  We consider localization of a random walk (RW) when attracted or repelled by
multiple extended manifolds of different dimensionalities. In particular, we
focus on $(d-1)$- and $(d-2)$-dimensional manifolds in $d$-dimensional space,
where attractive interactions are (fully or marginally) relevant. The RW can
then be in one of four phases where it is localized to neither, one, or both
manifolds. The four phases merge at a special multi-critical point where (away
from the manifolds) the RW spreads diffusively. Extensive numerical analyses on
two dimensional RWs confined inside or outside a rectangular wedge confirm
general features expected from a continuum theory, but also exhibit unexpected
attributes, such as a reentrant localization to the corner while repelled by
it.
