  We study analytically a simple random walk model on a one-dimensional
lattice, where at each time step the walker resets to the maximum of the
already visited positions (to the rightmost visited site) with a probability
$r$, and with probability $(1-r)$, it undergoes symmetric random walk, i.e., it
hops to one of its neighboring sites, with equal probability $(1-r)/2$. For
$r=0$, it reduces to a standard random walk whose typical distance grows as
$\sqrt{n}$ for large $n$. In presence of a nonzero resetting rate $0<r\le 1$,
we find that both the average maximum and the average position grow
ballistically for large $n$, with a common speed $v(r)$. Moreover, the
fluctuations around their respective averages grow diffusively, again with the
same diffusion coefficient $D(r)$. We compute $v(r)$ and $D(r)$ explicitly. We
also show that the probability distribution of the difference between the
maximum and the location of the walker, becomes stationary as $n\to \infty$.
However, the approach to this stationary distribution is accompanied by a
dynamical phase transition, characterized by a weakly singular large deviation
function. We also show that $r=0$ is a special `critical' point, for which the
growth laws are different from the $r\to 0$ case and we calculate the exact
crossover functions that interpolate between the critical $(r=0)$ and the
off-critical $(r\to 0)$ behavior for finite but large $n$.
