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.
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