Corrections to finite-size scaling in two-dimensional O(N) sigma-models

We have considered the corrections to the finite-size-scaling functions for a general class of $O(N)$ $\sigma$-models with two-spin interactions in two dimensions for $N=\mathrm{\infty }$. We have computed the leading corrections finding that they generically behave as $(f(z)\log L+g(z))\mathrm{/}{L}^2$ where $z=m(L)L$ and $m(L)$ is a mass scale; $f(z)$ vanishes for Symanzik improved actions for which the inverse propagator behaves as $q^2+O(q^6)$ for small $q$, but not for on-shell improved ones. We also discuss a model with four-spin interactions which shows a much more complicated behaviour.