Queuing transitions in the asymmetric simple exclusion process

Abstract

Stochastic driven flow along a channel can be modeled by the asymmetric simple exclusion process. We confirm numerically the presence of a dynamic queuing phase transition at a nonzero obstruction strength, and establish its scaling properties. Below the transition, the traffic jam is macroscopic in the sense that the length of the queue scales linearly with system size. Above the transition, only a power-law shaped queue remains. Its density profile scales as δρ ∼ x −ν with ν = 1 3, and x is the distance from the obstacle. We construct a heuristic argument, indicating that the exponent ν = 1 3 is universal and independent of the dynamic exponent of the underlying dynamic process. Fast bonds create only power-law shaped depletion queues, and with an exponent that could be equal to ν = 2 3, but the numerical results yield consistently somewhat smaller values ν ≃ 0.63(3). The implications of these results to faceting of growing interfaces and localization of directed polymers in random media, both in the presence of a columnar defect are pointed out as well.

Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals

“…As β is decreased there will be longer and longer excursions away from the diagonal until the critical point β c , when the directed polymer depins. It is conjectured that β c = 0 [20,7], but there are counterclaims mostly based on numerical simulations of the TASEP [18]. Unfortunately the source at x = 0 is not covered by our methods, since it does not respect the product structure.…”

“…As before, the distance between eigenvalues is of order 1. (6.6) becomes (8.17) and (6.9) is modified to 18) U N has to be chosen such that V = − log ψ g , where ψ g is the ground state of H N . U N depends only weakly on N. For the construction of the determinantal process one fills the first N levels of H N , which results in a Fermi energy E F = O(1).…”

Effect of a columnar defect on the shape of slow-combustion fronts

“…This model shows [11] a queueing transition at r = r c = 0.80 ± 0.02. In addition, the density profile displays a qualitatively similar asymmetry between the slow and fast defect-bond cases as the mean-field solution for the KPZ fronts between the advancing and retarding columnar-defect cases.…”

“…As the numerical data of [14] is not decisive, we have done [11] simulations on a totally asymmetric ASEP model with a fixed defect bond with hopping rate rp in the middle of the system, while the hopping rate at the other bonds was p. Open boundary conditions were imposed such that the hopping-in rate at the left boundary was αp, and the hopping-out rate at the right boundary was βp. In what follows we only consider the case…”

Polymer pinning in a random medium as influence percolation

“…On the basis of these data, in [8] the value λ c = 1 is conjectured. Recently this result has been challenged ( [6]) and λ c ∼ = 0.8 is claimed.…”