Vicious accelerating walkers

TL;DR

This paper investigates a system of N vicious accelerating walkers with Gaussian velocity fluctuations, computing survival probabilities and exploring their bounds, with implications for polymer physics and non-Markovian stochastic models.

Contribution

It introduces a numerical study of vicious accelerating walkers with Gaussian velocity fluctuations and proposes an upper bound for the survival probability exponent.

Findings

Survival exponent for N=3 is approximately 0.71.

Proposed upper bound for the exponent is 1/8N(N-1).

Numerical results for Levy flights show different exponents, e.g., 1.31 for N=3, μ=1.

Abstract

A vicious walker system consists of N random walkers on a line with any two walkers annihilating each other upon meeting. We study a system of N vicious accelerating walkers with the velocity undergoing Gaussian fluctuations, as opposed to the position. We numerically compute the survival probability exponent, {\alpha}, for this system, which characterizes the probability for any two walkers not to meet. For example, for N = 3, {\alpha} = 0.71 \pm 0.01. Based on our numerical data, we conjecture that 1/8N(N - 1) is an upper bound on {\alpha}. We also numerically study N vicious Levy flights and find, for instance, for N = 3 and a Levy index {\mu} = 1 that {\alpha} = 1.31 \pm 0.03. Vicious accelerating walkers relate to no-crossing configurations of semiflexible polymer brushes and may prove relevant for a non-Markovian extension of Dyson's Brownian motion model.