of a given size, and Rn has exponentially many automorphisms.

The Computational Complexity of the Tutte Plane: the Bipartite Case

Along different curves and at different points of the ( x, y )-plane the Tutte polynomial evaluates a wide range of quantities. Some of these, such as the number of spanning trees of a graph and the partition function of the planar Ising model, can be computed in polynomial time, others are # P -hard. Here we give a complete characterisation of which points and curves are easy/hard in the bipartite case.

Finite particle systems and infection models

Infinite particle systems on lattices have been extensively studied in recent years. The main questions of interest concern the ergodic and limiting behaviour of these processes, and their relationship with the dimension of the underlying lattice. A comprehensive review is given by Durrett( 6 ). One of the more tractable of these processes is the voter model introduced by Clifford and Sudbury( 3 ) and much studied since, see for example the monograph by Griffeath( 8 ), or the papers by Harris( 11 ), Holley and Liggett( 13 ), Bramson and Griffeath( 1 ) and ( 2 ) or, for a more general approach, Kelly( 16 ). In this paper we consider the case where the underlying spatial structure is finite and examine the transient behaviour of the voter process and also the infection process introduced by Williams and Bjerknes( 21 ).

A randomised 3-colouring algorithm

This paper describes a randomised algorithm for the NP-complete problem of 3-colouring the vertices of a graph. The method is based on a model of repulsion in interacting particle systems. Although it seems to work well on most random inputs there is a critical phenomenon apparent reminiscent of critical behaviour in other areas of statistical mechanics.

Polynomial time randomized approximation schemes for Tutte–Gröthendieck invariants: The dense case

The Tutte‐Grothendieck polynomial T(G; x, y) of a graph G encodes numerous interesting combinatorial quantities associated with the graph. Its evaluation in various points in the (x, y) plane give the number of spanning forests of the graph, the number of its strongly connected orientations, the number of its proper k‐colorings, the (all terminal) reliability probability of the graph, and various other invariants the exact computation of each of which is well known to be #P‐hard. Here we develop a general technique that supplies fully polynomial randomised approximation schemes for approximating the value of T(G; x, y) for any dense graph G, that is, any graph on n vertices whose minimum.

Complexity and Cryptography: An Introduction

Cryptography plays a crucial role in many aspects of todays world, from internet banking and ecommerce to email and web-based business processes. Understanding the principles on which it is based is an important topic that requires a knowledge of both computational complexity and a range of topics in pure mathematics. This book provides that knowledge, combining an informal style with rigorous proofs of the key results to give an accessible introduction. It comes with plenty of examples and exercises (many with hints and solutions), and is based on a highly successful course developed and taught over many years to undergraduate and graduate students in mathematics and computer science.

On the asymptotic proportion of connected matroids

Very little is known about the asymptotic behavior of classes of matroids. We make a number of conjectures about such behaviors. For example, we conjecture that asymptotically almost every matroid: has a trivial automorphism group; is arbitrarily highly connected; and is not representable over any field. We prove one result: the proportion of labeled n-element matroids that are connected is asymptotically at least 1/2.

Kruskal's theorem for matroids

Kruskals theorem for obtaining a minimal (maximal) spanning tree of a graph is shown to be a special case of a more general theorem for matroid spaces in which each element of the matroid has an associated weight. Since any finite subset of a vector space can be regarded as a matroid space this theorem gives an easy method of selecting a linearly independent set of vectors of minimal (maximal) weight.

Euler and bipartite matroids

Abstract We show that for binary matroids the properties of being Euler and bipartite are dual concepts, thus generalizing Eulers theorem for graphs.