Self-avoiding walks and connective constants

The connective constant $\mu(G)$ 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 $\mu$ in terms of the vertex-degree and girth of a transitive graph. $\bullet$ We discuss the question of whether $\mu\ge\phi$ for transitive cubic graphs (where $\phi$ 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 $\mu(G)$ 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.Comment: Accepted version. arXiv admin note: substantial text overlap with arXiv:1304.721

Equality of bond percolation critical exponents for pairs of dual lattices

For a certain class of two-dimensional lattices, lattice-dual pairs are shown to have the same bond percolation critical exponents. A computational proof is given for the martini lattice and its dual to illustrate the method. The result is generalized to a class of lattices that allows the equality of bond percolation critical exponents for lattice-dual pairs to be concluded without performing the computations. The proof uses the substitution method, which involves stochastic ordering of probability measures on partially ordered sets. As a consequence, there is an infinite collection of infinite sets of two-dimensional lattices, such that all lattices in a set have the same critical exponents.Comment: 10 pages, 7 figure

Ovarian and cervical cancer awareness: development of two validated measurement tools.

The aim of the study was to develop and validate measures of awareness of symptoms and risk factors for ovarian and cervical cancer (Ovarian and Cervical Cancer Awareness Measures)

FSA field test report, 1980 - 1982

Photovoltaic modules made of new and developing materials were tested in a continuing study of weatherability, compatibility, and corrosion protection. Over a two-year period, 365 two-cell submodules have been exposed for various intervals at three outdoor sites in Southern California or subjected to laboratory acceptance tests. Results to date show little loss of maximum power output, except in two types of modules. In the first of these, failure is due to cell fracture from the stresses that arise as water is regained from the surrounding air by a hardboard substrate, which shrank as it dried during its encapsulation in plastic film at 150 C in vacuo. In the second, the glass superstrate is sensitive to cracking, which also damages the cells electrostatically bonded to it; inadequate bonding of interconnects to the cells is also a problem in these modules. In a third type of module, a polyurethane pottant has begun to yellow, though as yet without significant effect on maximum power output

Predictions of bond percolation thresholds for the kagom\'e and Archimedean (3,122)(3,12^2) lattices

Here we show how the recent exact determination of the bond percolation threshold for the martini lattice can be used to provide approximations to the unsolved kagom\'e and (3,12^2) lattices. We present two different methods, one of which provides an approximation to the inhomogeneous kagom\'e and (3,12^2) bond problems, and the other gives estimates of $p_c$ for the homogeneous kagom\'e (0.5244088...) and (3,12^2) (0.7404212...) problems that respectively agree with numerical results to five and six significant figures.Comment: 4 pages, 5 figure

Approximate quantum error correction, random codes, and quantum channel capacity

We work out a theory of approximate quantum error correction that allows us to derive a general lower bound for the entanglement fidelity of a quantum code. The lower bound is given in terms of Kraus operators of the quantum noise. This result is then used to analyze the average error correcting performance of codes that are randomly drawn from unitarily invariant code ensembles. Our results confirm that random codes of sufficiently large block size are highly suitable for quantum error correction. Moreover, employing a lemma of Bennett, Shor, Smolin, and Thapliyal, we prove that random coding attains information rates of the regularized coherent information.Comment: 29 pages, final version to appear in Phys. Rev. A, improved lower bound for code entanglement fidelity, simplified proo

Security of Quantum Bit-String Generation

We consider the cryptographic task of bit-string generation. This is a generalisation of coin tossing in which two mistrustful parties wish to generate a string of random bits such that an honest party can be sure that the other cannot have biased the string too much. We consider a quantum protocol for this task, originally introduced in Phys. Rev. A {\bf 69}, 022322 (2004), that is feasible with present day technology. We introduce security conditions based on the average bias of the bits and the Shannon entropy of the string. For each, we prove rigorous security bounds for this protocol in both noiseless and noisy conditions under the most general attacks allowed by quantum mechanics. Roughly speaking, in the absence of noise, a cheater can only bias significantly a vanishing fraction of the bits, whereas in the presence of noise, a cheater can bias a constant fraction, with this fraction depending quantitatively on the level of noise. We also discuss classical protocols for the same task, deriving upper bounds on how well a classical protocol can perform. This enables the determination of how much noise the quantum protocol can tolerate while still outperforming classical protocols. We raise several conjectures concerning both quantum and classical possibilities for large n cryptography. An experiment corresponding to the scheme analysed in this paper has been performed and is reported elsewhere.Comment: 16 pages. No figures. Accepted for publication in Phys. Rev. A. A corresponding experiment is reported in quant-ph/040812

Enhancement of Entanglement Percolation in Quantum Networks via Lattice Transformations

We study strategies for establishing long-distance entanglement in quantum networks. Specifically, we consider networks consisting of regular lattices of nodes, in which the nearest neighbors share a pure, but non-maximally entangled pair of qubits. We look for strategies that use local operations and classical communication. We compare the classical entanglement percolation protocol, in which every network connection is converted with a certain probability to a singlet, with protocols in which classical entanglement percolation is preceded by measurements designed to transform the lattice structure in a way that enhances entanglement percolation. We analyze five examples of such comparisons between protocols and point out certain rules and regularities in their performance as a function of degree of entanglement and choice of operations.Comment: 12 pages, 17 figures, revtex4. changes from v3: minor stylistic changes for journal reviewer, minor changes to figures for journal edito

Simultaneous readout of two charge qubits

We consider a system of two solid state charge qubits, coupled to a single read-out device, consisting of a single-electron transistor (SET). The conductance of each tunnel junction is influenced by its neighboring qubit, and thus the current through the transistor is determined by the qubits' state. The full counting statistics of the electrons passing the transistor is calculated, and we discuss qubit dephasing, as well as the quantum efficiency of the readout. The current measurement is then compared to readout using real-time detection of the SET island's charge state. For the latter method we show that the quantum efficiency is always unity. Comparing the two methods a simple geometrical interpretation of the quantum efficiency of the current measurement appears. Finally, we note that full quantum efficiency in some cases can be achieved measuring the average charge of the SET island, in addition to the average current.Comment: 11 pages with 5 figure

Random-cluster representation of the Blume-Capel model

The so-called diluted-random-cluster model may be viewed as a random-cluster representation of the Blume--Capel model. It has three parameters, a vertex parameter $a$, an edge parameter $p$, and a cluster weighting factor $q$. Stochastic comparisons of measures are developed for the `vertex marginal' when $q\in[1,2]$, and the `edge marginal' when q\in[1,\oo). Taken in conjunction with arguments used earlier for the random-cluster model, these permit a rigorous study of part of the phase diagram of the Blume--Capel model