  The renormalized trajectory (RT) is determined from two different Monte Carlo
renormalization group techniques with $\delta$-function block spin
transformation in the multi-dimensional coupling parameter space of the
two-dimensional non-linear sigma model with O(3) symmetry. At a correlation
length $\xi \approx 3$-$5$, the RT is shown to break away from the straight
line of the fixed point trajectory (FPT) which is orthogonal to the critical
surface and originates from the ultraviolet fixed point (UVFP). The large $N$
calculation of the RT is also presented in the coupling parameter space of the
most general bilinear Hamiltonian. The RT in the large $N$ approximation
exhibits a similar shape with the sharp break occurring at a somewhat smaller
correlation length.
