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 rx,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.
Highlights
The one-dimensional totally asymmetric simple exclusion process (TASEP) is among the most studied particle systems
Using Theorem 1.6, we study the current for the TASEP on Galton–Watson trees
For every n ∈ N, we define a time window [tlow, tup] in which we study the current through the th n level of the tree, and where we see a number of particles proportional to n passing through Z n
Summary
The one-dimensional totally asymmetric simple exclusion process (TASEP) is among the most studied particle systems. In this article we define the TASEP on (directed) rooted trees This way the particle system retains the total asymmetry of its one-dimensional analogue, while having more space to explore. The dual question is to fix a generation window and see how many particles occupy sites in there, by a given time. In the subsection we give a formal introduction to the TASEP on trees and present our results on the disentanglement, the current and the large time behaviour of the particles. Our main results are Theorem 1.6, Theorem 1.17, Theorem 4.1, Theorem 4.3, Lemma 6.3, and Lemma 6.6
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