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

Dark matter halo assembly bias: environmental dependence in the non-Markovian excursion set theory