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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 $1/2$.

A lazy random walk remains at its current position with positive probability and otherwise makes an ordinary random walk step. A common convention replaces transition matrix $P$ by $P_{\mathrm{lazy}}=(I+P)/2$. This preserves every stationary distribution because $\pi P=\pi$ implies $\pi P_{\mathrm{lazy}}=\pi$. It also removes periodic alternation in an irreducible discrete-time chain: the self-loop permits return in one step. A lazy symmetric walk on the integers has increments $-1,0,1$ with probabilities $1/4,1/2,1/4$, so its step variance is $1/2$.