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Asymptotic Behaviour of the Simple Random Walk on the 2-dimensional Comb


 
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1. Title Title of document Asymptotic Behaviour of the Simple Random Walk on the 2-dimensional Comb
 
2. Creator Author's name, affiliation, country Daniela Bertacchi; Universita` di Milano-Bicocca
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Random Walk; Maximal Excursion; Generating Function; Comb; Brownian Motion
 
3. Subject Subject classification 60J10; 05A15; 60J65
 
4. Description Abstract We analyze the differences between the horizontal and the vertical component of the simple random walk on the 2-dimensional comb. In particular we evaluate by combinatorial methods the asymptotic behaviour of the expected value of the distance from the origin, the maximal deviation and the maximal span in $n$ steps, proving that for all these quantities the order is $n^{1/4}$ for the horizontal projection and $n^{1/2}$ for the vertical one (the exact constants are determined). Then we rescale the two projections of the random walk dividing by $n^{1/4}$ and $n^{1/2}$ the horizontal and vertical ones, respectively. The limit process is obtained. With similar techniques the walk dimension is determined, showing that the Einstein relation between the fractal, spectral and walk dimensions does not hold on the comb.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Italian 2004 PRIN project ``CAMPI ALEATORI''
 
7. Date (YYYY-MM-DD) 2006-12-07
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/377
 
10. Identifier Digital Object Identifier 10.1214/EJP.v11-377
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 11
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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