Methods in non-linear plasma theory by Ronald C. Davidson

By Ronald C. Davidson

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10. 4). 11. If Z ≥ 0 is a nonnegative random variable and g is a nonnegative increasing function, then E (Z g(Z)) ≥ EZ g(EZ/2) . 2 Proof. (from [65]) Let A be the event {Z ≥ EZ/2}. Then E(Z 1Ac ) ≤ EZ/2, so E(Z1A ) ≥ EZ/2. Therefore, E (Z g(2Z)) ≥ E (Z1A g(EZ)) ≥ EZ g(EZ) . 2 Let U = 2Z to get the result. It is fairly easy to translate these to mixing time bounds. 3. Conductance 51 bounds. 5) 2x(1 − x)(1 − C√z(1−z) (x))  π∗       1   1+ 2 /4 dx     4π∗ x(1 − x)(1 − C√ (x)) z(1−z) 1+3π∗ 1 to be convex.

PΛ(S, y) = z∈S ˆ ΛK(S, y) = S y Q(S, y) π(z) P(z, y) = π(S) π(S) π(y) π(y) ˆ = K(S, S) π(S ) π(S) The final equality is because Q(S, y)/π(y). S y K(S, S ) = S y Q(S, y) π(S) K(S, 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 . 4. Let E {x} then ˆ n πS (y) , Pn (x, y) = E n where πS (y) = on set S by π. 1S (y)π(y) π(S) denotes the probability distribution induced Proof. ˆ n )({x}, y) = E ˆ n πS (y) Pn (x, y) = (Pn Λ)({x}, y) = (ΛK n The final equality is because Λ(S, y) = πS (y).

5. Consider a finite Markov chain with stationary distribution π. Any distance dist(µ, π) which is convex in µ satisfies ˆ n dist(πS , π) dist(Pn (x, ·), π) ≤ E n whenever x ∈ Ω and S0 = {x}. Proof. 4 and convexity, ˆ n πS , π) ≤ E ˆ n dist(πS , π) . dist(Pn (x, ·), π) = dist(E n n In particular, if dist(µ, π) = Lπ πµ for a convex functional Lπ : (R+ )Ω → R then the distance is convex and the conditions of the theorem are satisfied. 6. 2 2 ˆ n (1 − π(Sn )), ≤ E ˆ n log 1 , ≤ E π(Sn ) ˆn ≤ E 1 − π(Sn ) .

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