arXiv subjects

Emmanuel Guitter

Publications and source records attributed to Emmanuel Guitter.

At least 37 records · Page 2Linked to original sources

arXiv2017

On a conjecture by Chapuy about Voronoi cells in large maps

In a recent paper, Chapuy conjectured that, for any positive integer k, the law for the fractions of total area covered by the k Voronoi cells defined by k points picked uniformly at random in the Brownian map of any fixed genus is the same law as that of a uniform k-division of the unit interval. For k=2, i.e. with two points chosen uniformly at random, it means that the law for the ratio of the area of one of the two Voronoi cells by the total area of the map is uniform between 0 and 1. Here, by a direct computation of the desired law, we show that this latter conjecture for k=2 actually holds in the case of large planar (genus 0) quadrangulations as well as for large general planar maps (i.e. maps whose faces have arbitrary degrees). This corroborates Chapuy's conjecture in its simplest realizations.

math.CO↗

arXiv2016

Refined universal laws for hull volumes and perimeters in large planar maps

We consider ensembles of planar maps with two marked vertices at distance k from each other and look at the closed line separating these vertices and lying at distance d from the first one (d<k). This line divides the map into two components, the hull at distance d which corresponds to the part of the map lying on the same side as the first vertex and its complementary. The number of faces within the hull is called the hull volume and the length of the separating line the hull perimeter. We study the statistics of the hull volume and perimeter for arbitrary d and k in the limit of infinitely large planar quadrangulations, triangulations and Eulerian triangulations. We consider more precisely situations where both d and k become large with the ratio d/k remaining finite. For infinitely large maps, two regimes may be encountered: either the hull has a finite volume and its complementary is infinitely large, or the hull itself has an infinite volume and its complementary is of finite size. We compute the probability for the map to be in either regime as a function of d/k as well as a number of universal statistical laws for the hull perimeter and volume when maps are conditioned to be in one regime or the other.

math.CO↗

arXiv2016

Eulerian triangulations: two-point function and hull perimeter statistics

We present a new derivation of the distance-dependent two-point function for planar Eulerian triangulations and give expressions for more refined generating functions where we also control hull perimeters. These results are obtained in the framework of a new recursion relation for slice generating functions and extend similar results obtained recently for triangulations and quadrangulations. A number of explicit formulas are given for the statistics of hull perimeters in infinitely large random planar Eulerian triangulations.

math.CO↗

arXiv2016

Some results on the statistics of hull perimeters in large planar triangulations and quadrangulations

The hull perimeter at distance d in a planar map with two marked vertices at distance k from each other is the length of the closed curve separating these two vertices and lying at distance d from the first one (d<k). We study the statistics of hull perimeters in large random planar triangulations and quadrangulations as a function of both k and d. Explicit expressions for the probability density of the hull perimeter at distance d, as well as for the joint probability density of hull perimeters at distances d1 and d2, are obtained in the limit of infinitely large k. We also consider the situation where the distance d at which the hull perimeter is measured corresponds to a finite fraction of k. The various laws that we obtain are identical for triangulations and for quadrangulations, up to a global rescaling. Our approach is based on recursion relations recently introduced by the author which determine the generating functions of so-called slices, i.e. pieces of maps with appropriate distance constraints. It is indeed shown that the map decompositions underlying these recursion relations are intimately linked to the notion of hull perimeters and provide a simple way to fully control them.

math.CO↗

arXiv2015

The distance-dependent two-point function of triangulations: a new derivation from old results

We present a new derivation of the distance-dependent two-point function of random planar triangulations. As it is well-known, this function is intimately related to the generating functions of so-called slices, which are pieces of triangulation having boundaries made of shortest paths of prescribed length. We show that the slice generating functions are fully determined by a direct recursive relation on their boundary length. Remarkably, the kernel of this recursion is some quantity introduced and computed by Tutte a long time ago in the context of a global enumeration of planar triangulations. We may thus rely on these old results to solve our new recursion relation explicitly in a constructive way.

math.CO↗

arXiv2015

Comparing two statistical ensembles of quadrangulations: a continued fraction approach

