Generic method for bijections between blossoming trees and planar maps

This article presents a unified bijective scheme between planar maps and blossoming trees, where a blossoming tree is defined as a spanning tree of the map decorated with some dangling half-edges that enable to reconstruct its faces. Our method generalizes a previous construction of Bernardi by loosening its conditions of applications so as to include annular maps, that is maps embedded in the plane with a root face different from the outer face. The bijective construction presented here relies deeply on the theory of \alpha-orientations introduced by Felsner, and in particular on the existence of minimal and accessible orientations. Since most of the families of maps can be characterized by such orientations, our generic bijective method is proved to capture as special cases all previously known bijections involving blossoming trees: for example Eulerian maps, m-Eulerian maps, non separable maps and simple triangulations and quadrangulations of a k-gon. Moreover, it also permits to obtain new bijective constructions for bipolar orientations and d-angulations of girth d of a k-gon. As for applications, each specialization of the construction translates into enumerative by-products, either via a closed formula or via a recursive computational scheme. Besides, for every family of maps described in the paper, the construction can be implemented in linear time. It yields thus an effective way to encode and generate planar maps. In a recent work, Bernardi and Fusy introduced another unified bijective scheme, we adopt here a different strategy which allows us to capture different bijections. These two approaches should be seen as two complementary ways of unifying bijections between planar maps and decorated trees.Comment: 45 pages, comments welcom

The Brownian continuum random tree as the unique solution to a fixed point equation

In this note, we provide a new characterization of Aldous' Brownian continuum random tree as the unique fixed point of a certain natural operation on continuum trees (which gives rise to a recursive distributional equation). We also show that this fixed point is attractive.Comment: 15 pages, 3 figure

Constellations and multicontinued fractions: application to Eulerian triangulations

We consider the problem of enumerating planar constellations with two points at a prescribed distance. Our approach relies on a combinatorial correspondence between this family of constellations and the simpler family of rooted constellations, which we may formulate algebraically in terms of multicontinued fractions and generalized Hankel determinants. As an application, we provide a combinatorial derivation of the generating function of Eulerian triangulations with two points at a prescribed distance.Comment: 12 pages, 4 figure

On the algebraic numbers computable by some generalized Ehrenfest urns

This article deals with some stochastic population protocols, motivated by theoretical aspects of distributed computing. We modelize the problem by a large urn of black and white balls from which at every time unit a fixed number of balls are drawn and their colors are changed according to the number of black balls among them. When the time and the number of balls both tend to infinity the proportion of black balls converges to an algebraic number. We prove that, surprisingly enough, not every algebraic number can be "computed" this way

Superconducting fluctuations and pseudogap in high-Tc cuprates

Large pulsed magnetic fields up to 60 Tesla are used to suppress the contribution of superconducting fluctuations (SCF) to the ab-plane conductivity above Tc in a series of YBa2Cu3O(6+x). These experiments allow us to determine the field H'c(T) and the temperature T'c above which the SCFs are fully suppressed. A careful investigation near optimal doping shows that T'c is higher than the pseudogap temperature T*, which is an unambiguous evidence that the pseudogap cannot be assigned to preformed pairs. Accurate determinations of the SCF contribution to the conductivity versus temperature and magnetic field have been achieved. They can be accounted for by thermal fluctuations following the Ginzburg-Landau scheme for nearly optimally doped samples. A phase fluctuation contribution might be invoked for the most underdoped samples in a T range which increases when controlled disorder is introduced by electron irradiation. Quantitative analysis of the fluctuating magnetoconductance allows us to determine the critical field Hc2(0) which is found to be be quite similar to H'c(0) and to increase with hole doping. Studies of the incidence of disorder on both T'c and T* allow us to propose a three dimensional phase diagram including a disorder axis, which allows to explain most observations done in other cuprate families.Comment: Paper presented at the "Eurasia-Pacific Summer School & Conference on Correlated Electrons", Turunc-Marmaris, Turkey, July 4-14, 201

Disorder, Metal-Insulator crossover and Phase diagram in high-Tc cuprates

We have studied the influence of disorder induced by electron irradiation on the normal state resistivities $\rho(T)$ of optimally and underdoped YBa2CuOx single crystals, using pulsed magnetic fields up to 60T to completely restore the normal state. We evidence that point defect disorder induces low T upturns of rho(T) which saturate in some cases at low T in large applied fields as would be expected for a Kondo-like magnetic response. Moreover the magnitude of the upturns is related to the residual resistivity, that is to the concentration of defects and/or their nanoscale morphology. These upturns are found quantitatively identical to those reported in lower Tc cuprates, which establishes the importance of disorder in these supposedly pure compounds. We therefore propose a realistic phase diagram of the cuprates, including disorder, in which the superconducting state might reach the antiferromagnetic phase in the clean limit.Comment: version 2 with minor change

Respective influences of pair breaking and phase fluctuations in disordered high Tc superconductors

Electron irradiation has been used to introduce point defects in a controlled way in the CuO2 planes of underdoped and optimally doped YBCO. This technique allows us to perform very accurate measurements of Tc and of the residual resistivity in a wide range of defect contents xd down to Tc=0. The Tc decrease does not follow the variation expected from pair breaking theories. The evolutions of Tc and of the transition width with xd emphasize the importance of phase fluctuations, at least for the highly damaged regime. These results open new questions about the evolution of the defect induced Tc depression over the phase diagram of the cupratesComment: 5 pages, 4 figure

Some families of increasing planar maps

Stack-triangulations appear as natural objects when one wants to define some increasing families of triangulations by successive additions of faces. We investigate the asymptotic behavior of rooted stack-triangulations with $2n$ faces under two different distributions. We show that the uniform distribution on this set of maps converges, for a topology of local convergence, to a distribution on the set of infinite maps. In the other hand, we show that rescaled by $n^{1/2}$, they converge for the Gromov-Hausdorff topology on metric spaces to the continuum random tree introduced by Aldous. Under a distribution induced by a natural random construction, the distance between random points rescaled by $(6/11)\log n$ converge to 1 in probability. We obtain similar asymptotic results for a family of increasing quadrangulations

Reply to Comment on "High-field studies of superconducting fluctuations in high-Tc cuprates: Evidence for a small gap distinct from the large pseudogap" by M.V. Ramallo et al

The experimental investigations done in our paper Phys.Rev.B84,014522(2011) allowed us to establish that the superconducting fluctuations (SCF) always die out sharply with increasing T. But contrary to the claim done in the comment of Ramallo et al., this sharp cutoff of SCF measured in YBa2Cu3O{6+x} depends on hole doping and/or disorder. So our data cannot be used to claim for a universality of the extended gaussian Ginzburg Landau theory proposed by the authors of the comment. Furthermore, to explain quantitatively our data near optimal doping using this model they need to consider that fluctuations in the two CuO2 planes of a bilayer are totally decoupled, which is not physically well justified. On the contrary a consistent interpretation of all our data (paraconductivity, Nernst effect and magnetoresistance) has been done by considering that the coupling between the two layers of the unit cell is dominant at least up to 1.1Tc.Comment: Reply to the comment published in Phys. Rev. B 85,106501 (2012