Disordered systems present multifractal properties at criticality. In particular, as discovered by Ludwig [A.W.W. Ludwig, Nucl. Phys. B 330, 639 (1990)] in the case of a diluted two-dimensional Potts model, the moments $\overline{{\rho }^q\left(r\right)}$ of the local order parameter $\rho \left(r\right)$ scale with a set $x\left(q\right)$ of nontrivial exponents $x\left(q\right)\ne qx\left(1\right)$. We reexamine these ideas to incorporate more recent findings: (i) whenever a multifractal measure $w\left(r\right)$ normalized over space ${\sum }_{r}w\left(r\right)=1$ occurs in a random system, it is crucial to distinguish between the typical values and the disorder-averaged values of the generalized moments $Y_q={\sum }_r{w}^q\left(r\right)$, since they may scale with different generalized dimensions $D\left(q\right)$ and $\tilde{D}\left(q\right)$, and (ii), as discovered by Wiseman and Domany [S. Wiseman and E. Domany, Phys. Rev. E 52, 3469 (1995)], the presence of an infinite correlation length induces a lack of self-averaging at critical points for thermodynamic observables, in particular for the order parameter. After this general discussion, valid for any random critical point, we apply these ideas to random polymer models that can be studied numerically for large sizes and good statistics over the samples. We study the bidimensional wetting or the Poland-Scheraga DNA model with loop exponents $c=1.5$ (marginal disorder) and $c=1.75$ (relevant disorder). Finally, we argue that the presence of finite Griffiths-ordered clusters at criticality determines the asymptotic value $x(q\to \infty )=d$ and the minimal value ${\alpha }_{min}=D(q\to \infty )=d-x\left(1\right)$ of the typical multifractal spectrum $f\left(\alpha \right)$.

• Received 3 May 2007

DOI:https://doi.org/10.1103/PhysRevE.76.021114