  We consider the dynamical properties of Quantum Walks defined on the
d-dimensional cubic lattice, or the homogeneous tree of coordination number 2d,
with site dependent random phases, further characterised by transition
probabilities between neighbouring sites equal to 1/(2d). We show that the
localisation length for these Balanced Random Quantum Walks can be expressed as
a combinatorial expression involving sums over weighted paths on the considered
graph. This expression provides lower bounds on the localisation length by
restriction to paths with weight 1, which allows us to prove the localisation
length diverges on the tree as d^2. On the cubic lattice, the method yields the
lower bound 1/ln(2) for all d, and allows us to bound the localisation length
from below by the correlation length of self-avoiding walks computed at 1/(2d)
