On the Markovian assumption in the excursion set approach: the
  approximation of Markov Velocities
  The excursion set approach uses the statistics of the density field smoothed
on a wide range of scales, to gain insight into a number of interesting
processes in nonlinear structure formation, such as cluster assembly, merging
and clustering. The approach treats the curve defined by the height of the
overdensity fluctuation field when changing the smoothing scale as a random
walk. The steps of the walks are often assumed to be uncorrelated, so that the
walk heights are a Markov process, even though this assumption is known to be
inaccurate for physically relevant filters. We develop a model in which the
walk steps, rather than heights, are a Markov process, and correlations between
steps arise because of nearest neighbour interactions. This model is a
particular case of a general class, which we call Markov Velocity models. We
show how these can approximate the walks generated by arbitrary power spectra
and filters, and, unlike walks with Markov heights, provide a very good
approximation to physically relevant models. We define a Markov Velocity Monte
Carlo algorithm to generate walks whose first crossing distribution is very
similar to that of TopHat-smoothed LCDM walks. Finally, we demonstrate that
Markov Velocity walks generically exhibit a simple but realistic form of
assembly bias, so we expect them to be useful in the construction of more
realistic merger history trees.
