On the Markovian assumption in the excursion set approach: the approximation of Markovian 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.