Vicious accelerating walkers

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, α, for this system, which characterizes the probability for any two walkers not to meet. For example, for N = 3, α = 0.71 ± 0.01. Based on our numerical data, we conjecture that 1 8 N (N − 1) is an upper bound on α. We also numerically study N vicious Levy flights and find, for instance, for N = 3 and a Levy index µ = 1 that α = 1.31 ± 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.

Persistence and first-passage properties in nonequilibrium systems

“…For N = 2, this problem is simply equivalent to a single random walker with an absorbing boundary at the origin and hence θ = 1/4 for N = 2. For N = 3, it was shown in [104] that the survival probability is equivalent to the one of a single particle performing a random acceleration process in two dimensions confined in a 60 • wedge geometry (Fig. 5), similarly to the case of three Brownian walkers discussed above in section 4.1.…”

“…In Ref. [103], the authors studied the case where the process terminates upon the first encounter between two walkers (note that one could alternatively consider that it terminates upon the first crossing of two walkers, which is a different situation for Lévy flights [104], see below). Such Lévy flights, in d dimensional space, perform Markov random walks where at each time step they jump in a random direction (with an angle which is uniformly distributed between 0 and 2π) while the length of the jump η is drawn from a pdf φ(η) with an algebraic tail φ(η) ∼ η −d−µ .…”

“…Hence there are two different ways to define vicious Lévy flights: one may allow jump-overs as discussed before [103] or instead prohibit them as it was studied for d = 1 in Ref. [104]. For N = 2, the survival probability decays algebraically, Q(t) ∼ t −1/2 , independently of µ -as a consequence of the Sparre Andersen theorem [2], see section 2.…”

“…For N ≥ 3, there is no known exact result but numerical simulations seem to indicate that θ actually depends on µ. For instance, for N = 4, θ = 2.3(1) for µ = 1 while θ = 2.91(9) for µ = 2 [104].…”

“…Finally, this question was extended to the case of N vicious walkers performing random acceleration processes, in dimension d = 1 [104]. For N = 2, this problem is simply equivalent to a single random walker with an absorbing boundary at the origin and hence θ = 1/4 for N = 2.…”