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We study a discrete model of an heterogeneous elastic line with internaldisorder, submitted to thermal fluctuations. The monomers are connected throughrandom springs with independent and identically distributed elastic constantsdrawn from $p(k)\sim k^{\mu-1}$ for $k\to0$. When $\mu>1$, the scaling of thestandard Edwards-Wilkinson model is recovered. When $\mu<1$, the elastic lineexhibits an anomalous scaling of the type observed in many growth models andexperiments. Here we derive and use the exact expression for the exactprobability distribution of the line shape at equilibrium, as well as thespectral properties of the matrix containing the random couplings, to fullycharacterize the sample to sample fluctuations. Our results lead to novelscaling predictions that partially disagree with previous works, but which arecorroborated by numerical simulations. We also provide a novel interpretationof the anomalous scaling in terms of the abrupt jumps in the line's shape thatdominate the average value of the observable.

Anomalous scaling of heterogeneous elastic lines: a new picture from sample to sample fluctuations