Parallel TASEP on a ring: blockage problem and non-analyticity of the current
The Totally Asymmetric Simple Exclusion Process (TASEP)
is an important example of a particle system driven by an irreversible Markov chain.
Assume \(n\) particles are placed on a ring of length \(2n\),
at each step particles free to move in the clockwise direction
occupy the free site independently with probability \(p\).
In this parallel framework
I will do a simple yet rigorous derivation
of the chain stationary measure.
I will next consider the blockage problem (a.k.a. slow bond problem),
deriving the exact expression of the current for an arbitrary blockage intensity \(\varepsilon\) in the case
of the so-called rule-184 cellular automaton,
i.e. a parallel TASEP with \(p=1\). Finally, I will
discuss through numerical experiments the conjecture that
for parallel updates other than rule-184 the current
may be non-analytic in the blockage intensity around the value \(\varepsilon=0\).