Spatial survival probability for one-dimensional fluctuating interfaces in the steady state

TL;DR

This paper investigates the spatial survival probability of one-dimensional fluctuating interfaces in steady state, combining numerical simulations and analytic methods to reveal scaling behavior and derive approximate formulas.

Contribution

It introduces a combined numerical and analytic approach to study spatial survival probabilities, including new scaling functions and approximate formulas.

Findings

Survival probability exhibits scaling with system size and sampling interval.

Analytic results from path-integral and Brownian motion models match numerical data well.

Derived closed-form expressions approximate survival probabilities effectively.

Abstract

We report numerical and analytic results for the spatial survival probability for fluctuating one-dimensional interfaces with Edwards-Wilkinson or Kardar-Parisi-Zhang dynamics in the steady state. Our numerical results are obtained from analysis of steady-state profiles generated by integrating a spatially discretized form of the Edwards-Wilkinson equation to long times. We show that the survival probability exhibits scaling behavior in its dependence on the system size and the `sampling interval' used in the measurement for both `steady-state' and `finite' initial conditions. Analytic results for the scaling functions are obtained from a path-integral treatment of a formulation of the problem in terms of one-dimensional Brownian motion. A `deterministic approximation' is used to obtain closed-form expressions for survival probabilities from the formally exact analytic treatment. The resulting approximate analytic results provide a fairly good description of the numerical data.