  In terms of the excursion set model, we used Monte Carlo methods in order to
study the non-Markovian stochastic evolution of the smoothed overdensity
$\delta$ at scale $S$. For a Gaussian density field, smoothed by the top-hat
filter, in real space, we used random walks, with the correct correlation
between scales, in order to calculate constrained probabilities and to connect
the number of structures with the overdensity of their environment. A constant
barrier is used, lower than the usual one but which improves the consistency of
our results with those of N-body simulations.We present comparisons of
multiplicity functions with those resulting from N-body simulations as well as
number densities of descendant haloes and of their progenitors and we connect
them to the density of their large environment.There exists a dependence of the
number of haloes on their environment which is different for different masses
of these haloes.This number increases for larger overdensity of the environment
for large haloes and decreases for small ones. The number of progenitors of
these haloes is insensitive to the overdensity of their large environment.
