  In the standard excursion set model for the growth of structure, the
statistical properties of halos are governed by the halo mass and are
independent of the larger scale environment in which the halos reside.
Numerical simulations, however, have found the spatial distributions of halos
to depend not only on their mass but also on the details of their assembly
history and environment. Here we present a theoretical framework for
incorporating this "assembly bias" into the excursion set model. Our
derivations are based on modifications of the path integral approach of
Maggiore & Riotto (2010) that models halo formation as a non-Markovian random
walk process. The perturbed density field is assumed to evolve stochastically
with the smoothing scale and exhibits correlated walks in the presence of a
density barrier. We write down conditional probabilities for multiple barrier
crossings, and derive from them analytic expressions for descendant and
progenitor halo mass functions and halo merger rates as a function of both halo
mass and the linear overdensity of the larger-scale environment of the halo.
Our results predict a higher halo merger rate and higher progenitor halo mass
function in regions of higher overdensity, consistent with the behavior seen in
N-body simulations.
