Facilitated exclusion process

Abstract

We study the Facilitated TASEP, an interacting particle system on the one dimensional integer lattice. We prove that starting from step initial condition, the position of the rightmost particle has Tracy Widom GSE statistics on a cube root time scale, while the statistics in the bulk of the rarefaction fan are GUE. This uses a mapping with last-passage percolation in a half-quadrant which is exactly solvable through Pfaffian Schur processes. Our results further probe the question of how first particles fluctuate for exclusion processes with downward jump discontinuities in their limiting density profiles. Through the Facilitated TASEP and a previously studied MADM exclusion process we deduce that cube-root time fluctuations seem to be a common feature of such systems. However, the statistics which arise are shown to be model dependent (here they are GSE, whereas for the MADM exclusion process they are GUE). We also discuss a two-dimensional crossover between GUE, GOE and GSE distribution by studying the multipoint distribution of the first particles when the rate of the first one varies. In terms of half-space last passage percolation, this corresponds to last passage times close to the boundary when the size of the boundary weights is simultaneously scaled close to the critical point.

v2: this version corrects mistakes in the definition of the crossover kernel in Section 5. v1: 26 pages. This paper results from splitting the v1 of arXiv:1606.00525 in two parts. This paper contains our results on the facilitated exclusion process, as well as a detailed derivation of crossover kernels. arXiv:1606.00525 now contains only our results on last passage percolation in a half-space

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• Correlation Functions of the Pfaffian Schur Process Using Macdonald Difference Operators
• Stationary States of the One-dimensional Facilitated Asymmetric Exclusion Process
• Stationary States of the One-Dimensional Discrete-Time Facilitated Symmetric Exclusion Process
• The Discrete-Time Facilitated Totally Asymmetric Simple Exclusion Process
• Tracy-Widom asymptotics for a river delta model
• Pinning, diffusive fluctuations, and Gaussian limits for half-space directed polymer models
• The Limiting Behavior of the FTASEP with Product Bernoulli Initial Distribution
• Exact Solution of the F-TASEP
• Approach to Hyperuniformity in the One-Dimensional Facilitated Exclusion Process
• The half-space Airy stat process
• The second class particle in the half-line open TASEP