By Ravi Montenegro, Prasad Tetali

Presents an creation to the analytical features of the idea of finite Markov chain blending instances and explains its advancements. This e-book seems to be at a number of theorems and derives them in uncomplicated methods, illustrated with examples. It contains spectral, logarithmic Sobolev thoughts, 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 in Theoretical Computer Science)

Example text

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 1 − 4/m2 ≥ 2/m2 √ m2 m . 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. 4. Modified conductance 289 ˜ let A be white vertices and B circled vertices.

The mixing time bounds are left to the interested reader. 3 Conductance The most common geometric tool for studying mixing time is the conductance Φ, a measure of the chance of leaving a set after a single step. The conductance profile can also be used to lower bound the various f congestion quantities Cf when the Markov chain is lazy. 3. 13, is used to bound C√a(1−a) (r) for the Thorp shuffle. The argument is fairly simple (see also [61]). 12. cave, then Cf (A) ≤ Given a lazy Markov chain, and f : R+ → R+ conf (π(A) + 2Q(A, Ac )) + f (π(A) − 2Q(A, Ac )) .

1 π(y)P rob(y ∈ Au ) = π(Au )du = 0 y∈Ω π(y) y∈Ω Q(A, y) = π(A) π(y) ˆ is a dual process of P. 3. If S ⊂ Ω, y ∈ Ω and Λ(S, y) = tion linkage, then π(y) π(S) 1S (y) is the projec- ˆ y) . PΛ(S, y) = ΛK(S, Proof. PΛ(S, y) = z∈S ˆ y) = ΛK(S, S y Q(S, y) π(z) P(z, y) = π(S) π(S) ˆ S ) π(y) = π(y) K(S, π(S ) π(S) The final equality Q(S, y)/π(y). is because S K(S, S ) = S y y K(S, S Q(S, y) π(S) ) = P rob(y ∈ S ) = With duality it becomes easy to write the n step transitions in terms ˆ of the walk K. ˆ n .

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