概率论 (Summer 2013)/Problem Set 5

Problem 2

Let [math]\displaystyle{ G(V,E) }[/math] be an undirected connected graph with maximum degree [math]\displaystyle{ \Delta }[/math].

• Design an efficient, time reversible, ergodic random walk on [math]\displaystyle{ G }[/math] whose stationary distribution is the uniform distribution.
• Let [math]\displaystyle{ \pi }[/math] be an arbitrary distribution on [math]\displaystyle{ V }[/math] such that [math]\displaystyle{ \pi(v)\gt 0 }[/math] for all [math]\displaystyle{ v\in V }[/math]. Design a time reversible, ergodic random walk on [math]\displaystyle{ G }[/math] whose stationary distribution is [math]\displaystyle{ \pi }[/math].

Problem 3

Consider the following random walk on [math]\displaystyle{ n }[/math]-dimensional hypercube: Assume we are now at the vertex [math]\displaystyle{ b_1b_2\dots b_n }[/math] where each [math]\displaystyle{ b_i\in\{0,1\} }[/math], then

• With probability [math]\displaystyle{ \frac{1}{n+1} }[/math], do nothing.
• Otherwise, with probability [math]\displaystyle{ \frac{1}{n+1} }[/math] for each coordinate [math]\displaystyle{ i }[/math], flip [math]\displaystyle{ b_i }[/math].

Prove by coupling that the mixing time of this markov chain is [math]\displaystyle{ O(n\ln n) }[/math]

Problem 4

Recall the markov chain we used in class to sample proper colorings: Let [math]\displaystyle{ G(V,E) }[/math] be an undirected graph with maximum degree [math]\displaystyle{ \Delta }[/math], [math]\displaystyle{ q\ge\delta+2 }[/math] is the number of colors. Assume we are currently given a proper coloring, then

• Pick a vertex [math]\displaystyle{ v\in V }[/math] uniformly at random and a color [math]\displaystyle{ c\in\{1,2,\dots,q\} }[/math] uniformly at random.
• Recolor [math]\displaystyle{ v }[/math] with [math]\displaystyle{ c }[/math] if this yields a proper coloring, else do nothing.

1. Prove that this markov chain is irreducible, aperiodic, time reversible and the stationary distribution is the uniform distribution.
2. Suppose [math]\displaystyle{ q\le\Delta+1 }[/math], show that this markov chain is no long always irreducible.