  We offer theoretical explanations for some recent observations in numerical
simulations of quantum random walks (QRW). Specifically, in the case of a QRW
on the line with one particle (walker) and two entangled coins, we explain the
phenomenon, called "localization", whereby the probability distribution of the
walker's position is seen to exhibit a persistent major "spike" (or "peak") at
the initial position and two other minor spikes which drift to infinity in
either direction. Another interesting finding in connection with QRW's of this
sort pertains to the limiting behavior of the position probability
distribution. It is seen that the probability of finding the walker at any
given location becomes eventually stationary and non-vanishing. We explain
these observations in terms of the degeneration of some eigenvalue of the time
evolution operator $U(k)$. An explicit general formula is derived for the
limiting probability, from which we deduce the limiting value of the height of
the observed spike at the origin. We show that the limiting probability
decreases {\em quadratically} for large values of the position $x$. We locate
the two minor spikes and demonstrate that their positions are determined by the
phases of non-degenerated eigenvalues of $U(k)$. Finally, for fixed time $t$
sufficiently large, we examine the dependence on $t$ of the probability of
finding a particle at a given location $x$.
