Integrable models and combinatorics

Relations between quantum integrable models solvable by the quantum inverse scattering method and some aspects of enumerative combinatorics and partition theory are discussed. The main example is the Heisenberg $XXZ$ spin chain in the limit cases of zero or infinite anisotropy. Form factors and some thermal correlation functions are calculated, and it is shown that the resulting form factors in a special $q$-parametrization are the generating functions for plane partitions and self-avoiding lattice paths. The asymptotic behaviour of the correlation functions is studied in the case of a large number of sites and a moderately large number of spin excitations. For sufficiently low temperature a relation is established between the correlation functions and the theory of matrix integrals.
Bibliography: 125 titles.