Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop

We consider quadrangulations with a boundary and derive explicit expressions for the generating functions of these maps with either a marked vertex at a prescribed distance from the boundary, or two boundary vertices at a prescribed mutual distance in the map. For large maps, this yields explicit formulas for the bulk-boundary and boundary-boundary correlators in the various encountered scaling regimes: a small boundary, a dense boundary and a critical boundary regime. The critical boundary regime is characterized by a one-parameter family of scaling functions interpolating between the Brownian map and the Brownian Continuum Random Tree. We discuss the cases of both generic and self-avoiding boundaries, which are shown to share the same universal scaling limit. We finally address the question of the bulk-loop distance statistics in the context of planar quadrangulations equipped with a self-avoiding loop. Here again, a new family of scaling functions describing critical loops is discovered.Comment: 55 pages, 14 figures, final version with minor correction

Combinatorics of bicubic maps with hard particles

We present a purely combinatorial solution of the problem of enumerating planar bicubic maps with hard particles. This is done by use of a bijection with a particular class of blossom trees with particles, obtained by an appropriate cutting of the maps. Although these trees have no simple local characterization, we prove that their enumeration may be performed upon introducing a larger class of "admissible" trees with possibly doubly-occupied edges and summing them with appropriate signed weights. The proof relies on an extension of the cutting procedure allowing for the presence on the maps of special non-sectile edges. The admissible trees are characterized by simple local rules, allowing eventually for an exact enumeration of planar bicubic maps with hard particles. We also discuss generalizations for maps with particles subject to more general exclusion rules and show how to re-derive the enumeration of quartic maps with Ising spins in the present framework of admissible trees. We finally comment on a possible interpretation in terms of branching processes.Comment: 41 pages, 19 figures, tex, lanlmac, hyperbasics, epsf. Introduction and discussion/conclusion extended, minor corrections, references adde

Distance statistics in large toroidal maps

We compute a number of distance-dependent universal scaling functions characterizing the distance statistics of large maps of genus one. In particular, we obtain explicitly the probability distribution for the length of the shortest non-contractible loop passing via a random point in the map, and that for the distance between two random points. Our results are derived in the context of bipartite toroidal quadrangulations, using their coding by well-labeled 1-trees, which are maps of genus one with a single face and appropriate integer vertex labels. Within this framework, the distributions above are simply obtained as scaling limits of appropriate generating functions for well-labeled 1-trees, all expressible in terms of a small number of basic scaling functions for well-labeled plane trees.Comment: 24 pages, 9 figures, minor corrections, new added reference

Reliability in digital systems with asymmetrical failure modes

Reliability schemes in digital systems with asymmetrical failure mode

Force distributions in a triangular lattice of rigid bars

We study the uniformly weighted ensemble of force balanced configurations on a triangular network of nontensile contact forces. For periodic boundary conditions corresponding to isotropic compressive stress, we find that the probability distribution for single-contact forces decays faster than exponentially. This super-exponential decay persists in lattices diluted to the rigidity percolation threshold. On the other hand, for anisotropic imposed stresses, a broader tail emerges in the force distribution, becoming a pure exponential in the limit of infinite lattice size and infinitely strong anisotropy.Comment: 11 pages, 17 figures Minor text revisions; added references and acknowledgmen

Binding Energy and the Fundamental Plane of Globular Clusters

A physical description of the fundamental plane of Galactic globular clusters is developed which explains all empirical trends and correlations in a large number of cluster observables and provides a small but complete set of truly independent constraints on theories of cluster formation and evolution in the Milky Way. Within the theoretical framework of single-mass, isotropic King models, it is shown that (1) 39 regular (non--core-collapsed) globulars with measured core velocity dispersions share a common V-band mass-to-light ratio of 1.45 +/- 0.10, and (2) a complete sample of 109 regular globulars reveals a very strong correlation between cluster binding energy and total luminosity, regulated by Galactocentric position: E_b \propto (L^{2.05} r_{\rm gc}^{-0.4}). The observational scatter about either of these two constraints can be attributed fully to random measurement errors, making them the defining equations of a fundamental plane for globular clusters. A third, weaker correlation, between total luminosity and the King-model concentration parameter, c, is then related to the (non-random) distribution of globulars on the plane. The equations of the FP are used to derive expressions for any cluster observable in terms of only L, r_{\rm gc}, and c. Results are obtained for generic King models and applied specifically to the globular cluster system of the Milky Way.Comment: 60 pages with 19 figures, submitted to Ap

Advances on Testing C-Planarity of Embedded Flat Clustered Graphs

We show a polynomial-time algorithm for testing c-planarity of embedded flat clustered graphs with at most two vertices per cluster on each face.Comment: Accepted at GD '1

Force distribution in a scalar model for non-cohesive granular material

We study a scalar lattice model for inter-grain forces in static, non-cohesive, granular materials, obtaining two primary results. (i) The applied stress as a function of overall strain shows a power law dependence with a nontrivial exponent, which moreover varies with system geometry. (ii) Probability distributions for forces on individual grains appear Gaussian at all stages of compression, showing no evidence of exponential tails. With regard to both results, we identify correlations responsible for deviations from previously suggested theories.Comment: 16 pages, 9 figures, Submitted to PR

Waves attractors in rotating fluids: a paradigm for ill-posed Cauchy problems

In the limit of low viscosity, we show that the amplitude of the modes of oscillation of a rotating fluid, namely inertial modes, concentrate along an attractor formed by a periodic orbit of characteristics of the underlying hyperbolic Poincar\'e equation. The dynamics of characteristics is used to elaborate a scenario for the asymptotic behaviour of the eigenmodes and eigenspectrum in the physically relevant r\'egime of very low viscosities which are out of reach numerically. This problem offers a canonical ill-posed Cauchy problem which has applications in other fields.Comment: 4 pages, 5 fi