The critical velocity of the bullet process appears pathwise

TL;DR

This work analyzes the bullet process with random firing speeds and interarrival delays, focusing on the critical velocity vc for the first bullet to survive. It introduces a pathwise surrogate hat{v}, defined via potential survivors in finite truncations, and proves hat{v} equals vc almost surely. A finite threats lemma and renewal arguments underpin a zero-one law and the main equivalence, with additional results showing infinitely many survivors when the velocity distribution has finite support and establishing continuity properties of the survival probability θ. The study also links the findings to one-sided ballistic annihilation and provides a practical means to approximate vc from a single process realization.

Abstract

In the bullet process, a gun fires bullets in the same direction at independent random speeds, and with independent random time delays between firings. When two bullets collide, they vanish. The critical velocity $v_c$ is the slowest speed the first bullet can take and still have positive probability of surviving forever. We characterize the critical velocity via a random variable determined by the sequence of speeds and delays, which we show almost surely equals $v_c$. In turn we prove other facts about the process, including that infinitely many bullets survive when the velocity distribution has finite support. Along the way we answer a question from Broutin--Marckert (2020), showing that if a bullet survives, it does so in all but finitely many truncations of the process.

Figures (7)

• Figure 1: The time-space diagram for a sample path of the bullet process up to time $t=20$ in the setting with $\mu = \operatorname{Unif}(.5,1.5)$ and unit delays, i.e. $\nu = \delta_1$. Time moves left to right, with distance to the origin depicted vertically.
• Figure 2: Approximation of $\theta$ and $v_c$. We approximate $\theta(v)$ in the context of the bullet problem where $\mu = \operatorname{Unif}(0,1)$ and $\nu = \delta_1$, using $N=2000$ simulations of $n=1$ billion bullets. Letting $\theta_n(v)$ be the probability the first bullet survives with velocity $v$ among $n$ bullets, and $\hat{\theta}_{n,N}(v)$ its empirical estimation based on $N$ simulations, we find $\hat{v}_c(n, N) \sim .7524$, the minimal $v$ with $\hat{\theta}_{n,N}(v) > 0$. By Theorem \ref{['thm:Marckert']} we have $\theta_n(v) \to \theta(v)$ as $n \to \infty$, and therefore $\lim_{N \to \infty}\lim_{n \to \infty}\hat{v}_c(n, N) = v_c$ in probability.
• Figure 3: Approximation of $\hat{v} = v_c$. We approximate $\hat{v}$ in two different simulations of the bullet process in its original setting using $n=10^8$ bullets each. We consider potential survivors fired between times $n/2+1$ and $n$. The yellow line displays their maximum velocity. A histogram of their velocities is shown in blue. We use buckets of width $.001$, and the height of each bar is the number of potential survivors with velocity in the bucket divided by $.001 (n/2)$, estimating the probability a velocity-$v$ bullet will be a potential survivor. While we have no guarantees on the rate of convergence for $\hat{v}$, the simulations appear to be stabilizing, giving a seemingly accurate estimate of $v_c$ in concordance with that from Figure \ref{['fig:theta']} obtained by estimating $\theta$.
• Figure 4: A bullet process with k=3 survivors, $S=\{b_{s_1}, b_{s_2}, b_{s_3}\}$. We have $S_{r_n} = S$ for each $r_n$.
• Figure 5: Above we depict $B(\omega)$ for $\omega \in F$. There are $4$ bullets in $PS_{>v}$, whose paths are drawn in red. Below, the path of $b^v_0$ is green, and in blue is $\vec{b}_{[1, 2m]}$, which shields $b^v_0$ from its $4$ threats.

Theorems & Definitions (29)

• Remark 1
• Theorem 2
• Theorem 3
• Theorem 4
• Remark 5
• Theorem 6
• Lemma 7
• proof
• Theorem 8
• proof