The idea of a random walk is to explore locally the space in the neighborhood of the current state z. The proposal transition kernel is symmetric:
The acceptance probability a of equation 16 reduces to:
A frequent choice consists in taking 
equal to a centered Gaussian distribution. The behaviour of the algorithm is then dependent on the variance. When this one is too low, the transitions are often accepted, but consecutive states are strongly correlated, resulting in a poor mixing within the chain. On the opposite, if it is too high, transitions are often rejected and the chain remains stuck in high 
probability states.
Figure 2: Transition in a bidimensional state space generated by a random walk. The target distribution is represented as black level lines, the proposal transition kernel 
in blue. The red transition is accepted (because 
), while the green one is likely to be rejected.
Figure 2 represents the transition of a Markov chain generated by a random walk.