Consider a Markov chain where every state has a $\delta$ probability of transitioning to every other state. Is the spectral gap of this Markov chain also at least $\delta$?
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If I remember correctly, the spectral gap is exactly 1. You can use the variational characterization of the spectral gap to prove this. Another way is to find all the eigenvalues of the transition kernel – LNT Jan 29 '24 at 03:51