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P. Di Francesco

Publications and source records attributed to P. Di Francesco.

At least 55 records · Page 3Linked to original sources
arXiv2003

Statistics of planar graphs viewed from a vertex: A study via labeled trees

We study the statistics of edges and vertices in the vicinity of a reference vertex (origin) within random planar quadrangulations and Eulerian triangulations. Exact generating functions are obtained for theses graphs with fixed numbers of edges and vertices at given geodesic distances from the origin. Our analysis relies on bijections with labeled trees, in which the labels encode the information on the geodesic distance from the origin. In the case of infinitely large graphs, we give in particular explicit formulas for the probabilities that the origin have given numbers of neighboring edges and/or vertices, as well as explicit values for the corresponding moments.

cond-mat.stat-mech↗
arXiv2003

Geodesic Distance in Planar Graphs

We derive the exact generating function for planar maps (genus zero fatgraphs) with vertices of arbitrary even valence and with two marked points at a fixed geodesic distance. This is done in a purely combinatorial way based on a bijection with decorated trees, leading to a recursion relation on the geodesic distance. The latter is solved exactly in terms of discrete soliton-like expressions, suggesting an underlying integrable structure. We extract from this solution the fractal dimensions at the various (multi)-critical points, as well as the precise scaling forms of the continuum two-point functions and the probability distributions for the geodesic distance in (multi)-critical random surfaces. The two-point functions are shown to obey differential equations involving the residues of the KdV hierarchy.

cond-mat.stat-mech↗
arXiv2002

Geometrically Constrained Statistical Models on Fixed and Random Lattices: From Hard Squares to Meanders

We review various combinatorial applications of field theoretical and matrix model approaches to equilibrium statistical physics involving the enumeration of fixed and random lattice model configurations. We show how the structures of the underlying lattices, in particular their colorability properties, become relevant when we consider hard-particles or fully-packed loop models on them. We show how a careful back-and-forth application of results of two-dimensional quantum gravity and matrix models allows to predict critical universality classes and consequently exact asymptotics for various numbers, counting in particular hard object configurations on fixed or random lattices and meanders.

cond-mat.stat-mech↗
arXiv2002

Combinatorics of Hard Particles on Planar Graphs

We revisit the problem of hard particles on planar random tetravalent graphs in view of recent combinatorial techniques relating planar diagrams to decorated trees. We show how to recover the two-matrix model solution to this problem in this purely combinatorial language.

cond-mat.stat-mech↗
arXiv2002

Rectangular Matrix Models and Combinatorics of Colored Graphs

We present applications of rectangular matrix models to various combinatorial problems, among which the enumeration of face-bicolored graphs with prescribed vertex degrees, and vertex-tricolored triangulations. We also mention possible applications to Interaction-Round-a-Face and hard-particle statistical models defined on random lattices.

cond-mat.stat-mech↗
arXiv2002

Census of Planar Maps: From the One-Matrix Model Solution to a Combinatorial Proof

We consider the problem of enumeration of planar maps and revisit its one-matrix model solution in the light of recent combinatorial techniques involving conjugated trees. We adapt and generalize these techniques so as to give an alternative and purely combinatorial solution to the problem of counting arbitrary planar maps with prescribed vertex degrees.

cond-mat.stat-mech↗
arXiv2002

Counting Colored Random Triangulations

We revisit the problem of enumeration of vertex-tricolored planar random triangulations solved in [Nucl. Phys. B 516 [FS] (1998) 543-587] in the light of recent combinatorial developments relating classical planar graph counting problems to the enumeration of decorated trees. We give a direct combinatorial derivation of the associated counting function, involving tricolored trees. This is generalized to arbitrary k-gonal tessellations with cyclic colorings and checked by use of matrix models.

cond-mat.stat-mech↗
arXiv2002

Critical and Tricritical Hard Objects on Bicolorable Random Lattices: Exact Solutions

We address the general problem of hard objects on random lattices, and emphasize the crucial role played by the colorability of the lattices to ensure the existence of a crystallization transition. We first solve explicitly the naive (colorless) random-lattice version of the hard-square model and find that the only matter critical point is the non-unitary Lee-Yang edge singularity. We then show how to restore the crystallization transition of the hard-square model by considering the same model on bicolored random lattices. Solving this model exactly, we show moreover that the crystallization transition point lies in the universality class of the Ising model coupled to 2D quantum gravity. We finally extend our analysis to a new two-particle exclusion model, whose regular lattice version involves hard squares of two different sizes. The exact solution of this model on bicolorable random lattices displays a phase diagram with two (continuous and discontinuous) crystallization transition lines meeting at a higher order critical point, in the universality class of the tricritical Ising model coupled to 2D quantum gravity.

