Symmetry relations for multifractal spectra at random critical points

Random critical points are generically characterized by multifractal properties. In the field of Anderson localization, Mirlin et al (2006 Phys. Rev. Lett. 97 046803) have proposed that the singularity spectrum f(alpha) of eigenfunctions satisfies the exact symmetry f(2d - alpha) = f(alpha) + d - alpha. In the present paper, we analyze the physical origin of this symmetry in relation to 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 for 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 moment exponents X(N) of the two-point correlation function (Ludwig 1990 Nucl. Phys. B 330 639), 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) = X(N) - NX(1). We present numerical tests favoring this symmetry for the 2D random Q-state Potts model with varying Q.