= Solution
Identifying $g$ with temperature, write $g=g_c^*+\delta g$ near the nonzero critical fixed point. The derivative of the beta function there is
$$
\beta'(g_c^*)=-\epsilon+2\frac{N-2}{2\pi}\Lambda^\epsilon g_c^*=\epsilon.
$$
Thus the temperature-like perturbation has renormalization-group eigenvalue $y_t=\epsilon$. Since the <correlation-length critical exponent> satisfies $\nu=1/y_t$,
$$
\boxed{\nu=\frac1\epsilon+O(1)}.
$$
The Gaussian fixed point $g=0$ is the stable ordered-phase fixed point for $d>2$, rather than the finite-temperature transition. In exactly two dimensions the flow instead gives an essential, exponential correlation-length divergence, corresponding formally to $\nu=\infty$ rather than a finite power-law exponent.
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