Someone argues: "A three-dimensional random walk always escapes to infinity with…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods

Error, Chance and the Machine, Ch Random walks and Markov chains

Problem

Someone argues: "A three-dimensional random walk always escapes to infinity with probability

0.6595

, regardless of the region it is confined to, because transience is a fixed property of three dimensions." They also claim: "every Markov chain has one and only one stationary distribution, since

\boldsymbol\pi\mathsf{P} = \boldsymbol\pi

is a linear equation and a linear equation has one solution." ((a)) A silicon atom is confined to a small nanocrystal (a bounded lattice region, with the atom reflected back whenever it would leave). Explain, without a detailed calculation, why the atom must return to its starting site with probability

1

, and say what this shows about the first claim. ((b)) A four-state chain is built from two closed pairs,

\{1,2\}

and

\{3,4\}

, with transition matrix

\mathsf{P} = \begin{pmatrix} 0.5 & 0.5 & 0 & 0 \\ 0.5 & 0.5 & 0 & 0 \\ 0 & 0 & 0.5 & 0.5 \\ 0 & 0 & 0.5 & 0.5 \end{pmatrix}

Verify that

\boldsymbol\pi_A = (\tfrac12, \tfrac12, 0, 0)

\boldsymbol\pi_B = (0,0,\tfrac12,\tfrac12)

and

\boldsymbol\pi_C = (\tfrac14,\tfrac14,\tfrac14,\tfrac14)

are all stationary, and say what this shows about the second claim. ((c)) State the one property this chain lacks that the Perron-Frobenius theorem needs to guarantee a unique stationary distribution.

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