The asymmetric ABAB matrix model

In this letter, it is pointed out that the two matrix model defined by the action S=(1/2)(tr A^2+tr B^2)-(alpha_A/4) tr A^4-(alpha_B/4) tr B^4-(beta/2) tr(AB)^2 can be solved in the large N limit using a generalization of the solution of Kazakov and Zinn-Justin (who considered the symmetric case alpha_A=alpha_B). This model could have useful applications to 3D Lorentzian gravity.Comment: 7 pages, 1 figur

Proof of Razumov-Stroganov conjecture for some infinite families of link patterns

We prove the Razumov--Stroganov conjecture relating ground state of the O(1) loop model and counting of Fully Packed Loops in the case of certain types of link patterns. The main focus is on link patterns with three series of nested arches, for which we use as key ingredient of the proof a generalization of the Mac Mahon formula for the number of plane partitions which includes three series of parameters

Jucys-Murphy elements and Weingarten matrices

We provide a compact proof of the recent formula of Collins and Matsumoto for the Weingarten matrix of the orthogonal group using Jucys-Murphy elements.Comment: v2: added a referenc

The General O(n) Quartic Matrix Model and its application to Counting Tangles and Links

The counting of alternating tangles in terms of their crossing number, number of external legs and connected components is presented here in a unified framework using quantum field-theoretic methods applied to a matrix model of colored links. The overcounting related to topological equivalence of diagrams is removed by means of a renormalization scheme of the matrix model; the corresponding ``renormalization equations'' are derived. Some particular cases are studied in detail and solved exactly.Comment: 21 page

Littlewood--Richardson coefficients and integrable tilings

We provide direct proofs of product and coproduct formulae for Schur functions where the coefficients (Littlewood--Richardson coefficients) are defined as counting puzzles. The product formula includes a second alphabet for the Schur functions, allowing in particular to recover formulae of [Molev--Sagan '99] and [Knutson--Tao '03] for factorial Schur functions. The method is based on the quantum integrability of the underlying tiling model.Comment: 29 pages, color figures. v2: corrected minor misprints and added short appendix. v3: fig 16 fixe