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.

Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line

“…where B k,m is given in (37) and D k i ,m i ,k j ,m j is given in (28) and where we recall the constraint ns j=1 m j = n. This formula is identical to the one obtained by a completely different calculation in [28] for the problem of the droplet initial condition with any A, consistent with the theorem of Ref. [39].…”

“…We obtain an explicit formula for the transition kernel in the critical region, A + 1 2 = ǫt −1/3 → 0 with fixed ǫ = O(1), continuously depending on ǫ. Although it is different from the expression obtained in [36,37], we conjecture that the two expressions are in fact equivalent, and check it by a systematic expansion in traces to third order. In the limit ǫ → 0 we obtain that the PDF converges to the GOE TW distribution.…”

“…For the latter an explicit Fredholm Pfaffian formula was obtained in Ref. [36,37]. Although the kernel there is different from ours, we have checked by expansion in traces to third order that for ǫ > 0 their Fredholm Pfaffian coincides (see Appendix C).…”

“…Finally, although we do not address the case A < −1/2 or ǫ < 0 the conjectured equivalence with the kernel of Ref. [36,37] (which holds for ǫ > 0) suggests that for any A < − 1 2 , the one-point KPZ height has Gaussian fluctuations.…”

“…This transition PDF was obtained as a solution of Painlevé type system of equations. Further results were obtained recently using Pfaffian-Schur processes, for variants of TASEP models and last passage percolation in a half-quadrant [36,37]. Not only the one-point, but also the multi-point height distributions where studied (the extended process) and for arbitrary positions with respect to the wall.…”