  We analyze the differences between the horizontal and the vertical component
of the simple random walk on the 2-dimensional comb. In particular we evaluate
by combinatorial methods the asymptotic behaviour of the expected value of the
distance from the origin, the maximal deviation and the maximal span in $n$
steps, proving that for all these quantities the order is $n^{1/4}$ for the
horizontal projection and $n^{1/2}$ for the vertical one (the exact constants
are determined). Then we rescale the two projections of the random walk
dividing by $n^{1/4}$ and $n^{1/2}$ the horizontal and vertical ones,
respectively. The limit process is obtained. As a corollary of the estimate of
the expected value of the maximal deviation, the walk dimension is determined,
showing that the Einstein relation between the fractal, spectral and walk
dimensions does not hold on the comb.
