Finite average lengths in critical loop models

A relation between the average length of loops and their free energy is obtained for a variety of O(n)-type models on two-dimensional lattices, by extending to finite temperatures a calculation due to Kast. We show that the (number) averaged loop length L stays finite for all non-zero fugacities n, and in particular it does not diverge upon entering the critical regime (n → 2+). Fully packed loop (FPL) models with n = 2 seem to obey the simple relation L = 3Lmin, where Lmin is the smallest loop length allowed by the underlying lattice. We demonstrate this analytically for the FPL model on the honeycomb lattice and for the 4-state Potts model on the square lattice, and based on numerical estimates obtained from a transfer matrix method we conjecture that this is also true for the two-flavour FPL model on the square lattice. We present in addition numerical results for the average loop length on the three critical branches (compact, dense and dilute) of the O(n) model on the honeycomb lattice, and discuss the limit n → 0. Contact is made with the predictions for the distribution of loop lengths obtained by conformal invariance methods. PACS numbers: 05.50.+q, 64.60.-i, 75.10.Hk Laboratoire associé aux universités Paris 6, Paris 7 et au CNRS.