In one and two dimensions, all roads lead to Rome, but not in three dimensions.
Specifically, given a random walk (equally likely to move in any direction) on the integers in one or two dimensions, then no matter the starting point, with probability one (a.k.a. almost surely), the random walk will eventually get to a specific designated point ("Rome").
However, for three or more dimensions, the probability of getting to "Rome" is less than one; with the probability decreasing as the number of dimensions increases.
So for instance, if conducting a stochastic (Monte Carlo) simulation of a random walk starting at "Rome", which will stop when Rome is returned to, then in one and two dimensions, you can be assured of eventually making it back to Rome and stopping the simulation - so easy. In three dimensions, you may never make it back, so hard.
To visualize the two-dimensional case, one can imagine a person
walking randomly around a city. The city is effectively infinite and
arranged in a square grid of sidewalks. At every intersection, the
person randomly chooses one of the four possible routes (including the
one originally travelled from). Formally, this is a random walk on the
set of all points in the plane with integer coordinates.
Will the person ever get back to the original starting point of the
walk? This is the 2-dimensional equivalent of the level crossing
problem discussed above. In 1921 George Pólya proved that the person
almost surely would in a 2-dimensional random walk, but for 3
dimensions or higher, the probability of returning to the origin
decreases as the number of dimensions increases. In 3 dimensions, the
probability decreases to roughly 34%
See http://mathworld.wolfram.com/PolyasRandomWalkConstants.html for numerical values.