KTH/Nordita/SU seminar in Theoretical Physics

Critical percolation and qKZ equations

by Bernard Nienhuis (University of Amsterdam)

FA31 ()


Based on the work of Razumov and Stroganov it was noted that many exact expressions for certain correlation functions for critical bond percolation could be found for arbitrary system sizes and distances involved. These results were obtained by the study of the Perron-Frobenius eigenvector of the transfer matrix for the cylinder or strip. It turned out far more difficult to obtain similar results for site percolation model on the triangular lattice.

Here we present an approach to both site- and bond percolation, applicable to arbitrary rhombus tilings and to isoradial lattices respectively. It makes use of relations known as q-Knizhnik-Zamolodchikov equations (qKZ). These relations are satisfied by the correlations in the models. In some geometries these equations can be solved, yielding the correlation functions. For some specific correlation functions, these results can be extrapolated to arbitrary sizes.