Strong law of large numbers for a branching random walk among Bernoulli traps

TL;DR

The work analyzes a discrete-time BRW on $\mathbb{Z}^d$ in an i.i.d. Bernoulli-trap environment with hard killing, proving a quenched SLLN for the total mass conditional on ultimate survival within the infinite trap-free cluster $\mathcal{C}$ when $d\ge 2$ and $p>p_d$. It introduces a two-colored BRW to control mass via trap interactions, and establishes large-deviation bounds for the free BRW mass, enabling precise upper and lower bounds. The lower bound hinges on constructing a typical environment with large and huge clearings, and a bootstrap growth mechanism inside clearings that yields exponential mass growth; the upper bound uses percolation and survival estimates combined with conditioning and Borel-Cantelli. Collectively, the results extend LLN-type phenomena from continuum obstacle problems to a discrete BRWRE with hard killing, offering a rigorous quenched LLN for mass growth in random media with hard barriers.

Abstract

We study a $d$-dimensional branching random walk (BRW) in an i.i.d. random environment on $\mathbb{Z}^d$ in discrete time. A Bernoulli trap field is attached to $\mathbb{Z}^d$, where each site, independently of the others, is a trap with a fixed probability. The interaction between the BRW and the trap field is given by the hard killing rule. Given a realization of the environment, over each time step, each particle first moves according to a simple symmetric random walk to a nearest neighbor, and immediately afterwards, splits into two particles if the new site is not a trap or is killed instantly if the new site is a trap. Conditional on the ultimate survival of the BRW, we prove a strong law of large numbers for the total mass of the process. Our result is quenched, that is, it holds in almost every environment in which the starting point of the BRW is inside the infinite connected component of trap-free sites.

Figures (2)

• Figure 1: Illustration of a typical environment $\omega$ in $\Omega_3\cap\Omega_4$ at a large time $n$ in $d=2$. Red sites represent traps and white sites are vacant. The smaller balls with grey boundaries represent the 'large' accessible clearings of radius $r(n)=(R_0/2)(\log\rho(n))^{1/d}$. The larger ball with grey boundary represents the 'huge' accessible clearing of radius $R(n)=R_0(\log n)^{1/d}-2\sqrt{d}$. Observe that since the balls with grey boundaries represent accessible clearings, they do not contain any red sites which are traps, and moreover each can be reached from the origin by a nearest neighbor path that only visits vacant sites.
• Figure 2: Illustration of a snapshot at time $m(n)$ of the strategy described in Part $2$. The small grey ball is $B(z_0,r(m(n)))$, which is a 'large' accessible clearing and contains at least $e^{\delta_1 m(n)}$ particles at time $m(n)$. The bigger red ball is the 'huge' accessible clearing of radius $R(m(n))$ and has at least one particle on its center at time $\lfloor(1+\delta_2)m(n)\rfloor$.

Theorems & Definitions (17)

• Theorem 1: Survival probability for BRW among Bernoulli traps
• Theorem 2: Quenched SLLN for BRW among Bernoulli traps
• Remark
• Definition 1: Ancestral line segment
• Proposition 1: Expected mass
• proof
• Proposition 2
• proof
• Proposition 3
• proof