Exact Solution for a Class of Random Walk on the Hypercube |
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Authors: | Benedetto Scoppola |
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Institution: | (1) University of Southern California, Los Angeles, CA 90089-2532, USA |
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Abstract: | A class of families of Markov chains defined on the vertices of the n-dimensional hypercube, Ω
n
={0,1}
n
, is studied. The single-step transition probabilities P
n,ij
, with i,j∈Ω
n
, are given by
Pn,ij=\frac(1-a)dij(2-a)nP_{n,ij}=\frac{(1-{\alpha})^{d_{ij}}}{(2-{\alpha})^{n}}, where α∈(0,1) and d
ij
is the Hamming distance between i and j. This corresponds to flip independently each component of the vertex with probability
\frac1-a2-a\frac{1-{\alpha}}{2-{\alpha}}. The m-step transition matrix Pn,ijmP_{n,ij}^{m} is explicitly computed in a close form. The class is proved to exhibit cutoff. A model-independent result about the vanishing
of the first m terms of the expansion in α of Pn,ijmP_{n,ij}^{m} is also proved. |
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Keywords: | |
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