# Irrelevant variables

We may also include the irrelevant variables in the scaling law of ${\displaystyle f}$:

${\displaystyle f(t,h,k_{3},k_{4},\dots )=\ell ^{-d}f(t\lambda _{\ell }^{t},h\lambda _{\ell }^{h},k_{3}\lambda _{\ell }^{(3)},k_{4}\lambda _{\ell }^{(4)}\dots )}$
where ${\displaystyle \lambda _{\ell }^{t},\lambda _{\ell }^{h}>1}$ and ${\displaystyle \lambda _{\ell }^{(i\geq 3)},\dots <1}$. After ${\displaystyle n}$ iterations we have:
${\displaystyle f(t,h,k_{3},k_{4},\dots )=\ell ^{-dn}f(t\ell ^{ny_{t}},h\ell ^{ny_{h}},k_{3}\ell ^{ny_{3}},k_{4}\ell ^{ny_{4}},\dots )}$
where ${\displaystyle y_{i\geq 3}<0}$ in accordance with the fact that ${\displaystyle \lambda _{\ell }^{(i\geq 3)}<1}$. Setting ${\displaystyle \ell ^{ny_{t}}t=b}$ we have:
${\displaystyle f(t,h,k_{3},k_{4}\dots )=b^{-d}t^{d/y_{t}}f(b^{y_{t}},b^{y_{h}}ht^{-y_{h}/y_{t}},b^{y_{3}}k_{3}t^{-y_{3}/y_{t}},k_{4}t^{-y_{4}/y_{t}},\dots )}$
For ${\displaystyle t\to 0}$ the terms involving the irrelevant variables become vanishingly small, so we get:
${\displaystyle f(t,h,k_{3},k_{4},\dots )=t^{d/y_{t}}b^{-d}f(b^{y_{t}},b^{y_{h}}ht^{-y_{h}/y_{t}},0,0,\dots )}$
Note that in the last step we have implicitly assumed that ${\displaystyle f}$ is analytic in the limit ${\displaystyle k_{i\geq 3}\to 0}$. This assumption is however frequently false! When this happens, i.e. when the free energy density is singular in the limit ${\displaystyle k_{j}\to 0}$ for a particular irrelevant variable ${\displaystyle k_{j}}$, 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:
${\displaystyle {\mathcal {L}}=\int \left({\frac {a}{2}}tm^{2}+{\frac {b}{4}}m^{4}+{\frac {k}{2}}\left({\vec {\nabla }}m\right)^{2}-hm\right)d^{d}{\vec {r}}}$
the parameter ${\displaystyle b}$ 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).