84 citations · 521 across the 32 of their papers we have counts for
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Integrable Combinatorics
Philippe Di Francesco
We explore various combinatorial problems mostly borrowed from physics, that share the property of being continuously or discretely integrable, a feature that guarantees the existe…
Integrable Combinatorics
Philippe Di Francesco
We review various combinatorial problems with underlying classical or quantum integrable structures. (Plenary talk given at the International Congress of Mathematical Physics, Aalb…
Quantum Q-system solutions as q-multinomial series
Philippe Di Francesco
We derive explicit expressions for the generating series of the fundamental solutions of the quantum -system of Ref. [P. Di Francesco and R. Kedem, arXiv:1006.4774 [math-p…
Discrete non-commutative integrability: the proof of a conjecture by M. Kontsevich
P. Di Francesco, R. Kedem
We prove a conjecture of Kontsevich regarding the solutions of rank two recursion relations for non-commutative variables which, in the commutative case, reduce to rank two cluster…
Quantum Knizhnik-Zamolodchikov Equation, Totally Symmetric Self-Complementary Plane Partitions and Alternating Sign Matrices
P. Di Francesco, P. Zinn-Justin
We present multiresidue formulae for partial sums in the basis of link patterns of the polynomial solution to the level 1 U_q(\hat sl_2) quantum Knizhnik--Zamolodchikov equation at…
Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups
A. Prats Ferrer, B. Eynard, P. Di Francesco +1
The Harish-Chandra correlation functions, i.e. integrals over compact groups of invariant monomials prod tr{X^{p_1} Omega Y^{q_1} Omega^dagger X^{p_2} ... with the weight exp tr{X…
Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain
P. Di Francesco, P. Zinn-Justin, J. -B. Zuber
The sums of components of the ground states of the O(1) loop model on a cylinder or of the XXZ quantum spin chain at Delta=-1/2 (of size L) are expressed in terms of combinatorial…
From Orbital Varieties to Alternating Sign Matrices
P. Di Francesco, P. Zinn-Justin
We study a one-parameter family of vector-valued polynomials associated to each simple Lie algebra. When this parameter equals -1 one recovers Joseph polynomials, whereas at $q…
Boundary qKZ equation and generalized Razumov-Stroganov sum rules for open IRF models
P. Di Francesco
We find higher rank generalizations of the Razumov--Stroganov sum rules at for models with open boundaries, by constructing polynomial solutions of l…
Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials
P. Di Francesco, P. Zinn-Justin
We prove higher rank analogues of the Razumov--Stroganov sum rule for the groundstate of the O(1) loop model on a semi-infinite cylinder: we show that a weighted sum of components…
Inhomogeneous loop models with open boundaries
P. Di Francesco
We consider the crossing and non-crossing O(1) dense loop models on a semi-infinite strip, with inhomogeneities (spectral parameters) that preserve the integrability. We compute th…
Inhomogenous model of crossing loops and multidegrees of some algebraic varieties
P. Di Francesco, P. Zinn-Justin
We consider a quantum integrable inhomogeneous model based on the Brauer algebra B(1) and discuss the properties of its ground state eigenvector. In particular we derive various su…
Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops
P. Di Francesco, P. Zinn-Justin, J. -B. Zuber
We present determinant formulae for the number of tilings of various domains in relation with Alternating Sign Matrix and Fully Packed Loop enumeration.
2D Quantum Gravity, Matrix Models and Graph Combinatorics
P. Di Francesco
Lecture notes given at the summer school ``Applications of random matrices to physics", Les Houches, June 2004.
Matrix Model Combinatorics: Applications to Folding and Coloring
P. Di Francesco
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…