On some relations between local height probabilities and conformal invariance

We observe that the expressions of order parameters (“local height probabilities”) in the regime III of integrable A, D lattice models are closely related to the partition functions of the associated minimal critical theories in a finite box with appropriate boundary conditions. This is more precisely stated in terms of corner transfer matrix properties. Our work thus points out a relation between integrable systems off critically and in the thermodynamic limit with models at critically but in a finite box, once the nome which parametrizes the elliptic Boltzmann weights in the former case is identified with the modular parameter in the latter. As an application we are able to obtain some new properties of the LHP for D N or E 6 , E 7 , E 8 models using conformal invariance techniques only.