Finite average lengths in critical loop models

A relation between the average length of loops and their free energy is obt ained for a variety of O(n)-type models on two-dimensional lattices, by ext ending to finite temperatures a calculation due to Kast. We show that the ( number) averaged loop length (L) over bar stays finite for all nonzero fuga cities 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) over bar = 3L(min), where L-min, is the smallest l oop length allowed by the underlying lattice. We demonstrate this analytica lly for the FPL model on the honeycomb lattice and for the 4-state Potts mo del 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-fl avour FPL model on the square lattice. We present, in addition, numerical r esults 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 distributio nof loop lengths obtained by conformal invariance methods.