  Quantum algorithms for searching one or more marked items on a d-dimensional
lattice provide an extension of Grover's search algorithm including a spatial
component. We demonstrate that these lattice search algorithms can be viewed in
terms of the level dynamics near an avoided crossing of a one-parameter family
of quantum random walks. We give approximations for both the level-splitting at
the avoided crossing and the effectively two-dimensional subspace of the full
Hilbert space spanning the level crossing. This makes it possible to give the
leading order behaviour for the search time and the localisation probability in
the limit of large lattice size including the leading order coefficients. For
d=2 and d=3, these coefficients are calculated explicitly. Closed form
expressions are given for higher dimensions.
