Matchings on infinite graphs

Elek and Lippner (2010) showed that the convergence of a sequence of bounded-degree graphs implies the existence of a limit for the proportion of vertices covered by a maximum matching. We provide a characterization of the limiting parameter via a local recursion defined directly on the limit of the graph sequence. Interestingly, the recursion may admit multiple solutions, implying non-trivial long-range dependencies between the covered vertices. We overcome this lack of correlation decay by introducing a perturbative parameter (temperature), which we let progressively go to zero. This allows us to uniquely identify the correct solution. In the important case where the graph limit is a unimodular Galton-Watson tree, the recursion simplifies into a distributional equation that can be solved explicitly, leading to a new asymptotic formula that considerably extends the well-known one by Karp and Sipser for Erd\"os-R\'enyi random graphs.Comment: 23 page

Dynamical Properties of one dimensional Mott Insulators

At low energies the charge sector of one dimensional Mott insulators can be described in terms of a quantum Sine-Gordon model. Using exact results derived from integrability it is possible to determine dynamical properties like the frequency dependent optical conductivity. We compare the exact results to perturbation theory and renormalisation group calculations. We also discuss the application of our results to experiments on quasi-1D organic conductors.Comment: 17 pages, 5 figures, to appear in the proceedings of the NATO ASI/EC summer school "New Theoretical Approaches to Strongly Correlated Systems" Newton Institute for Mathematical Sciences, Cambridge UK, April 200

recA mediated spontaneous deletions of the icaADBC operon of clinical Staphylococcus epidermidis isolates: a new mechanism of phenotypic variations

Phenotypic variation of Staphylococcus epidermidis involving the slime related ica operon results in heterogeneity in surface characteristics of individual bacteria in axenic cultures. Five clinical S. epidermidis isolates demonstrated phenotypic variation, i.e. both black and red colonies on Congo Red agar. Black colonies displayed bi-modal electrophoretic mobility distributions at pH 2, but such phenotypic variation was absent in red colonies of the same strain as well as in control strains without phenotypic variation. All red colonies had lost ica and the ability to form biofilms, in contrast to black colonies of the same strain. Real time PCR targeting icaA indicated a reduction in gene copy number within cultures exhibiting phenotypic variation, which correlated with phenotypic variations in biofilm formation and electrophoretic mobility distribution of cells within a culture. Loss of ica was irreversible and independent of the mobile element IS256. Instead, in high frequency switching strains, spontaneous mutations in lexA were found which resulted in deregulation of recA expression, as shown by real time PCR. RecA is involved in genetic deletions and rearrangements and we postulate a model representing a new mechanism of phenotypic variation in clinical isolates of S. epidermidis. This is the first report of S. epidermidis strains irreversibly switching from biofilm-positive to biofilm-negative phenotype by spontaneous deletion of icaADBC

On the Absence of Phase Transition in the Monomer-Dimer Model

Suppose we cover the set of vertices of a graph $G$ by non-overlapping monomers (singleton sets) and dimers (pairs of vertices corresponding to an edge). Each way to do this is called a monomer-dimer configuration. If $G$ is finite and $lambda > 0$, we define the monomer-dimer distribution for $G$ (with parameter $lambda$) as the probability distribution which assigns to each monomer-dimer configuration a probability proportional to lambda^{|mbox{dimers|, where |mbox{dimers| is the number of dimers in that configuration. If the graph is infinite, monomer-dimer distributions can be constructed in the standard way, by taking weak limits. We are particularly interested in the monomer-dimer model on (subgraphs of) the $d$-dimensional cubic lattice. Heilmann and Lieb (1972) prove absence of phase transition, in terms of smoothness properties of certain thermodynamic functions. They do this by studying the location in the complex plane of the zeros of the partition function. We present a different approach and show, by probabilistic arguments, that boundary effects become negligible as the distance to the boundary goes to $infty$. This gives absence of phase transition in a related, but generally not equivalent sense as above. However, the decay of boundary effects appears to occur in such a strong way that, by results on general Gibbs systems of Dobrushin and Shlosman (1987) and Dobrushin and Warstat (1990), smoothness properties of thermodynamic functions follow. More precisely we show that, in their terminology, the model is {em completely analytic