LIC · Loop models, integrability and combinatorics

The purpose of this proposal is to investigate new connections whichhave emerged in the recent years between problems from statisticalmechanics, namely two-dimensional exactly solvable models, and a varietyof combinatorial problems, among which: the enumeration of plane partitions,alternating sign matrices and related objects;combinatorial properties of certainalgebro-geometric objects such as orbital varieties or the Brauer loop scheme;or finally certain problems in free probability. One of the key methodsthat emerged in recent years is the useof quantum integrability and more precisely the quantum Knizhnik--Zamolodchikovequation, which itself is related to many deep results in representation theory.The fruitful interaction between all these ideas has led to many advancesin the last few years, including proofs of some old conjectures butalso completely new results. More specifically, loop modelsare a class of statistical models where the PI has madesignificant progress, in particular in relation to the so-calledRazumov--Stroganov conjecture (now Cantini--Sportiello theorem).New directions that should be pursued include:further applications to enumerative combinatorics such as proofs of variousopen conjectures relating Alternating Sign Matrices, Plane Partitionsand their symmetry classes;a full understanding of the quantum integrability of theFully Packed Loop model,a specific loop model at the heart of the Razumov--Stroganov correspondence;a complete description of the Brauer loop scheme, including itsdefining equations, and of the underlying poset; the extensionof the work on Di Francesco and Zinn-Justin on the loop model/6-vertex vertexrelation to the case of the 8-vertex model(corresponding to elliptic solutions of the Yang--Baxter equation);the study of solvable tilings models, in relation togeneralizations of the Littlewood--Richardson rule, and the determinationof their limiting shapes.