Scaling universalities of kth-nearest neighbor distances on closed manifolds

Take N sites distributed randomly and uniformly on a smooth closed surface. We express the expected distance from an arbitrary point on the surface to its kth-nearest neighboring site, in terms of the function A(l) giving the area of a disc of radius l about that point. We then find two universalities. First, for a flat surface, where A(l)=\pi l^2, the k-dependence and the N-dependence separate in . All kth-nearest neighbor distances thus have the same scaling law in N. Second, for a curved surface, the average \int d\mu over the surface is a topological invariant at leading and subleading order in a large N expansion. The 1/N scaling series then depends, up through O(1/N), only on the surface's topology and not on its precise shape. We discuss the case of higher dimensions (d>2), and also interpret our results using Regge calculus.Comment: 14 pages, 2 figures; submitted to Advances in Applied Mathematic

Glass models on Bethe lattices

We consider ``lattice glass models'' in which each site can be occupied by at most one particle, and any particle may have at most l occupied nearest neighbors. Using the cavity method for locally tree-like lattices, we derive the phase diagram, with a particular focus on the vitreous phase and the highest packing limit. We also study the energy landscape via the configurational entropy, and discuss different equilibrium glassy phases. Finally, we show that a kinetic freezing, depending on the particular dynamical rules chosen for the model, can prevent the equilibrium glass transitions.Comment: 24 pages, 11 figures; minor corrections + enlarged introduction and conclusio

Cut Size Statistics of Graph Bisection Heuristics

We investigate the statistical properties of cut sizes generated by heuristic algorithms which solve approximately the graph bisection problem. On an ensemble of sparse random graphs, we find empirically that the distribution of the cut sizes found by ``local'' algorithms becomes peaked as the number of vertices in the graphs becomes large. Evidence is given that this distribution tends towards a Gaussian whose mean and variance scales linearly with the number of vertices of the graphs. Given the distribution of cut sizes associated with each heuristic, we provide a ranking procedure which takes into account both the quality of the solutions and the speed of the algorithms. This procedure is demonstrated for a selection of local graph bisection heuristics.Comment: 17 pages, 5 figures, submitted to SIAM Journal on Optimization also available at http://ipnweb.in2p3.fr/~martin

The random link approximation for the Euclidean traveling salesman problem

The traveling salesman problem (TSP) consists of finding the length of the shortest closed tour visiting N ``cities''. We consider the Euclidean TSP where the cities are distributed randomly and independently in a d-dimensional unit hypercube. Working with periodic boundary conditions and inspired by a remarkable universality in the kth nearest neighbor distribution, we find for the average optimum tour length = beta_E(d) N^{1-1/d} [1+O(1/N)] with beta_E(2) = 0.7120 +- 0.0002 and beta_E(3) = 0.6979 +- 0.0002. We then derive analytical predictions for these quantities using the random link approximation, where the lengths between cities are taken as independent random variables. From the ``cavity'' equations developed by Krauth, Mezard and Parisi, we calculate the associated random link values beta_RL(d). For d=1,2,3, numerical results show that the random link approximation is a good one, with a discrepancy of less than 2.1% between beta_E(d) and beta_RL(d). For large d, we argue that the approximation is exact up to O(1/d^2) and give a conjecture for beta_E(d), in terms of a power series in 1/d, specifying both leading and subleading coefficients.Comment: 29 pages, 6 figures; formatting and typos correcte

Entanglement of localized states

We derive exact expressions for the mean value of Meyer-Wallach entanglement Q for localized random vectors drawn from various ensembles corresponding to different physical situations. For vectors localized on a randomly chosen subset of the basis, tends for large system sizes to a constant which depends on the participation ratio, whereas for vectors localized on adjacent basis states it goes to zero as a constant over the number of qubits. Applications to many-body systems and Anderson localization are discussed.Comment: 6 pages, 4 figure

Do Childhood Vaccines Have Non-Specific Effects on Mortality

A recent article by Kristensen et al. suggested that measles vaccine and bacille Calmette–Guérin (BCG) vaccine might\ud reduce mortality beyond what is expected simply from protection against measles and tuberculosis. Previous reviews of the potential effects of childhood vaccines on mortality have not considered methodological features of reviewed studies. Methodological considerations play an especially important role in observational assessments, in which selection factors for vaccination may be difficult to ascertain. We reviewed 782 English language articles on vaccines and childhood mortality and found only a few whose design met the criteria for methodological rigor. The data reviewed suggest that measles vaccine delivers its promised reduction in mortality, but there is insufficient evidence to suggest a mortality benefit above that caused by its effect on measles disease and its sequelae. Our review of the available data in the literature reinforces how difficult answering these considerations has been and how important study design will be in determining the effect of specific vaccines on all-cause mortality.\u