arXiv subjects

Geoffrey R. Grimmett

Publications and source records attributed to Geoffrey R. Grimmett.

At least 19 recordsLinked to original sources

arXiv2026

Connective constants of Grigorchuk graphs

The connective constant $μ(G)$ of a graph $G$ is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. We prove upper and lower bounds for the connective constants of Cayley graphs $G_ω$ of a general Grigorchuk group encoded by a sequence $ω\in\{0,1,2\}^{\Bbb N}$. In particular, $μ(G_ω) > ϕ$ for any such Cayley graph (subject to a simple condition on $ω$), where $ϕ:= \frac12(1+\sqrt 5)$ is the golden mean. This extends earlier work of the author and Zhongyang Li in "Cubic graphs and the golden mean'', Discrete Math. 343 (2020), article 111638, where it was conjectured that $μ(G)\geϕ$ for all infinite, vertex-transitive, cubic graphs. The current work includes an analysis of the proportions of appearances of given label-sequences in the orbital Schreier graphs of general Grigorchuk groups.

math.CO↗

arXiv2026

Coalescence in Markov chains

A Markov chain $X^i$ on a finite state space $S$ has transition matrix $P$ and initial state $i$. We may run the chains $(X^i: i\in S)$ in parallel, while insisting that any two such chains coalesce whenever they are simultaneously at the same state. There are $|S|$ trajectories which evolve separately, but not necessarily independently, prior to coalescence. What can be said about the number $k(μ)$ of coalescence classes of the process, and what is the set $K(P)$ of such numbers $k(μ)$, as the coupling $μ$ of the chains ranges over couplings that are consistent with $P$? We continue earlier work of the authors ('Non-coupling from the past', $\textit{In and Out of Equilibrium 3}$, Springer, 2021) on these two fundamental questions, which have special importance for the 'coupling from the past' algorithm. We concentrate partly on a family of couplings termed block measures, which may be viewed as couplings of lumpable chains with coalescing lumps. Constructions of such couplings are presented, and also of non-block measure with similar properties.

math.PR↗

arXiv2026

On counting polygons in a crystal

How many $n$-step polygons exist that contain a given vertex of an infinite quasi-transitive graph $G$? The exponential growth rate of such polygons is identified as the connective constant when $G$ has sub-exponential growth and possesses a so-called square graph height function. The last condition amounts to the requirement that $G$ has a certain ${\Bbb Z}^2$ action of automorphisms. The main theorem extends a result of Hammersley (Proc. Cambridge Philos. Soc. 57 (1961) 516--523) and others for the hypercubic lattice, and responds to Hammersley's challenge to prove such a result for more general "crystals''.

math.CO↗

arXiv2025

Non-coupling from the past

The method of 'coupling from the past' permits exact sampling from the invariant distribution of a Markov chain on a finite state space. The coupling is successful whenever the stochastic dynamics are such that there is coalescence of all trajectories. The issue of the coalescence or non-coalescence of trajectories of a finite state space Markov chain is investigated in this note. The notion of the 'coalescence number' $k(μ)$ of a Markovian coupling $μ$ is introduced, and results are presented concerning the set $K(P)$ of coalescence numbers of couplings corresponding to a given transition matrix $P$. Note: This is a revision of the original published version, in which part of Theorem 6 has been removed. A correction may be found in Thm 5.3 of arXiv:2510.13572.

math.PR↗

arXiv2025

Alice and Bob on $\Bbb X$: reversal, coupling, renewal

A neat question involving coin flips surfaced on $\Bbb X$, and generated an intensive `storm' of `social mathematics'. In a sequence of flips of a fair coin, Alice wins a point at each appearance of two consecutive heads, and Bob wins a point whenever a head is followed immediately by a tail. Who is more likely to win the game? The subsequent discussion illustrated conflicting intuitions, and concluded with the correct answer (it is a close thing). It is explained here why the context of the question is interesting and how it may be answered in a quantitative manner using the probabilistic techniques of reversal, coupling, and renewal.

math.PR↗

arXiv2025

Non-self-touching paths in plane graphs

A path in a graph $G$ is called non-self-touching if two vertices are neighbours in the path if and only if they are neighbours in the graph. We investigate the existence of doubly infinite non-self-touching paths in infinite plane graphs. The matching graph $G_*$ of an infinite plane graph $G$ is obtained by adding all diagonals to all faces, and it plays an important role in the theory of site percolation on $G$. The main result of this paper is a necessary and sufficient condition on $G$ for the existence of a doubly infinite non-self-touching path in $G_*$ that traverses some diagonal. This is a key step in proving, for quasi-transitive $G$, that the critical points of site percolation on $G$ and $G_*$ satisfy the strict inequality $p_c(G_*) < p_c(G)$, and it complements the earlier result of Grimmett and Li (Random Struct. Alg. 65 (2024) 832--856), proved by different methods, concerning the case of transitive graphs. Furthermore it implies, for quasi-transitive graphs, that $p_u(G) + p_c(G) \ge 1$, with equality if and only if the graph $G_Δ$, obtained from $G$ by emptying all separating triangles, is a triangulation. Here, $p_u$ is the critical probability for the existence of a unique infinite open cluster.

