  We present an analytical approach to study simple symmetric random walks
(RWs) on a crossing geometry consisting of a plane square lattice crossed by
$n_l$ number of lines that all meet each other at a single point (the origin)
on the plane. The probability density to find the walker at a given distance
from the origin either in a line or in the plane geometry is exactly calculated
as a function of time t. We find that the large time asymptotic behavior of the
walker for any arbitrary number $n_l$ of lines is eventually governed by the
plane geometry after a crossover time approximately given by $t_c\propto
n_l^2$. We show that this competition can be changed in favor of the line
geometry by switching on an arbitrarily small perturbation of a drift term in
which even a weak biased walk is able to drain the whole probability density
into the line at long time limit. We also present the results of our extensive
simulations of the model which perfectly support our analytical predictions.
Our method can, however, be simply extended to other crossing geometries with a
single common point.
