  The percolation properties of equatorial strips of the two dimensional O(3)
nonlinear $\sigma$ model are investigated numerically. Convincing evidence is
found that a sufficently narrow strip does not percolate at arbitrarily low
temperatures. Rigorous arguments are used to show that this result implies both
the presence of a massless phase at low temperature and lack of asymptotic
freedom in the massive continuum limit. A heuristic estimate of the transition
temperature is given which is consistent with the numerical data.
