arXiv subjects

Geoffrey Grimmett

Publications and source records attributed to Geoffrey Grimmett.

At least 19 recordsLinked to original sources

arXiv2019

Bounded entanglement entropy in the quantum Ising model

A rigorous proof is presented of the boundedness of the entanglement entropy of a block of spins for the ground state of the one-dimensional quantum Ising model with sufficiently strong transverse field. This is proved by a refinement of the arguments in the earlier work by the same authors (J. Statist. Phys. 131 (2008) 305-339). The proof is geometrical, and utilises a transformation to a model of classical probability called the continuum random-cluster model. Our method of proof is fairly robust, and applies also to certain disordered systems.

quant-ph↗

arXiv2015

Influence in product spaces

The theory of influence and sharp threshold is a key tool in probability and probabilistic combinatorics, with numerous applications. One significant aspect of the theory is directed at identifying the level of generality of the product probability space that accommodates the event under study. We derive the influence inequality for a completely general product space, by establishing a relationship to the Lebesgue cube studied by Bourgain, Kahn, Kalai, Katznelson, and Linial (BKKKL) in 1992. This resolves one of the assertions of BKKKL. Our conclusion is valid also in the setting of the generalized influences of Keller.

math.PR↗

arXiv2013

Bond percolation on isoradial graphs: criticality and universality

In an investigation of percolation on isoradial graphs, we prove the criticality of canonical bond percolation on isoradial embeddings of planar graphs, thus extending celebrated earlier results for homogeneous and inhomogeneous square, triangular, and other lattices. This is achieved via the star-triangle transformation, by transporting the box-crossing property across the family of isoradial graphs. As a consequence, we obtain the universality of these models at the critical point, in the sense that the one-arm and 2j-alternating-arm critical exponents (and therefore also the connectivity and volume exponents) are constant across the family of such percolation processes. The isoradial graphs in question are those that satisfy certain weak conditions on their embedding and on their track system. This class of graphs includes, for example, isoradial embeddings of periodic graphs, and graphs derived from rhombic Penrose tilings.

math.PR↗

arXiv2012

Three theorems in discrete random geometry

These notes are focused on three recent results in discrete random geometry, namely: the proof by Duminil-Copin and Smirnov that the connective constant of the hexagonal lattice is \sqrt{2+\sqrt 2}; the proof by the author and Manolescu of the universality of inhomogeneous bond percolation on the square, triangular, and hexagonal lattices; the proof by Beffara and Duminil-Copin that the critical point of the random-cluster model on Z^2 is \sqrt q/(1+\sqrt q). Background information on the relevant random processes is presented on route to these theorems. The emphasis is upon the communication of ideas and connections as well as upon the detailed proofs.

math.PR↗

arXiv2011

A power-weighted variant of the EU27 Cambridge Compromise

The Cambridge Compromise composition of the European Parliament allocates five base seats to each Member State's citizenry, and apportions the remaining seats proportionately to population figures using the divisor method with rounding upwards and observing a 96 seat capping. The power-weighted variant avoids the capping step, proceeding instead by a progressive non-linear downweighting of the population figures until the largest State is allocated exactly 96 seats. The pertinent calculations of the variant are described, and its relative constitutional merits are discussed.

math.HO↗

arXiv2011

Sharp thresholds for the random-cluster and Ising models

A sharp-threshold theorem is proved for box-crossing probabilities on the square lattice. The models in question are the random-cluster model near the self-dual point $p_{\mathrm {sd}}(q)=\sqrt{q}/(1+\sqrt{q})$, the Ising model with external field, and the colored random-cluster model. The principal technique is an extension of the influence theorem for monotonic probability measures applied to increasing events with no assumption of symmetry.

math.PR↗

arXiv2009

Three problems for the clairvoyant demon

A number of tricky problems in probability are discussed, having in common one or more infinite sequences of coin tosses, and a representation as a problem in dependent percolation. Three of these problems are of `Winkler' type, that is, they ask about what can be achieved by a clairvoyant demon.

math.PR↗

arXiv2009

Random even graphs

We study a random even subgraph of a finite graph $G$ with a general edge-weight $p\in(0,1)$. We demonstrate how it may be obtained from a certain random-cluster measure on $G$, and we propose a sampling algorithm based on coupling from the past. A random even subgraph of a planar lattice undergoes a phase transition at the parameter-value $\frac 12 \pc$, where $\pc$ is the critical point of the $q=2$ random-cluster model on the dual lattice. The properties of such a graph are discussed, and are related to Schramm--Löwner evolutions (SLE).

math.PR↗

arXiv2008

David George Kendall, a biographical account

This biographical account of the life and work of David Kendall includes details of his personal and professional activities. Kendall is probably best known for his work in applied probability, especially queueing theory, and in stochastic analysis and spatial statistics.

