Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 46 / 2 / c / Solution

Diagonalizing the linearized blocking map gives perturbations $v_i^{\prime }={\lambda }_i{v}_i=b^{y_i}{v}_i$. A relevant operator has $y_i>0$, equivalently $|{\lambda }_i|>1$ for a discrete scaling multiplier: its perturbation grows under repeated coarse-graining and must be tuned to reach the critical fixed point. An irrelevant operator has $y_i<0$ and decays, usually contributing corrections to scaling. A marginal operator has $y_i=0$, so linearization alone is insufficient; nonlinear flow can make it marginally relevant or irrelevant, or exactly marginal.

For an ordinary spin-inversion-symmetric critical point the temperature and magnetic field are relevant. Fixing $h=0$ by symmetry leaves the temperature to tune. At a tricritical point the quartic coupling is another relevant tuning direction. A dangerously irrelevant coupling is an exception to simply discarding small variables: setting it to zero can destabilize the ordered potential or make its scaling function singular, even though it flows toward zero.