Large time asymptotics of growth models on space-like paths I: PushASEP

Abstract

We consider a new interacting particle system on the onedimensional lattice that interpolates between TASEP and Toom's model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way.We prove that for flat and step initial conditions, the large time fluctuations of the height function of the associated growth model along any space-like path are described by the Airy 1 and Airy 2 processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle's trajectory as special cases.

Anisotropic Growth of Random Surfaces in 2 + 1 Dimensions

“…The row {x m m } m≥1 also evolves as a Markov chain on Z that is sometimes called "long range TASEP"; it was also called PushASEP in [7]. It is convenient to view {x m m + m} m≥1 as particle locations in Z.…”

“…In the Markov chain considered above one can make the particle jump rates depend on the upper index m in an arbitrary way. One can also allow the particles jump both right and left, with ratio of left and right jump rates possibly changing in time [7].…”

“…As was mentioned above, rows {x m 1 } m≥1 and {x m m } m≥1 can be viewed as one-dimensional growth models on their own, and their asymptotic behavior was studied in [7] using essentially the same Theorem 1.1. This is exactly the edge behavior of our two-dimensional growth model.…”

“…These computations lead to the asymptotic results of [7][8][9][10][11]48] for one-dimensional growth models with more general types of initial conditions.…”

“…We are then in a position to apply Theorem 4.2 of [7]. For convenience of the reader, this theorem can be found in Appendix A.…”