Loop model with mixed boundary conditions, qKZ equation and alternating sign matrices

The integrable loop model with mixed boundary conditions based on the 1-boundary extended Temperley--Lieb algebra with loop weight 1 is considered. The corresponding qKZ equation is introduced and its minimal degree solution described. As a result, the sum of the properly normalized components of the ground state in size L is computed and shown to be equal to the number of Horizontally and Vertically Symmetric Alternating Sign Matrices of size 2L+3. A refined counting is also considered

Ground-state properties of a supersymmetric fermion chain

We analyze the ground state of a strongly interacting fermion chain with a supersymmetry. We conjecture a number of exact results, such as a hidden duality between weak and strong couplings. By exploiting a scale free property of the perturbative expansions, we find exact expressions for the order parameters, yielding the critical exponents. We show that the ground state of this fermion chain and another model in the same universality class, the XYZ chain along a line of couplings, are both written in terms of the same polynomials. We demonstrate this explicitly for up to N = 24 sites, and provide consistency checks for large N. These polynomials satisfy a recursion relation related to the Painlev\'e VI differential equation, and using a scale-free property of these polynomials, we derive a simple and exact formula for their limit as N goes to infinity.Comment: v2: added more information on scaling function, fixed typo

Finite-size left-passage probability in percolation

We obtain an exact finite-size expression for the probability that a percolation hull will touch the boundary, on a strip of finite width. Our calculation is based on the q-deformed Knizhnik--Zamolodchikov approach, and the results are expressed in terms of symplectic characters. In the large size limit, we recover the scaling behaviour predicted by Schramm's left-passage formula. We also derive a general relation between the left-passage probability in the Fortuin--Kasteleyn cluster model and the magnetisation profile in the open XXZ chain with diagonal, complex boundary terms.Comment: 21 pages, 8 figure