The TASEP on Galton-Watson trees

Abstract

We study the totally asymmetric simple exclusion process (TASEP) on trees where particles are generated at the root. Particles can only jump away from the root, and they jump from $x$ to $y$ at rate $r_{x,y}$ provided $y$ is empty. Starting from the all empty initial condition, we show that the distribution of the configuration at time $t$ converges to an equilibrium. We study the current and give conditions on the transition rates such that the current is of linear order or such that there is zero current, i.e. the particles block each other. A key step, which is of independent interest, is to bound the first generation at which the particle trajectories of the first $n$ particles decouple.

Figures (7)

• Figure 1: Snapshot of a Tree-TASEP evolution. Particles enter at the root at rate $\lambda$ and then move down the tree, i.e. their distance from the root can only grow. They attempt a jump when the Poisson clock of an edge in front of them rings and the target site will be the child associated to the edge. The jump is suppressed if the target site is occupied (e.g. look at particle 1 attempting to jump at the occupied child) otherwise the jump is performed (e.g. particle 2).
• Figure 2: The $3$-regular (or binary) tree satisfying a flow rule with $r_{j}^{\min} = r_{j}^{\max}=2^{-j-1}$.
• Figure 3: Visualization of the key idea for the proof of the a priori bound on the disentanglement. When \ref{['def:UniformElliptic']} holds, the probability that particle $i$ follows the blue trajectory of particle $j$ is at most $( \frac{1}{1+\varepsilon} )^2$.
• Figure 4: A core at the left-hand side and one of its corresponding Galton--Watson trees on the right-hand side. We obtain the Galton--Watson tree from the core (the core from the Galton--Watson tree) by adding (removing) the smaller vertices depicted in gray.
• Figure 5: Visualization of the TASEP on trees and the different generations $\mathscr{D}_n$ and $\mathscr M_n$ involved in the proof for $n=4$. The particles are drawn in red. Note that it depends on the next successful jump of the particle at generation $3$, if the first $4$ particles are disentangled at generation $\mathscr M_n=4$, i.e. they will disentangle if the particle jumps at the location indicated by the arrow.

Theorems & Definitions (24)

• proof : Proof of Proposition \ref{['lem:Early-sep']}
• proof
• proof : Proof of Lemma \ref{['lem:depthboundRoot']}
• proof : Proof of Lemma \ref{['lem:depth-ren-root']}
• proof : Proof of Proposition \ref{['pro:EnteringParticles']}
• proof
• proof
• proof : Proof of Theorem \ref{['thm:DisentanglementGWT']}
• proof
• proof : Proof of Lemma \ref{['lem:ExpoLPP']}