Consider the following Markov chain on permutations of length $n$. At each time step we choose a random position. If the letter at that position is smaller than the letter immediately to the left (cyclically) then these letters swap positions. Otherwise nothing happens, corresponding to a loop in the Markov chain. This is the circular TASEP. We compute the average proportion of time the chain spends at the identity permutation (and, in greater generality, at sorted words). This answers a conjecture by Thomas Lam.
We prove a new shift-invariance property of the colored TASEP. From the shift-invariance of the colored stochastic six-vertex model (proved in Borodin-Gorin-Wheeler or Galashin), one can get a shift-invariance property of the colored TASEP at one time, and our result generalizes this to multiple times. Our proof takes the single-time shift-invariance as an input, and uses analyticity of the probability functions and induction arguments. We apply our shift-invariance to prove a distributional identity between the finishing times of the oriented swap process and the point-to-line passage times i…
We introduce a new interacting particle system on $\mathbb{Z}$, \emph{slowed $t$-TASEP}. It may be viewed as a $q$-TASEP with additional position-dependent slowing of jump rates depending on a parameter $t$, which leads to discrete and nonuniversal asymptotics at large time. If on the other hand $t \to 1$ as $\text{time} \to \infty$, we prove (1) a law of large numbers for particle positions, (2) a central limit theorem, with convergence to the fixed-time Gaussian marginal of a stationary solution to SDEs derived from the particle jump rates, and (3) a bulk limit to a certain explicit stationa…
We obtain a new relation between the distributions $μ_t$ at different times $t\ge 0$ of the continuous-time TASEP (Totally Asymmetric Simple Exclusion Process) started from the step initial configuration. Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions $μ_t$ backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to a stationary Markov dynamics p…
The oriented swap process is a natural directed random walk on the symmetric group that can be interpreted as a multi-species version of the Totally Asymmetric Simple Exclusion Process (TASEP) on a finite interval. An open problem from a 2008 paper of Angel, Holroyd, and Romik asks for the limiting distribution of the absorbing time of the process. We resolve this question by proving that this random variable satisfies GOE Tracy-Widom asymptotics. Our starting point is a distributional identity relating the behavior of the oriented swap process to last passage percolation, conjectured in a rec…
Many integrable stochastic particle systems in one space dimension (such as TASEP - totally asymmetric simple exclusion process - and its various deformations, with a notable exception of ASEP) remain integrable when we equip each particle $x_i$ with its own jump rate parameter $ν_i$. It is a consequence of integrability that the distribution of each particle $x_n(t)$ in a system started from the step initial configuration depends on the parameters $ν_j$, $j\le n$, in a symmetric way. A transposition $ν_n \leftrightarrow ν_{n+1}$ of the parameters thus affects only the distribution of $x_n(t)$…