math.HO↗

arXiv2007

Random graphs with forbidden vertex degrees

We study the random graph G_{n,λ/n} conditioned on the event that all vertex degrees lie in some given subset S of the non-negative integers. Subject to a certain hypothesis on S, the empirical distribution of the vertex degrees is asymptotically Poisson with some parameter \mux given as the root of a certain `characteristic equation' of S that maximises a certain function \psis(μ). Subject to a hypothesis on S, we obtain a partial description of the structure of such a random graph, including a condition for the existence (or not) of a giant component. The requisite hypothesis is in many cases benign, and applications are presented to a number of choices for the set S including the sets of (respectively) even and odd numbers. The random \emph{even} graph is related to the random-cluster model on the complete graph K_n.

math.PR↗

arXiv2007

Space-time percolation

The contact model for the spread of disease may be viewed as a directed percolation model on $\ZZ \times \RR$ in which the continuum axis is oriented in the direction of increasing time. Techniques from percolation have enabled a fairly complete analysis of the contact model at and near its critical point. The corresponding process when the time-axis is unoriented is an undirected percolation model to which now standard techniques may be applied. One may construct in similar vein a random-cluster model on $\ZZ \times \RR$, with associated continuum Ising and Potts models. These models are of independent interest, in addition to providing a path-integral representation of the quantum Ising model with transverse field. This representation may be used to obtain a bound on the entanglement of a finite set of spins in the quantum Ising model on $\ZZ$, where this entanglement is measured via the entropy of the reduced density matrix. The mean-field version of the quantum Ising model gives rise to a random-cluster model on $K_n \times \RR$, thereby extending the Erdos-Renyi random graph on the complete graph $K_n$.

math.PR↗

arXiv2007

Entanglement in the quantum Ising model

We study the asymptotic scaling of the entanglement of a block of spins for the ground state of the one-dimensional quantum Ising model with transverse field. When the field is sufficiently strong, the entanglement grows at most logarithmically in the number of spins. The proof utilises a transformation to a model of classical probability called the continuum random-cluster model, and is based on a property of the latter model termed ratio weak-mixing. Our proof applies equally to a large class of disordered interactions.

quant-ph↗

arXiv2006

John Michael Hammersley (1920-2004)

In writing this biographical memoir of John Hammersley, we have tried to communicate something of the character of the person, and of the impact of his scientific achievements across lattice models (for example, percolation, self-avoiding walks, first-passage percolation, dimer models), stochastic processes (including subadditive ergodic theory), Monte Carlo methods, applied probability, statistics, and other areas to which he contributed. There is also an extended account of his earlier life, taken from autobiographical notes written around 1994, together with a list of his published work.

math.PR↗

arXiv2006

Uniqueness and multiplicity of infinite clusters

The Burton--Keane theorem for the almost-sure uniqueness of infinite clusters is a landmark of stochastic geometry. Let $μ$ be a translation-invariant probability measure with the finite-energy property on the edge-set of a $d$-dimensional lattice. The theorem states that the number $I$ of infinite components satisfies $μ(I\in\{0,1\})=1$. The proof is an elegant and minimalist combination of zero--one arguments in the presence of amenability. The method may be extended (not without difficulty) to other problems including rigidity and entanglement percolation, as well as to the Gibbs theory of random-cluster measures, and to the central limit theorem for random walks in random reflecting labyrinths. It is a key assumption on the underlying graph that the boundary/volume ratio tends to zero for large boxes, and the picture for non-amenable graphs is quite different.

math.PR↗

arXiv2005

Flows and ferromagnets

The two-point correlation function of a Potts model on a graph $G$ may be expressed in terms of the flow polynomials of `Poissonian' random graphs derived from $G$ by replacing each edge by a Poisson-distributed number of copies of itself. This fact extends to Potts models the so-called random-current expansion of the Ising model.

math.PR↗

arXiv2004

Branching Processes, and Random-Cluster Measures on Trees

Random-cluster measures on infinite regular trees are studied in conjunction with a general type of `boundary condition', namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of random-cluster measures are explored for certain classes of equivalence relations. In proving uniqueness, the following problem concerning branching processes is encountered and answered. Consider bond percolation on the family-tree $T$ of a branching process. What is the probability that every infinite path of $T$, beginning at its root, contains some vertex which is itself the root of an infinite open sub-tree?

math.PR↗

arXiv2003

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 $n \to \infty$ 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.

quant-ph↗

arXiv2003

The Random-Cluster Model

The class of random-cluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, and of electrical networks. Much (but not all) of the physical theory of Ising/Potts models is best implemented in the context of the random-cluster representation. This systematic summary of random-cluster models includes accounts of the fundamental methods and inequalities, the uniqueness and specification of infinite-volume measures, the existence and nature of the phase transition, and the structure of the subcritical and supercritical phases. The theory for two-dimensional lattices is better developed than for three and more dimensions. There is a rich collection of open problems, including some of substantial significance for the general area of disordered systems, and these are highlighted when encountered. Amongst the major open questions, there is the problem of ascertaining the exact nature of the phase transition for general values of the cluster-weighting factor q, and the problem of proving that the critical random-cluster model in two dimensions, with 1\le q\le 4, converges when re-scaled to a stochastic Löwner evolution. Overall the emphasis is upon the random-cluster model for its own sake, rather than upon its applications to Ising and Potts systems.

math.PR↗