math.CO↗

arXiv2024

Hyperbolic site percolation

Several results are presented for site percolation on quasi-transitive, planar graphs $G$ with one end, when properly embedded in either the Euclidean or hyperbolic plane. If $(G_1,G_2)$ is a matching pair derived from some quasi-transitive mosaic $M$, then $p_u(G_1)+p_c(G_2)=1$, where $p_c$ is the critical probability for the existence of an infinite cluster, and $p_u$ is the critical value for the existence of a unique such cluster. This fulfils and extends to the hyperbolic plane an observation of Sykes and Essam(1964), and it extends to quasi-transitive site models a theorem of Benjamini and Schramm (Theorem 3.8, J. Amer. Math. Soc. 14 (2001) 487--507) for transitive bond percolation. It follows that $p_u (G)+p_c (G_*)=p_u(G_*)+p_c(G)=1$, where $G_*$ denotes the matching graph of $G$. In particular, $p_u(G)+p_c(G)\ge 1$ and hence, when $G$ is amenable we have $p_c(G)=p_u(G) \ge \frac12$. When combined with the main result of the companion paper by the same authors ("Percolation critical probabilities of matching lattice-pairs", Random Struct. Alg. 2024), we obtain for transitive $G$ that the strict inequality $p_u(G)+p_c(G)> 1$ holds if and only if $G$ is not a triangulation. A key technique is a method for expressing a planar site percolation process on a matching pair in terms of a dependent bond process on the corresponding dual pair of graphs. Amongst other things, the results reported here answer positively two conjectures of Benjamini and Schramm (Conjectures 7 and 8, Electron. Comm. Probab. 1 (1996) 71--82) in the case of quasi-transitive graphs.

math.PR↗

arXiv2024

Dominic Welsh (1938-2023)

This biographical and scientific memoir of Dominic Welsh includes summaries of his important contributions to probability and combinatorics. With John Hammersley, he introduced first-passage percolation, and in so doing they formulated and proved the first subadditive ergodic theorem. Welsh has numerous results in matroid theory, and wrote the first monograph on the topic. He worked on computational complexity and particularly the complexity of computing the Tutte polynomial. He was an inspirational teacher and advisor who helped to develop a community of scholars in combinatorics.

math.HO↗

arXiv2024

Percolation critical probabilities of matching lattice-pairs

A necessary and sufficient condition is established for the strict inequality $p_c(G_*)<p_c(G)$ between the critical probabilities of site percolation on a quasi-transitive, plane graph $G$ and on its matching graph $G_*$. It is assumed that $G$ is properly embedded in either the Euclidean or the hyperbolic plane. When $G$ is transitive, strict inequality holds if and only if $G$ is not a triangulation. The basic approach is the standard method of enhancements, but its implemention has complexity arising from the non-Euclidean (hyperbolic) space, the study of site (rather than bond) percolation, and the generality of the assumption of quasi-transitivity. This result is complementary to the work of the authors ("Hyperbolic site percolation", arXiv:2203.00981) on the equality $p_u(G) + p_c(G_*) = 1$, where $p_u$ is the critical probability for the existence of a unique infinite open cluster. It implies for transitive $G$ that $p_u(G) + p_c(G) \ge 1$, with equality if and only if $G$ is a triangulation.

math.PR↗

arXiv2022

The work of Hugo Duminil-Copin

This article is an account of the scientific work of Hugo Duminil-Copin at the time of his award in 2022 of the Fields Medal "for solving longstanding problems in the probabilistic theory of phase transitions in statistical physics, especially in dimensions three and four''.

math.PR↗

arXiv2022

Brownian snails with removal: epidemics in diffusing populations

Two stochastic models of susceptible/infected/removed (SIR) type are introduced for the spread of infection through a spatially-distributed population. Individuals are initially distributed at random in space, and they move continuously according to independent diffusion processes. The disease may pass from an infected individual to an uninfected individual when they are sufficiently close. Infected individuals are permanently removed at some given rate $α$. Such processes are reminiscent of so-called frog models, but differ through the action of removal, as well as the fact that frogs jump whereas snails slither. Two models are studied here, termed the `delayed diffusion' and the `diffusion' models. In the first, individuals are stationary until they are infected, at which time they begin to move; in the second, all individuals start to move at the initial time $0$. Using a perturbative argument, conditions are established under which the disease infects a.s. only finitely many individuals. It is proved for the delayed diffusion model that there exists a critical value $α_c\in(0,\infty)$ for the survival of the epidemic.