We use a continued fraction approach to compare two statistical ensembles of quadrangulations with a boundary, both controlled by two parameters. In the first ensemble, the quadrangulations are bicolored and the parameters control their numbers of vertices of both colors. In the second ensemble, the parameters control instead the number of vertices which are local maxima for the distance to a given vertex, and the number of those which are not. Both ensembles may be described either by their (bivariate) generating functions at fixed boundary length or, after some standard slice decomposition, by their (bivariate) slice generating functions. We first show that the fixed boundary length generating functions are in fact equal for the two ensembles. We then show that the slice generating functions, although different for the two ensembles, simply correspond to two different ways of encoding the same quantity as a continued fraction. This property is used to obtain explicit expressions for the slice generating functions in a constructive way.

math.CO↗

arXiv2014

The two-point function of bicolored planar maps

We compute the distance-dependent two-point function of vertex-bicolored planar maps, i.e., maps whose vertices are colored in black and white so that no adjacent vertices have the same color. By distance-dependent two-point function, we mean the generating function of these maps with both a marked oriented edge and a marked vertex which are at a prescribed distance from each other. As customary, the maps are enumerated with arbitrary degree-dependent face weights, but the novelty here is that we also introduce color-dependent vertex weights. Explicit expressions are given for vertex-bicolored maps with bounded face degrees in the form of ratios of determinants of fixed size. Our approach is based on a slice decomposition of maps which relates the distance-dependent two-point function to the coefficients of the continued fraction expansions of some distance-independent map generating functions. Special attention is paid to the case of vertex-bicolored quadrangulations and hexangulations, whose two-point functions are also obtained in a more direct way involving equivalences with hard dimer statistics. A few consequences of our results, as well as some extension to vertex-tricolored maps, are also discussed.

math.CO↗

arXiv2014

The three-point function of general planar maps

We compute the distance-dependent three-point function of general planar maps and of bipartite planar maps, i.e., the generating function of these maps with three marked vertices at prescribed pairwise distances. Explicit expressions are given for maps counted by their number of edges only, or by both their numbers of edges and faces. A few limiting cases and applications are discussed.

math.CO↗

arXiv2013

On the two-point function of general planar maps and hypermaps

We consider the problem of computing the distance-dependent two-point function of general planar maps and hypermaps, i.e. the problem of counting such maps with two marked points at a prescribed distance. The maps considered here may have faces of arbitrarily large degree, which requires new bijections to be tackled. We obtain exact expressions for the following cases: general and bipartite maps counted by their number of edges, 3-hypermaps and 3-constellations counted by their number of dark faces, and finally general and bipartite maps counted by both their number of edges and their number of faces.

math.CO↗

arXiv2007

Mass distribution exponents for growing trees

We investigate the statistics of trees grown from some initial tree by attaching links to preexisting vertices, with attachment probabilities depending only on the valence of these vertices. We consider the asymptotic mass distribution that measures the repartition of the mass of large trees between their different subtrees. This distribution is shown to be a broad distribution and we derive explicit expressions for scaling exponents that characterize its behavior when one subtree is much smaller than the others. We show in particular the existence of various regimes with different values of these mass distribution exponents. Our results are corroborated by a number of exact solutions for particular solvable cases, as well as by numerical simulations.

cond-mat.stat-mech↗

arXiv2001

Critical and Multicritical Semi-Random (1+d)-Dimensional Lattices and Hard Objects in d Dimensions

We investigate models of (1+d)-D Lorentzian semi-random lattices with one random (space-like) direction and d regular (time-like) ones. We prove a general inversion formula expressing the partition function of these models as the inverse of that of hard objects in d dimensions. This allows for an exact solution of a variety of new models including critical and multicritical generalized (1+1)-D Lorentzian surfaces, with fractal dimensions $d_F=k+1$, k=1,2,3,..., as well as a new model of (1+2)-D critical tetrahedral complexes, with fractal dimension $d_F=12/5$. Critical exponents and universal scaling functions follow from this solution. We finally establish a general connection between (1+d)-D Lorentzian lattices and directed-site lattice animals in (1+d) dimensions.

cond-mat.stat-mech↗

arXiv2000

Exact Meander Asymptotics: a Numerical Check

This note addresses the meander enumeration problem: "Count all topologically inequivalent configurations of a closed planar non self-intersecting curve crossing a line through a given number of points". We review a description of meanders introduced recently in terms of the coupling to gravity of a two-flavored fully-packed loop model. The subsequent analytic predictions for various meandric configuration exponents are checked against exact enumeration, using a transfer matrix method, with an excellent agreement.

cond-mat.stat-mech↗

arXiv1999

Monte Carlo Results for Projected Self-Avoiding Polygons: A Two-dimensional Model for Knotted Polymers

