We study memory based random walk models to understand diffusive motion in
crowded heterogeneous environment. The models considered are non-Markovian as
the current move of the random walk models is determined by randomly selecting
a move from history. At each step, particle can take right, left or stay moves
which is correlated with the randomly selected past step. There is a perfect
stay-stay correlation which ensures that the particle does not move if the
randomly selected past step is a stay move. The probability of traversing the
same direction as the chosen history or reversing it depends on the current
time and the time or position of the history selected. The time or position
dependent biasing in moves implicitly corresponds to the heterogeneity of the
environment and dictates the long-time behavior of the dynamics that can be
diffusive, sub or super diffusive. A combination of analytical solution and
Monte Carlo simulation of different random walk models gives rich insight on
the effects of correlations on the dynamics of a system in heterogeneous
environment
