= Holding probability
{title2=$P(x,x)$}
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 by
$$
P_\alpha=\alpha I+(1-\alpha)P,\qquad 0<\alpha<1.
$$
Its <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 $1/2$.
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