# Characteristic Sign Renewals of Kardar–Parisi–Zhang Fluctuations

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## Abstract

Tracking the *sign* of fluctuations governed by the \((1+1)\)-dimensional Kardar–Parisi–Zhang (KPZ) universality class, we show, both experimentally and numerically, that its evolution has an unexpected link to a simple stochastic model called the renewal process, studied in the context of aging and ergodicity breaking. Although KPZ and the renewal process are fundamentally different in many aspects, we find remarkable agreement in some of the time correlation properties, such as the recurrence time distributions and the persistence probability, while the two systems can be different in other properties. Moreover, we find inequivalence between long-time and ensemble averages in the fraction of time occupied by a specific sign of the KPZ-class fluctuations. The distribution of its long-time average converges to nontrivial broad functions, which are found to differ significantly from that of the renewal process, but instead be characteristic of KPZ. Thus, we obtain a new type of ergodicity breaking for such systems with many-body interactions. Our analysis also detects qualitative differences in time-correlation properties of circular and flat KPZ-class interfaces, which were suggested from previous experiments and simulations but still remain theoretically unexplained.

## Keywords

Growth phenomenon Scaling laws KPZ universality class Renewal theory Stochastic process Weak ergodicity breaking## Notes

### Acknowledgments

We acknowledge fruitful discussions with E. Barkai, I. Dornic, C. Godrèche, and S. N. Majumdar. This work is supported in part by KAKENHI from JSPS (No. JP25707033 and No. JP25103004), the JSPS Core-to-Core Program “Non-equilibrium dynamics of soft matter and information”, and the National Science Foundation under Grant No. NSF PHY11-25915.

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