Resetting of fluctuating interfaces at power-law times

Abstract

What happens when the time evolution of a fluctuating interface is interrupted with resetting to a given initial configuration after random time intervals $\tau$ distributed as a power-law $\sim {\tau }^{-(1+\alpha )};\alpha >0$? For an interface of length $L$ in one dimension, and an initial flat configuration, we show that depending on $\alpha$, the dynamics as $L\to \infty$ exhibits a rich long-time behavior. Without resetting, the interface width grows unbounded with time as $t^{\beta }$, where $\beta$ is the so-called growth exponent. We show that introducing resetting induces for $\alpha >1$ and at long times fluctuations that are bounded in time. Corresponding to such a stationary state is a distribution of fluctuations that is strongly non-Gaussian, with tails decaying as a power-law. The distribution exhibits a cusp for small argument, implying that the stationary state is out of equilibrium. For $\alpha <1$, resetting is unable to counter the otherwise unbounded growth of fluctuations in time, so that the distribution of fluctuations remains time dependent with an ever-increasing width even at long times. Although stationary for $\alpha >1$, the width of the interface grows forever with time as a power-law for $1<\alpha <{\alpha }^{(\mathrm{w})}$, and converges to a finite constant only for larger $\alpha$, thereby exhibiting a crossover at ${\alpha }^{(\mathrm{w})}=1+2\beta$. The time-dependent distribution of fluctuations for $\alpha <1$ exhibits for small argument another interesting crossover behavior, from cusp to divergence, across ${\alpha }^{(\mathrm{d})}=1-\beta$. We demonstrate these results by exact analytical results for the paradigmatic Edwards-Wilkinson (EW) dynamical evolution of the interface, and further corroborate our findings by extensive numerical simulations of interface models in the EW and the Kardar-Parisi-Zhang universality class.

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