Statistics of Largest Loops in a Random Walk

TL;DR

This paper analyzes the size distribution of the largest neutral segments in a random walk, providing bounds, numerical results, and exploring potential singularities, with implications for understanding random walk return properties.

Contribution

It offers analytical bounds and extensive numerical analysis of the largest loop size distribution in random walks, focusing on the large N limit and potential singularities.

Findings

Distribution exhibits an essential singularity at small segment sizes

Numerical results extend down to very low probabilities (~10^{-15})

Potential singularities at specific ratios of segment size to total length

Abstract

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.