Geodesic
The range of a simple random walk on \(\mathbb{Z}\): an elementary combinatorial approach
Bernhard A. Moser1
1Software Competence Center Hagenberg
The electronic journal of combinatorics, Tome 21 (2014) no. 4
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Two different elementary approaches for deriving an explicit formula for the distribution of the range of a simple random walk on $\mathbb{Z}$ of length $n$ are presented. Both of them rely on Hermann Weyl's discrepancy norm, which equals the maximal partial sum of the elements of a sequence. By this the original combinatorial problem on $\mathbb{Z}$ can be turned into a known path-enumeration problem on a bounded lattice. The solution is provided by means of the adjacency matrix $\mathbf Q_d$ of the walk on a bounded lattice $(0,1,\ldots,d)$. The second approach is algebraic in nature, and starts with the adjacency matrix $\mathbf{Q_d}$. The powers of the adjacency matrix are expanded in terms of products of non-commutative left and right shift matrices. The representation of such products by means of the discrepancy norm reveals the solution directly.
DOI : 10.37236/4106
Classification : 05C81, 60G50
Mots-clés : random walk, discrepancy norm, lattice path enumeration

Bernhard A. Moser  1

1 Software Competence Center Hagenberg 
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     title = {The range of a simple random walk on {\(\mathbb{Z}\):} an elementary combinatorial approach},
     journal = {The electronic journal of combinatorics},
     year = {2014},
     volume = {21},
     number = {4},
     doi = {10.37236/4106},
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Bernhard A. Moser. The range of a simple random walk on \(\mathbb{Z}\): an elementary combinatorial approach. The electronic journal of combinatorics, Tome 21 (2014) no. 4. doi: 10.37236/4106

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