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.

**Read or Download Mathematical Aspects of Mixing Times in Markov Chains (Foundations and Trends(r) in Theoretical Computer Science) PDF**

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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.