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Arctic curves of the T-system with Slanted Initial Data
Philippe Di Francesco, Hieu Trung Vu
We study the T-system of type , also known as the octahedron recurrence/equation, viewed as a 2+1-dimensional discrete evolution equation. Generalizing the study of [P. D…
Ruijsenaars wavefunctions as modular group matrix coefficients
Philippe Di Francesco, Rinat Kedem, Sergey Khoroshkin +2
We give a description of the Hallnäs--Ruijsenaars eigenfunctions of the 2-particle hyperbolic Ruijsenaars system as matrix coefficients for the order 4 element $S\in SL(2,\mathbb{Z…
Duality and Macdonald difference operators
Philippe Di Francesco, Rinat Kedem
This note summarizes certain properties common to Macdonald, Koornwinder and Arthamonov-Shakirov -difference operators, relating to the duality or bi-spectrality properties of t…
Arctic curves of the 20V model on a triangle
Philippe Di Francesco
We apply the Tangent Method of Colomo and Sportiello to predict the arctic curves of the Twenty Vertex model with specific domain wall boundary conditions on a triangle, in the Dis…
Exponents for Hamiltonian paths on random bicubic maps and KPZ
Philippe Di Francesco, Bertrand Duplantier, Olivier Golinelli +1
We evaluate the configuration exponents of various ensembles of Hamiltonian paths drawn on random planar bicubic maps. These exponents are estimated from the extrapolations of exac…
Arctic Curves in path models from the Tangent Method
Philippe Di Francesco, Matthew F. Lapa
Recently, Colomo and Sportiello introduced a powerful method, known as the \emph{Tangent Method}, for computing the arctic curve in statistical models which have a (non- or weakly-…
(t,q) Q-systems, DAHA and quantum toroidal algebras via generalized Macdonald operators
Philippe Di Francesco, Rinat Kedem
We introduce difference operators on the space of symmetric functions which are a natural generalization of the -Macdonald operators. In the limit, they satisfy…
Arctic curves of the octahedron equation
P. Di Francesco, R. Soto-Garrido
We study the octahedron relation (also known as the -system), obeyed in particular by the partition function for dimer coverings of the Aztec Diamond graph. For a s…
T-systems, networks and dimers
P. Di Francesco
We study the solutions of the T-system for type A, also known as the octahedron equation, viewed as a 2+1-dimensional discrete evolution equation. These may be expressed entirely i…
The solution of the quantum T-system for arbitrary boundary
Philippe Di Francesco, Rinat Kedem
We solve the quantum version of the -system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cl…
Discrete integrable systems, positivity, and continued fraction rearrangements
Philippe Di Francesco
In this review article, we present a unified approach to solving discrete, integrable, possibly non-commutative, dynamical systems, including the - and -systems based on $A_r…
Noncommutative integrability, paths and quasi-determinants
Philippe Di Francesco, Rinat Kedem
In previous work, we showed that the solution of certain systems of discrete integrable equations, notably and -systems, is given in terms of partition functions of positive…