Statistics of Largest Loops in a Random Walk

Details Statistics of Largest Loops in a Random Walk Statistics of Largest Loops in a Random Walk

1996-09-09

arxiv.org/abs/cond-mat/9609085v1

Phys. Rev. E55, 261-265 (1997)

Physics

Condensed Matter

5 pages and 4 eps figures, requires RevTeX, epsf and multicol. Postscript file also available at http://cmtw.harvard.edu/~de

Scientific paper

10.1103/PhysRevE.55.261

We report further findings on the size distribution of the largest neutral segments in a sequence of N randomly charged monomers [D. Ertas and Y. Kantor, Phys. Rev. E53, 846 (1996); cond-mat/9507005]. Upon mapping to one--dimensional random walks (RWs), this corresponds to finding the probability distribution for the size L of the largest segment that returns to its starting position in an N--step RW. We primarily focus on the large N, \ell = L/N << 1 limit, which exhibits an essential singularity. We establish analytical upper and lower bounds on the probability distribution, and numerically probe the distribution down to \ell \approx 0.04 (corresponding to probabilities as low as 10^{-15}) using a recursive Monte Carlo algorithm. We also investigate the possibility of singularities at \ell=1/k for integer k.

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