Integrability and combinatorics: loop models, alternating sign matrices and multidegree of algebraic varieties

Philippe di Francesco


We show how integrability of various loop models allows to relate them to purely combinatorial settings. As a by-product, we prove the ``Razumov-Stroganov" sum rule, that relates the partition function of the dense O(1) loop model on a cylinder to the number of alternating sign matrices of same size. We also interpret the ``Brauer" crossing O(1) loop model in terms of degrees of matrix varieties, computed via matrix integrals.