  Random critical points are generically characterized by multifractal
properties. In the field of Anderson localization, Mirlin, Fyodorov,
Mildenberger and Evers [Phys. Rev. Lett 97, 046803 (2006)] have proposed that
the singularity spectrum $f(\alpha)$ of eigenfunctions satisfies the exact
symmetry $f(2d-\alpha)=f(\alpha)+d-\alpha$ at any Anderson transition. In the
present paper, we analyse the physical origin of this symmetry in relation with
the Gallavotti-Cohen fluctuation relations of large deviation functions that
are well-known in the field of non-equilibrium dynamics: the multifractal
spectrum of the disordered model corresponds to the large deviation function of
the rescaling exponent $\gamma=(\alpha-d)$ along a renormalization trajectory
in the effective time $t=\ln L$. We conclude that the symmetry discovered on
the specific example of Anderson transitions should actually be satisfied at
many other random critical points after an appropriate translation. For
many-body random phase transitions, where the critical properties are usually
analyzed in terms of the multifractal spectrum $H(a)$ and of the moments
exponents X(N) of two-point correlation function [A. Ludwig, Nucl. Phys. B330,
639 (1990)], the symmetry becomes $H(2X(1) -a)= H(a) + a-X(1)$, or equivalently
$\Delta(N)=\Delta(1-N)$ for the anomalous parts $\Delta(N) \equiv X(N)-NX(1)$.
We present numerical tests in favor of this symmetry for the 2D random
$Q-$state Potts model with various $Q$.