math.PR↗

arXiv2022

Selected problems in probability theory

This celebratory article contains a personal and idiosyncratic selection of a few open problems in discrete probability theory. These include certain well known questions concerning Lorentz scatterers and self-avoiding walks, and also some problems of percolation-type. The author hopes the reader will find something to leaven winter evenings, and perhaps even a project for the longer term.

math.PR↗

arXiv2021

Alignment percolation

The existence (or not) of infinite clusters is explored for two stochastic models of intersecting line segments in $d \ge 2$ dimensions. Salient features of the phase diagram are established in each case. The models are based on site percolation on ${\mathbb Z}^d$ with parameter $p\in (0,1]$. For each occupied site $v$, and for each of the $2d$ possible coordinate directions, declare the entire line segment from $v$ to the next occupied site in the given direction to be either blue or not blue according to a given stochastic rule. In the one-choice model, each occupied site declares one of its $2d$ incident segments to be blue. In the independent model, the states of different line segments are independent.

math.PR↗

arXiv2020

Harry Kesten's work in probability theory

We survey the published work of Harry Kesten in probability theory, with emphasis on his contributions to random walks, branching processes, percolation, and related topics. A complete bibliography is included of his publications.

math.PR↗

arXiv2020

Harry Kesten (1931-2019), a personal and scientific tribute

The mathematical achievements of Harry Kesten since the mid-1950s have revolutionized probability theory as a subject in its own right and in its associations with aspects of algebra, analysis, geometry, and statistical physics. Through his personality and scientific ability, he has framed the modern subject to a degree exceeded by no other. The impact of his work and personality is summarised in this memoir.

math.HO↗

arXiv2019

The lost boarding pass, and other practical problems

The reader is reminded of several puzzles involving randomness. These may be ill-posed, and if well-posed there is sometimes a solution that uses probabilistic intuition in a special way. Various examples are presented including the well known problem of the lost boarding pass: what is the probability that the last passenger boarding a fully booked plane sits in the assigned seat if the first passenger has occupied a randomly chosen seat? This problem, and its striking answer of $\frac12$, has attracted a good deal of attention since around 2000. We review elementary solutions to this, and to the more general problem of finding the probability the $m$th passenger sits in the assigned seat when in the presence of some number $k$ of passengers with lost boarding passes. A simple proof is presented of the independence of the occupancy status of different seats, and a connection to the Poisson--Dirichlet distribution is mentioned.

math.PR↗

arXiv2019

Cubic graphs and the golden mean

The connective constant $μ(G)$ of a graph $G$ is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. We investigate the validity of the inequality $μ\ge ϕ$ for infinite, transitive, simple, cubic graphs, where $ϕ:= \frac12(1+\sqrt 5)$ is the golden mean. The inequality is proved for several families of graphs including (i) Cayley graphs of infinite groups with three generators and strictly positive first Betti number, (ii) infinite, transitive, topologically locally finite (TLF) planar, cubic graphs, and (iii) cubic Cayley graphs with two ends. Bounds for $μ$ are presented for transitive cubic graphs with girth either $3$ or $4$, and for certain quasi-transitive cubic graphs.

math.CO↗

arXiv2019

Self-avoiding walks and connective constants

The connective constant $μ(G)$ of a quasi-transitive graph $G$ is the asymptotic growth rate of the number of self-avoiding walks (SAWs) on $G$ from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph $G$. $\bullet$ We present upper and lower bounds for $μ$ in terms of the vertex-degree and girth of a transitive graph. $\bullet$ We discuss the question of whether $μ\geϕ$ for transitive cubic graphs (where $ϕ$ denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vertices by triangles). $\bullet$ We present strict inequalities for the connective constants $μ(G)$ of transitive graphs $G$, as $G$ varies. $\bullet$ As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a further generator. $\bullet$ We describe so-called graph height functions within an account of "bridges" for quasi-transitive graphs, and indicate that the bridge constant equals the connective constant when the graph has a unimodular graph height function. $\bullet$ A partial answer is given to the question of the locality of connective constants, based around the existence of unimodular graph height functions. $\bullet$ Examples are presented of Cayley graphs of finitely presented groups that possess graph height functions (that are, in addition, harmonic and unimodular), and that do not. $\bullet$ The review closes with a brief account of the "speed" of SAW.

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