= Relevant direction of a fixed point
= Relevant directions of a fixed point
{synonym}
A perturbation of a <renormalization-group fixed point> is relevant if it grows as the coarse-graining length increases. For a continuous flow $d\delta g/d\ell=M\delta g$, an <eigenvector> of the linearized <matrix> with <eigenvalue> $\sigma$ grows as $e^{\sigma\ell}$; a positive real part is relevant, a negative real part is irrelevant and a zero real part requires nonlinear analysis. For a discrete transformation with length factor $b>1$, its <Jacobian matrix> multiplier $\lambda$ instead gives $\delta g_n=\lambda^n\delta g_0$: $|\lambda|>1$ is relevant, $|\lambda|<1$ is irrelevant and $|\lambda|=1$ requires nonlinear analysis. For a positive multiplier the scaling exponent is $y=\log\lambda/\log b$. Thus a positive discrete <eigenvalue> smaller than one is an <irrelevant operator> direction, while the sign criterion belongs to the continuous generator.
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