We present a scheme to accurately calculate the persistence probabilities on sequences of $n$ heights above a level $h$ from the measured $n+2$ points of the height-height correlation function of a fluctuating interface. The calculated persistence pr
obabilities compare very well with the measured persistence probabilities of a fluctuating phase-separated colloidal interface for the whole experimental range.
Fluctuations of the interface between coexisting colloidal fluid phases have been measured with confocal microscopy. Due to a very low surface tension, the thermal motions of the interface are so slow, that a record can be made of the positions of th
e interface. The theory of the interfacial height fluctuations is developed. For a host of correlation functions, the experimental data are compared with the theoretical expressions. The agreement between theory and experiment is remarkably good.
We report on the residence times of capillary waves above a given height $h$ and on the typical waiting time in between such fluctuations. The measurements were made on phase separated colloid-polymer systems by laser scanning confocal microscopy. Du
e to the Brownian character of the process, the stochastics vary with the chosen measurement interval $Delta t$. In experiments, the discrete scanning times are a practical cutoff and we are able to measure the waiting time as a function of this cutoff. The measurement interval dependence of the observed waiting and residence times turns out to be solely determined by the time dependent height-height correlation function $g(t)$. We find excellent agreement with the theory presented here along with the experiments.
