arXiv subjects

Philippe Di Francesco

Publications and source records attributed to Philippe Di Francesco.

41 records · Page 3Linked to original sources

arXiv2008

Q-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property

We define the cluster algebra associated with the Q-system for the Kirillov-Reshetikhin characters of the quantum affine algebra $U_q(\hat{\g})$ for any simple Lie algebra g, generalizing the simply-laced case treated in [Kedem 2007]. We describe some special properties of this cluster algebra, and explain its relation to the deformed Q-systems which appeared on our proof of the combinatorial-KR conjecture. We prove that the polynomiality of the cluster variables in terms of the ``initial cluster seeds'', including solutions of the Q-system, is a consequence of the Laurent phenomenon and the boundary conditions. We also give a formulation of both Q-systems and generalized T-systems as cluster algebras with coefficients. This provides a proof of the polynomiality of solutions of generalized T-systems with appropriate boundary conditions.

math.RT↗

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↗

arXiv1996

Discrete Folding

Models of folding of a triangular lattice embedded in a discrete space are studied as simple models of the crumpling transition of fixed-connectivity membranes. Both the case of planar folding and three-dimensional folding on a face-centered-cubic lattice are treated. The 3d-folding problem corresponds to a 96-vertex model and exhibits a first-order folding transition from a crumpled phase to a completely flat phase as the bending rigidity increases.

cond-mat.stat-mech↗