We introduce a two-dimensional lattice model for the description of knotted polymer rings. A polymer configuration is modeled by a closed polygon drawn on the square diagonal lattice, with possible crossings describing pairs of strands of polymer passing on top of each other. Each polygon configuration can be viewed as the two- dimensional projection of a particular knot. We study numerically the statistics of large polygons with a fixed knot type, using a generalization of the BFACF algorithm for self-avoiding walks. This new algorithm incorporates both the displacement of crossings and the three types of Reidemeister transformations preserving the knot topology. Its ergodicity within a fixed knot type is not proven here rigorously but strong arguments in favor of this ergodicity are given together with a tentative sketch of proof. Assuming this ergodicity, we obtain numerically the following results for the statistics of knotted polygons: In the limit of a low crossing fugacity, we find a localization along the polygon of all the primary factors forming the knot. Increasing the crossing fugacity gives rise to a transition from a self-avoiding walk to a branched polymer behavior.

cond-mat.stat-mech↗

arXiv1998

Statistical and dynamical properties of the discrete Sinai model at finite times

We study the Sinai model for the diffusion of a particle in a one dimensional quenched random energy landscape. We consider the particular case of discrete energy landscapes made of random +/- 1 jumps on the semi infinite line Z+ with a reflecting wall at the origin. We compare the statistical distribution of the successive local minima of the energy landscapes, which we derive explicitly, with the dynamical distribution of the position of the diffusing particle, which we obtain numerically. At high temperature, the two distributions match only in the large time asymptotic regime. At low temperature however, we find even at finite times a clear correspondence between the statistical and dynamical distributions, with additional interesting oscillatory behaviors.

cond-mat.stat-mech↗

arXiv1997

Effects of Self-Avoidance on the Tubular Phase of Anisotropic Membranes

We study the tubular phase of self-avoiding anisotropic membranes. We discuss the renormalizability of the model Hamiltonian describing this phase and derive from a renormalization group equation some general scaling relations for the exponents of the model. We show how particular choices of renormalization factors reproduce the Gaussian result, the Flory theory and the Gaussian Variational treatment of the problem. We then study the perturbative renormalization to one loop in the self-avoiding parameter using dimensional regularization and an epsilon-expansion about the upper critical dimension, and determine the critical exponents to first order in epsilon.

cond-mat.stat-mech↗

arXiv1997

The Poisson ratio of crystalline surfaces

A remarkable theoretical prediction for a crystalline (polymerized) surface is that its Poisson ratio (σ) is negative. Using a large scale Monte Carlo simulation of a simple model of such surfaces we show that this is indeed true. The precise numerical value we find is (σ\simeq -0.32) on a (128^2) lattice at bending rigidity (kappa = 1.1). This is in excellent agreement with the prediction (σ= -1/3) following from the self-consistent screening approximation of Le Doussal and Radzihovsky.

cond-mat.stat-mech↗

arXiv1997

Renormalization Theory for the Self-Avoiding Polymerized Membranes

We prove the renormalizability of the generalized Edwards model for self-avoiding polymerized membranes. This is done by use of a short distance multilocal operator product expansion, which extends the methods of local field theories to a large class of models with non-local singular interactions. This ensures the existence of scaling laws for crumpled self-avoiding membranes, and validates the direct renormalization method used for polymers and membranes. This also provides a framework for explicit perturbative calculations. We discuss hyperscaling relations for the configuration exponent and contact exponents. We finally consider membranes with long range interactions and at the Theta-point.

cond-mat.soft↗

arXiv1996

Geometrical Folding Transitions of the Triangular Lattice in the Face-Centred Cubic Lattice

We study the folding of the regular two-dimensional triangular lattice embedded in the regular three-dimensional Face-Centred Cubic lattice, a discrete model for the crumpling of membranes. Possible folds are complete planar folds, folds with the angle of a regular tetrahedron (71 degrees) or with that of a regular octahedron (109 degrees). We study this model in the presence of a negative bending rigidity K, which favours the folding process. We use both a cluster variation method (CVM) approximation and a transfer matrix approach. The system is shown to undergo two separate geometrical transitions with increasing |K|: a first discontinuous transition separates a phase where the triangular lattice is preferentially wrapped around octahedra from a phase where it is preferentially wrapped around tetrahedra. A second continuous transition separates this latter phase from a phase of complete folding of the lattice on top of a single triangle.

cond-mat.stat-mech↗