Holding probability ($P(x,x)$)

For a random walk with transition matrix $P$, its holding probability at state $x$ is $P(x,x)$, the conditional probability of staying at $x$ for the next step. It is the holding probability of a Markov chain specialized to the walk. A lazy random walk can be constructed byIts holding probability is $\alpha +(1-\alpha )P(x,x)$, which equals $\alpha$ only if the original walk cannot stay at $x$.

This change preserves every stationary distribution: $\pi P=\pi$ implies $\pi P_{\alpha }=\pi$. A positive holding probability permits return in one step, giving period one at that state; for an irreducible Markov chain, this removes periodicity throughout the class. Laziness does not connect previously disconnected classes. For a loop-free walk with $\alpha =1/2$, the conventional holding probability is therefore exactly $1/2$.