Mathematical Aspects of Mixing Times in Markov Chains by Montenegro R. By Montenegro R.

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3) for lazy Markov chains. 15 for an example of how to use conductance. Conductance is inappropriate for non-lazy chains because it cannot distinguish a periodic chain from an aperiodic one. 3 it was found that for a discrete time chain the L2 mixing time is closely related to λPP∗ , and via Cheeger’s inequality it is thus related to ΦPP∗ . In fact, the same holds for the evolving set bound on L2 mixing. 13. 1 − C√z(1−z) (A) ≥ 1 − 4 1 − Φ2PP∗ (A) ≥ 1 2 Φ ∗ (A) 4 PP Proof. Given A, B ⊂ Ω, and u, w ∈ [0, 1] chosen uniformly at random, then QP (A, x)P∗ (x, B) QPP∗ (A, B) = x π(x) P rob(x ∈ Au ) P rob(x ∈ Bw ) = x = Eπ(Au ∩ Bw ) .

8. ˆ n+1 f (π(Sn+1 )) − E ˆ n f (π(Sn )) = −E ˆ n f (π(Sn )) (1 − Czf (z) (Sn )) E ˆ n f (π(Sn )) ≤ −(1 − Czf (z) ) E Proof. The inequality is because 1−Czf (z) ≤ 1−Czf (z) (S) for all S ⊂ Ω. 2. 9. In discrete time n dist(Pn (x, ·), π) ≤ Czf (z) f (π(x)) and 1 f (π∗ ) log 1 − Czf (z) τ( ) ≤ . ˆ n+1 f (π(Sn+1 )) ≤ Czf (z) E ˆ n f (π(Sn )), and by Proof. 8, E ˆ n f (π(Sn )) ≤ C n induction E zf (z) f (π(S0 )). Solving for when this drops to and using the approximation log Czf (z) ≤ −(1 − Czf (z) ), gives the corollary.

The spectral profile Λ : [π∗ , ∞) → R is given by Λ(r) = inf π∗ ≤π(S)≤r λ1 (S). The spectral profile is a natural extension of spectral gap λ, and we will now see that it can be used to improve on the basic bound E(f, f ) ≥ λVar(f ) used earlier. 2. 9. For every non-constant function f : Ω → R+ , E(f, f ) ≥ 1 Λ 2 4(Ef )2 Var f Var(f ) . Proof. Given a ∈ R use the notation a+ = max{a, 0} to denote the positive part. For c constant, E(f, f ) = E(f −c, f −c). Also, E(f −c, f − c) ≥ E((f − c)+ , (f − c)+ ) because ∀a, b ∈ R : (a − b)2 ≥ (a+ − b+ )2 .