Diffusion-limited annihilating-coalescing systems

TL;DR

The paper analyzes diffusion-limited annihilating-coalescing systems (DLACS) with two particle types on transitive unimodular graphs, incorporating caps M and N and potentially distinct diffusion rates. It establishes a phase transition in A-particle survival at a critical density p_c and provides sharp occupation-time estimates for the root, including a linear lower bound in supercritical regimes and a logarithmic bound when caps are infinite. In the symmetric no-cap case (M=N=∞, λ_A=λ_B, p=1/2), the work proves that the root occupation probability scales as P(ξ_t(0)≠0) ∼ (2/3) P(ζ_t(0)=1), connecting DLACS to coalescing random walk via a robust 2/3 thinning argument. Additionally, a lower bound ρ_t(d) ≥ C t^{-5/4} is established at criticality on ℤ^d, and the results are obtained through a combination of mass-transport principles, tracer arguments, and voter-model duality, resolving an open case and extending diffusion-limited interaction models beyond prior DLAS work.

Abstract

We study a family of interacting particle systems with annihilating and coalescing reactions. Two types of particles are interspersed throughout a transitive unimodular graph. Both types diffuse as simple random walks with possibly different jump rates. Upon colliding, like particles coalesce up to some cap and unlike particles annihilate. We describe a phase transition as the initial particle density is varied and provide estimates for the expected occupation time of the root. For the symmetric setting with no cap on coalescence, we prove that the limiting occupation probability of the root is asymptotic to 2/3 the occupation probability for classical coalescing random walk. This addresses an open problem from Stephenson.

Figures (4)

• Figure 1: Simulations of DLACS with $\lambda_B=0$ and $M=\infty$ on a cycle with 2000 vertices. $A$-particles perform discrete time random walk for 2000 time steps. The top graphic has $p=.6$, the middle graphic has $p=.7$, and the bottom graphic has $p=.8$. Determining the critical value is an open problem.
• Figure 2: Graphical construction of the coalescing random walk $\zeta_t$ on a subset of $\mathbb{Z}$. Arrows along each oriented edge occur at times of independent Poisson processes with rate $1/2$. Particles move from the tail to the head of an arrow, and when two particles meet they coalesce. Arrows in bold signify coalescence events. Bold (red or blue) lines indicate the presence of particles.
• Figure 3: Graphical constructions of the DLACS $\xi_t$ on a subset of $\mathbb{Z}$. The arrows are the same as in Figure \ref{['fig:CRW']}, and only the labels of the bold arrows are shown, as the labels on the remaining arrows are irrelevant. Bold (red or blue) lines indicate the presence of particles. The difference between the left and right realizations is the label of the leftmost arrow; note the effect of this label on the presence of the particle at the middle site.
• Figure 4: The dual trees $\mathcal{T}^*_t$ corresponding to the middle vertex (red component) in Figure \ref{['fig:CRW']} with labels from Figure \ref{['fig:DLACS']}. The nodes act as logical OR and XOR gates with inputs of $1=\texttt{True}$ at the leaves and output $0$ or $1$ at the root.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• proof
• Lemma 6
• Lemma 7
• proof
• proof : Proof of \ref{['thm:main']}