# Irrelevant variables

We may also include the irrelevant variables in the scaling law of $f$ :

$f(t,h,k_{3},k_{4},\dots )=\ell ^{-d}f(t\lambda _{\ell }^{t},h\lambda _{\ell }^{h},k_{3}\lambda _{\ell }^{(3)},k_{4}\lambda _{\ell }^{(4)}\dots )$ where $\lambda _{\ell }^{t},\lambda _{\ell }^{h}>1$ and $\lambda _{\ell }^{(i\geq 3)},\dots <1$ . After $n$ iterations we have:
$f(t,h,k_{3},k_{4},\dots )=\ell ^{-dn}f(t\ell ^{ny_{t}},h\ell ^{ny_{h}},k_{3}\ell ^{ny_{3}},k_{4}\ell ^{ny_{4}},\dots )$ where $y_{i\geq 3}<0$ in accordance with the fact that $\lambda _{\ell }^{(i\geq 3)}<1$ . Setting $\ell ^{ny_{t}}t=b$ we have:
$f(t,h,k_{3},k_{4}\dots )=b^{-d}t^{d/y_{t}}f(b^{y_{t}},b^{y_{h}}ht^{-y_{h}/y_{t}},b^{y_{3}}k_{3}t^{-y_{3}/y_{t}},k_{4}t^{-y_{4}/y_{t}},\dots )$ For $t\to 0$ the terms involving the irrelevant variables become vanishingly small, so we get:
$f(t,h,k_{3},k_{4},\dots )=t^{d/y_{t}}b^{-d}f(b^{y_{t}},b^{y_{h}}ht^{-y_{h}/y_{t}},0,0,\dots )$ Note that in the last step we have implicitly assumed that $f$ is analytic in the limit $k_{i\geq 3}\to 0$ . This assumption is however frequently false! When this happens, i.e. when the free energy density is singular in the limit $k_{j}\to 0$ for a particular irrelevant variable $k_{j}$ , that variable is termed dangerous irrelevant variable. For example, considering the Landau free energy of the Ising model obtained as a saddle-point approximation of the general functional partition function:
${\mathcal {L}}=\int \left({\frac {a}{2}}tm^{2}+{\frac {b}{4}}m^{4}+{\frac {k}{2}}\left({\vec {\nabla }}m\right)^{2}-hm\right)d^{d}{\vec {r}}$ the parameter $b$ of the quartic term is a dangerous irrelevant variable (we have seen in Coarse graining procedure for the Ising model that problems arise when we try to treat it as a perturbative parameter).