  We consider a discrete-time random walk where the random increment at time
step $t$ depends on the full history of the process. We calculate exactly the
mean and variance of the position and discuss its dependence on the initial
condition and on the memory parameter $p$. At a critical value $p_c^{(1)}=1/2$
where memory effects vanish there is a transition from a weakly localized
regime (where the walker returns to its starting point) to an escape regime.
Inside the escape regime there is a second critical value where the random walk
becomes superdiffusive. The probability distribution is shown to be governed by
a non-Markovian Fokker-Planck equation with hopping rates that depend both on
time and on the starting position of the walk. On large scales the memory
organizes itself into an effective harmonic oscillator potential for the random
walker with a time-dependent spring constant $k = (2p-1)/t$. The solution of
this problem is a Gaussian distribution with time-dependent mean and variance
which both depend on the initiation of the process.

Statistical Mechanics