# 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.

## Notes

### Acknowledgements

G.B and I.C. would like to thank Sidney Redner for drawing their attention to the Facilitated TASEP. Part of this work was done during the stay of J.B, G.B and I.C. at the Kavli Institute of Theoretical Physics and supported by the National Science Foundation under Grant No. NSF PHY11-25915. J.B. was supported in part by NSF grant DMS-1361782, DMS- 1664692 and DMS-1664531, and the Simons Fellows program. G.B. was partially supported by the Laboratoire de Probabilités et Modèles Aléatoires UMR CNRS 7599, Université Paris-Diderot–Paris 7 and the Packard Foundation through I.C.’s Packard Fellowship. I.C. was partially supported by the NSF through DMS-1208998 and DMS-1664650, the Clay Mathematics Institute through a Clay Research Fellowship, the Institute Henri Poincaré through the Poincaré Chair, and the Packard Foundation through a Packard Fellowship for Science and Engineering.

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