  The mass, accretion rate and formation time of dark matter haloes near
proto-filaments (identified as saddle points of the potential) are analytically
predicted using a conditional version of the excursion set approach in its
so-called "upcrossing" approximation. The model predicts that at fixed mass,
mass accretion rate and formation time vary with orientation and distance from
the saddle, demonstrating that assembly bias is indeed influenced by the tides
imposed by the cosmic web. Starved, early forming haloes of smaller mass lie
preferentially along the main axis of filaments, while more massive and younger
haloes are found closer to the nodes. Distinct gradients for distinct tracers
such as typical mass and accretion rate occur because the saddle condition is
anisotropic, and because the statistics of these observables depend on both the
conditional means and their covariances. The theory is extended to other
critical points of the potential field. The response of the mass function to
variations of the matter density field (the so-called large scale bias) is
computed, and its trend with accretion rate is shown to invert along the
filament. The signature of this model should correspond at low redshift to an
excess of reddened galactic hosts at fixed mass along preferred directions, as
recently reported in spectroscopic and photometric surveys and in
hydrodynamical simulations. The anisotropy of the cosmic web emerges therefore
as a significant ingredient to describe jointly the dynamics and physics of
galaxies, e.g. in the context of intrinsic alignments or morphological
diversity.