cond-mat.stat-mech↗
arXiv2001

Generalized Lorentzian Triangulations and the Calogero Hamiltonian

We introduce and solve a generalized model of 1+1D Lorentzian triangulations in which a certain subclass of outgrowths is allowed, the occurrence of these being governed by a coupling constant β. Combining transfer matrix-, saddle point- and path integral techniques we show that for β<1 it is possible to take a continuum limit in which the model is described by a 1D quantum Calogero Hamiltonian. The coupling constant βsurvives the continuum limit and appears as a parameter of the Calogero potential.

hep-th↗
arXiv2000

Meanders: Exact Asymptotics

We conjecture that meanders are governed by the gravitational version of a c=-4 two-dimensional conformal field theory, allowing for exact predictions for the meander configuration exponent α=\sqrt{29}(\sqrt{29}+\sqrt{5})/12, and the semi-meander exponent {\barα}=1+\sqrt{11}(\sqrt{29}+\sqrt{5})/24. This result follows from an interpretation of meanders as pairs of fully packed loops on a random surface, described by two c=-2 free fields. The above values agree with recent numerical estimates. We generalize these results to a score of meandric numbers with various geometries and arbitrary loop fugacities.

cond-mat.stat-mech↗
arXiv1999

Matrix Model Combinatorics: Applications to Folding and Coloring

We present a detailed study of the combinatorial interpretation of matrix integrals, including the examples of tessellations of arbitrary genera, and loop models on random surfaces. After reviewing their methods of solution, we apply these to the study of various folding problems arising from physics, including: the meander (or polymer folding) problem ``enumeration of all topologically inequivalent closed non-intersecting plane curves intersecting a line through a given number of points" and a fluid membrane folding problem reformulated as that of ``enumerating all vertex-tricolored triangulations of arbitrary genus, with given numbers of vertices of either color".

math-ph↗
arXiv1999

Integrable 2D Lorentzian Gravity and Random Walks

We introduce and solve a family of discrete models of 2D Lorentzian gravity with higher curvature, which possess mutually commuting transfer matrices, and whose spectral parameter interpolates between flat and curved space-times. We further establish a one-to-one correspondence between Lorentzian triangulations and directed Random Walks. This gives a simple explanation why the Lorentzian triangulations have fractal dimension 2 and why the curvature model lies in the universality class of pure Lorentzian gravity. We also study integrable generalizations of the curvature model with arbitrary polygonal tiles. All of them are found to lie in the same universality class.

hep-th↗
arXiv1999

Fully Packed O(n=1) Model on Random Eulerian Triangulations

We introduce a matrix model describing the fully-packed O(n) model on random Eulerian triangulations (i.e. triangulations with all vertices of even valency). For n=1 the model is mapped onto a particular gravitational 6-vertex model with central charge c=1, hence displaying the expected shift c -> c+1 when going from ordinary random triangulations to Eulerian ones. The case of arbitrary n is also discussed.

cond-mat.stat-mech↗
arXiv1998

New Integrable Lattice Models From Fuss-Catalan Algebras

We construct new trigonometric solutions of the Yang-Baxter equation, using the Fuss-Catalan algebras, a set of multi-colored versions of the Temperley-Lieb algebra, recently introduced by Bisch and Jones. These lead to new two-dimensional integrable lattice models, describing dense gases of colored loops.

hep-th↗
arXiv1998

Folding Transitions of the Square-Diagonal Lattice

We address the problem of "phantom" folding of the tethered membrane modelled by the two-dimensional square lattice, with bonds on the edges and diagonals of each face. Introducing bending rigidities $K_1$ and $K_2$ for respectively long and short bonds, we derive the complete phase diagram of the model, using transfer matrix calculations. The latter displays two transition curves, one corresponding to a first order (ferromagnetic) folding transition, and the other to a continuous (anti-ferromagnetic) unfolding transition.

cond-mat↗
arXiv1998

Folding the Square-Diagonal Lattice

We study the problem of "phantom" folding of the two-dimensional square lattice, in which the edges and diagonals of each face can be folded. The non-vanishing thermodynamic folding entropy per face $s \simeq .2299(1)$ is estimated both analytically and numerically, by successively mapping the model onto a dense loop model, a spin model and a new 28 Vertex, 4-color model. Higher dimensional generalizations are investigated, as well as other foldable lattices.

cond-mat.stat-mech↗
arXiv1997

Coloring Random Triangulations

We introduce and solve a two-matrix model for the tri-coloring problem of the vertices of a random triangulation. We present three different solutions: (i) by orthogonal polynomial techniques (ii) by use of a discrete Hirota bilinear equation (iii) by direct expansion. The model is found to lie in the universality class of pure two-dimensional quantum gravity, despite the non-polynomiality of its potential.

cond-mat↗