Mathematical Aspects of Mixing Times in Markov Chains by Ravi Montenegro, Prasad Tetali

By Ravi Montenegro, Prasad Tetali

Presents an creation to the analytical elements of the speculation of finite Markov chain blending instances and explains its advancements. This publication seems to be at a number of theorems and derives them in uncomplicated methods, illustrated with examples. It comprises spectral, logarithmic Sobolev ideas, the evolving set method, and problems with nonreversibility.

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Extra info for Mathematical Aspects of Mixing Times in Markov Chains (Foundations and Trends(r) in Theoretical Computer Science)

Example text

Modified Conductance 55 Define φ(A) similarly but without π(Ac ) in the denominator. ˜ Observe that for a lazy chain Ψ(A) = Q(A, Ac ), and so φ(A) = ˜ Φ(A) ≥ Φ(A), showing that modified conductance bounds extend conductance to the non-lazy case. 15. Consider a random walk on a cycle of even length, Z/mZ. 2, and so we can conclude that 1 − C√z(1−z) ≥ 1 − and τ2 ( ) ≤ 1 − 4/m2 ≥ 2/m2 √ m m2 log . 2 shows the worst case of A and B, with Ψ(A) = Q(A, B) = 0, and so φ˜ = 0. The conductance of the lazy version of this chain was good, and we correctly found that the chain converged, while the non-lazy version is periodic and has zero modified conductance, so modified conductance captured the key differences between these chains.

8. For a non-empty subset S ⊂ Ω the first Dirichlet eigenvalue on S is given by E(f, f ) λ1 (S) = inf Var(f ) f ∈c+ 0 (S) where c+ 0 (S) = {f ≥ 0 : supp(f ) ⊂ S} is the set of non-negative functions supported on S. 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 ) .

As mentioned earlier, the argument is based on improving on the elementary relation E(f, f ) ≥ λVar(f ), and instead making a relation depending on the size of the support of f . 8. For a non-empty subset S ⊂ Ω the first Dirichlet eigenvalue on S is given by E(f, f ) λ1 (S) = inf Var(f ) f ∈c+ 0 (S) where c+ 0 (S) = {f ≥ 0 : supp(f ) ⊂ S} is the set of non-negative functions supported on S. 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.

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