Universal correlations between shocks in the ground state of elastic interfaces in disordered media

The ground state of an elastic interface in a disordered medium undergoes collective jumps upon variation of external parameters. These mesoscopic jumps are called shocks, or static avalanches. Submitting the interface to a parabolic potential centered at $w$, we study the avalanches which occur as $w$ is varied. We are interested in the correlations between the avalanche sizes $S_1$ and $S_2$ occurring at positions $w_1$ and $w_2$. Using the functional renormalization group (FRG), we show that correlations exist for realistic interface models below their upper critical dimension. Notably, the connected moment ${\langle S_1{S}_2\rangle }^c$ is up to a prefactor exactly the renormalized disorder correlator, itself a function of $|w_2-w_1|$. The latter is the universal function at the center of the FRG; hence, correlations between shocks are universal as well. All moments and the full joint probability distribution are computed to first nontrivial order in an $\epsilon$ expansion below the upper critical dimension. To quantify the local nature of the coupling between avalanches, we calculate the correlations of their local jumps. We finally test our predictions against simulations of a particle in random-bond and random-force disorder, with surprisingly good agreement.

Figure 1

Illustrations of the typical shape of the renormalized disorder correlator $\mathrm{\Delta }\left(W\right)$ (black dashed line) and of its second-order derivative ${\mathrm{\Delta }}^{″}\left(W\right)$ (red line) for the random-field (left) and random-bond (right) universality classes (not to scale). Our results predict that the shock sizes are always negatively correlated in the random-field universality class, whereas the random-bond universality class exhibits a richer structure with negatively (positively) correlated shock sizes at small (large) distances.

Figure 2

Renormalized disorder $\mathrm{\Delta }\left(u\right)$ measured in the $d=0$ RB toy model. (Inset) Its second derivative ${\mathrm{\Delta }}^{″}\left(u\right)$, computed using a numerical fit of the measured $\mathrm{\Delta }\left(u\right)$.

Figure 3

Comparison between the measurement of the normalized moment $\frac{{\langle S_1{S}_2\rangle }_{{\rho }_W^c}}{{\langle S\rangle }_{\rho }^2}$ (blue dots) and the prediction from the exact result (40) using the measurement of $\mathrm{\Delta }\left(u\right)$ (red curve) in the RB toy model. The agreement is perfect, as expected.

Figure 4

Comparison between the measurement of ${\rho }_2\left(W\right)$ (blue dots) and the prediction from the $O\left(\epsilon \right)$ result (75) using the measurement of $\mathrm{\Delta }\left(u\right)$ (red curve) in the RB toy model. We obtain a surprisingly good agreement.

Figure 5

Comparison between the measurement of the normalized moment $\frac{{\langle S_1^2{S}_2\rangle }_{{\rho }_W^c}}{{\langle S^2\rangle }_{\rho }{\langle S\rangle }_{\rho }}$ (blue dots) and the prediction from the exact result (40) using the measurement of $\mathrm{\Delta }\left(u\right)$ (red curve) in the RB toy model.

Figure 6

Renormalized disorder $\mathrm{\Delta }\left(u\right)$ measured in the $d=0$ RF toy model. (Inset) Its second derivative ${\mathrm{\Delta }}^{″}\left(u\right)$, computed using a numerical fit of the measured $\mathrm{\Delta }\left(u\right)$.

Figure 7

Comparison between the measurement of the normalized moment $\frac{{\langle S_1{S}_2\rangle }_{{\rho }_W^c}}{{\langle S\rangle }_{\rho }^2}$ (blue dots) and the prediction from the exact result (40) using the measurement of $\mathrm{\Delta }\left(u\right)$ (red curve) in the RF toy model. The agreement is perfect, as expected.

Figure 8

Comparison between the measurement of ${\rho }_2\left(W\right)$ (blue dots) and the prediction from the $O\left(\epsilon \right)$ result (75) using the measurement of $\mathrm{\Delta }\left(u\right)$ (red curve) in the RF toy model. The agreement is surprisingly good.

Figure 9

Comparison between the measurement of the normalized moment $\frac{{\langle S_1^2{S}_2\rangle }_{{\rho }_W^c}}{{\langle S^2\rangle }_{\rho }{\langle S\rangle }_{\rho }}$ (blue dots) and the prediction from the $O\left(\epsilon \right)$ result (40) using the measurement of $\mathrm{\Delta }\left(u\right)$ (red curve) in the RF toy model.