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Loop models, integrability and combinatorics

Objective

The purpose of this proposal is to investigate new connections which
have emerged in the recent years between problems from statistical
mechanics, namely two-dimensional exactly solvable models, and a variety
of combinatorial problems, among which: the enumeration of plane partitions,
alternating sign matrices and related objects;
combinatorial properties of certain
algebro-geometric objects such as orbital varieties or the Brauer loop scheme;
or finally certain problems in free probability. One of the key methods
that emerged in recent years is the use
of quantum integrability and more precisely the quantum Knizhnik--Zamolodchikov
equation, which itself is related to many deep results in representation theory.
The fruitful interaction between all these ideas has led to many advances
in the last few years, including proofs of some old conjectures but
also completely new results. More specifically, loop models
are a class of statistical models where the PI has made
significant progress, in particular in relation to the so-called
Razumov--Stroganov conjecture (now Cantini--Sportiello theorem).

New directions that should be pursued include:
further applications to enumerative combinatorics such as proofs of various
open conjectures relating Alternating Sign Matrices, Plane Partitions
and their symmetry classes;
a full understanding of the quantum integrability of the
Fully Packed Loop model,
a specific loop model at the heart of the Razumov--Stroganov correspondence;
a complete description of the Brauer loop scheme, including its
defining equations, and of the underlying poset; the extension
of the work on Di Francesco and Zinn-Justin on the loop model/6-vertex vertex
relation 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 to
generalizations of the Littlewood--Richardson rule, and the determination
of their limiting shapes.

Field of science

  • /natural sciences/physical sciences/classical mechanics/statistical mechanics
  • /natural sciences/mathematics/applied mathematics/statistics and probability

Call for proposal

ERC-2011-StG_20101014
See other projects for this call

Funding Scheme

ERC-SG - ERC Starting Grant

Host institution

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
Address
Rue Michel Ange 3
75794 Paris
France
Activity type
Research Organisations
EU contribution
€ 840 120
Principal investigator
Paul Georges Zinn-Justin (Dr.)
Administrative Contact
Julie Zittel (Ms.)

Beneficiaries (1)

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
France
EU contribution
€ 840 120
Address
Rue Michel Ange 3
75794 Paris
Activity type
Research Organisations
Principal investigator
Paul Georges Zinn-Justin (Dr.)
Administrative Contact
Julie Zittel (Ms.)