The topological structure of scaling limits of large planar maps

We discuss scaling limits of large bipartite planar maps. If p is a fixed integer strictly greater than 1, we consider a random planar map M(n) which is uniformly distributed over the set of all 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space M(n) equipped with the graph distance rescaled by the factor n to the power -1/4 converges in distribution as n tends to infinity towards a limiting random compact metric space, in the sense of the Gromov-Hausdorff distance. We prove that the topology of the limiting space is uniquely determined independently of p, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the limit is almost surely equal to 4.Comment: 45 pages Second version with minor modification

A simple proof of Duquesne's theorem on contour processes of conditioned Galton-Watson trees

We give a simple new proof of a theorem of Duquesne, stating that the properly rescaled contour function of a critical aperiodic Galton-Watson tree, whose offspring distribution is in the domain of attraction of a stable law of index $\theta \in (1,2]$, conditioned on having total progeny $n$, converges in the functional sense to the normalized excursion of the continuous-time height function of a strictly stable spectrally positive L\'evy process of index $\theta$. To this end, we generalize an idea of Le Gall which consists in using an absolute continuity relation between the conditional probability of having total progeny exactly $n$ and the conditional probability of having total progeny at least $n$. This new method is robust and can be adapted to establish invariance theorems for Galton-Watson trees having $n$ vertices whose degrees are prescribed to belong to a fixed subset of the positive integers.Comment: 16 pages, 2 figures. Published versio

Quantum Algorithms for Matrix Products over Semirings

In this paper we construct quantum algorithms for matrix products over several algebraic structures called semirings, including the (max,min)-matrix product, the distance matrix product and the Boolean matrix product. In particular, we obtain the following results. We construct a quantum algorithm computing the product of two n x n matrices over the (max,min) semiring with time complexity O(n^{2.473}). In comparison, the best known classical algorithm for the same problem, by Duan and Pettie, has complexity O(n^{2.687}). As an application, we obtain a O(n^{2.473})-time quantum algorithm for computing the all-pairs bottleneck paths of a graph with n vertices, while classically the best upper bound for this task is O(n^{2.687}), again by Duan and Pettie. We construct a quantum algorithm computing the L most significant bits of each entry of the distance product of two n x n matrices in time O(2^{0.64L} n^{2.46}). In comparison, prior to the present work, the best known classical algorithm for the same problem, by Vassilevska and Williams and Yuster, had complexity O(2^{L}n^{2.69}). Our techniques lead to further improvements for classical algorithms as well, reducing the classical complexity to O(2^{0.96L}n^{2.69}), which gives a sublinear dependency on 2^L. The above two algorithms are the first quantum algorithms that perform better than the $\tilde O(n^{5/2})$-time straightforward quantum algorithm based on quantum search for matrix multiplication over these semirings. We also consider the Boolean semiring, and construct a quantum algorithm computing the product of two n x n Boolean matrices that outperforms the best known classical algorithms for sparse matrices. For instance, if the input matrices have O(n^{1.686...}) non-zero entries, then our algorithm has time complexity O(n^{2.277}), while the best classical algorithm has complexity O(n^{2.373}).Comment: 19 page

Invariant Peano curves of expanding Thurston maps

We consider Thurston maps, i.e., branched covering maps $f\colon S^2\to S^2$ that are postcritically finite. In addition, we assume that $f$ is expanding in a suitable sense. It is shown that each sufficiently high iterate $F=f^n$ of $f$ is semi-conjugate to $z^d\colon S^1\to S^1$, where $d$ is equal to the degree of $F$. More precisely, for such an $F$ we construct a Peano curve $\gamma\colon S^1\to S^2$ (onto), such that $F\circ \gamma(z) = \gamma(z^d)$ (for all $z\in S^1$).Comment: 63 pages, 12 figure

Statistics of geodesics in large quadrangulations

We study the statistical properties of geodesics, i.e. paths of minimal length, in large random planar quadrangulations. We extend Schaeffer's well-labeled tree bijection to the case of quadrangulations with a marked geodesic, leading to the notion of "spine trees", amenable to a direct enumeration. We obtain the generating functions for quadrangulations with a marked geodesic of fixed length, as well as with a set of "confluent geodesics", i.e. a collection of non-intersecting minimal paths connecting two given points. In the limit of quadrangulations with a large area n, we find in particular an average number 3*2^i of geodesics between two fixed points at distance i>>1 from each other. We show that, for generic endpoints, two confluent geodesics remain close to each other and have an extensive number of contacts. This property fails for a few "exceptional" endpoints which can be linked by truly distinct geodesics. Results are presented both in the case of finite length i and in the scaling limit i ~ n^(1/4). In particular, we give the scaling distribution of the exceptional points.Comment: 37 pages, 18 color figures, improved version with several clarifications (mostly in sections 2.1 and 2.4) and one added section (3.1) on ensembles of random quadrangulation

Packing and Hausdorff measures of stable trees

In this paper we discuss Hausdorff and packing measures of random continuous trees called stable trees. Stable trees form a specific class of L\'evy trees (introduced by Le Gall and Le Jan in 1998) that contains Aldous's continuum random tree (1991) which corresponds to the Brownian case. We provide results for the whole stable trees and for their level sets that are the sets of points situated at a given distance from the root. We first show that there is no exact packing measure for levels sets. We also prove that non-Brownian stable trees and their level sets have no exact Hausdorff measure with regularly varying gauge function, which continues previous results from a joint work with J-F Le Gall (2006).Comment: 40 page

Confluence of geodesic paths and separating loops in large planar quadrangulations

We consider planar quadrangulations with three marked vertices and discuss the geometry of triangles made of three geodesic paths joining them. We also study the geometry of minimal separating loops, i.e. paths of minimal length among all closed paths passing by one of the three vertices and separating the two others in the quadrangulation. We concentrate on the universal scaling limit of large quadrangulations, also known as the Brownian map, where pairs of geodesic paths or minimal separating loops have common parts of non-zero macroscopic length. This is the phenomenon of confluence, which distinguishes the geometry of random quadrangulations from that of smooth surfaces. We characterize the universal probability distribution for the lengths of these common parts.Comment: 48 pages, 33 color figures. Final version, with one concluding paragraph and one reference added, and several other small correction

Benefit of the Vittel criteria to determine the need for whole body scanning in a severe trauma patient.

OBJECTIVE: To evaluate the use of the Vittel criteria in addition to a clinical examination to determine the need for a whole body scan (WBS) in a severe trauma patient. MATERIALS AND METHODS: Between December 2008 and November 2009, 339 severe trauma patients with at least one Vittel criterion were prospectively evaluated with a WBS. The following data were collected: the Vittel criteria present, circumstances of the accident, traumatic injury on the WBS, and irradiation. The original intent to prescribe a computed tomography (CT) scan (whole body or a targeted region), based solely on clinical signs, was specified. RESULTS: Injuries were diagnosed in 55.75% of the WBS (n=189). The most common Vittel criteria were "global assessment" (n=266), "thrown, run over" (n=116), and "ejected from vehicle" (n=94). The multivariate analysis used the following as independent criteria for predicting severe traumatic injury on the WBS: Glasgow score less than 13, penetrating trauma, and colloid resuscitation greater than 11. Based solely on clinical factors, 164 patients would not have had any scan or (only) a targeted scan. In that case, 15% of the severe injuries would have been missed. CONCLUSION: Using the Vittel criteria to determine the need for a WBS in a severe trauma patient makes it possible to find serious injuries not suspected on the clinical examination, but at the cost of an increased number of normal scans

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