Excursion Set Theory for Correlated Random Walks
  We present a new method to compute the first crossing distribution in
excursion set theory for the case of correlated random walks. We use a
combination of the path integral formalism of Maggiore & Riotto, and the
integral equation solution of Zhang & Hui, and Benson et al. to find a
numerically robust and convenient algorithm to derive the first crossing
distribution in terms of a perturbative expansion around the limit of an
uncorrelated random walk. We apply this methodology to the specific case of a
Gaussian random density field filtered with a Gaussian smoothing function. By
comparing our solutions to results from Monte Carlo calculations of the first
crossing distribution we demonstrate that our method accurate for power spectra
$P(k)\propto k^n$ for $n=1$, becoming less accurate for smaller values of $n$.
It is therefore complementary to the method of Musso & Sheth, which will
therefore be more useful for standard $\Lambda$CDM power spectra. Our approach
is quite general, and can be adapted to other smoothing functions, and also to
non-Gaussian density fields.
