## Electronic Journal of Probability

### Exceedingly large deviations of the totally asymmetric exclusion process

#### Abstract

Consider the Totally Asymmetric Simple Exclusion Process (TASEP) on the integer lattice $\mathbb{Z}$. We study the functional Large Deviations of the integrated current $\mathsf{h} (t,x)$ under the hyperbolic scaling of space and time by $N$, i.e., $\mathsf{h} _{N}(t,\xi ) := \frac{1} {N}\mathsf{h} (Nt,N\xi )$. As hinted by the asymmetry in the upper- and lower-tail large deviations of the exponential Last Passage Percolation, the TASEP exhibits two types of deviations. One type of deviations occur with probability $\exp (-O(N))$, referred to as speed-$N$; while the other with probability $\exp (-O(N^{2}))$, referred to as speed-$N^2$. In this work we study the speed-$N^2$ functional Large Deviation Principle (LDP) of the TASEP, and establish (non-matching) large deviation upper and lower bounds.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 16, 71 pp.

Dates
Accepted: 9 February 2019
First available in Project Euclid: 22 February 2019

https://projecteuclid.org/euclid.ejp/1550826099

Digital Object Identifier
doi:10.1214/19-EJP278

Subjects
Primary: 60F10: Large deviations

#### Citation

Olla, Stefano; Tsai, Li-Cheng. Exceedingly large deviations of the totally asymmetric exclusion process. Electron. J. Probab. 24 (2019), paper no. 16, 71 pp. doi:10.1214/19-EJP278. https://projecteuclid.org/euclid.ejp/1550826099

#### References

• [BCG16] A. Borodin, I. Corwin, and V. Gorin. Stochastic six-vertex model. Duke Math J, 165(3):563–624, 2016.
• [BD06] T. Bodineau and B. Derrida. Current large deviations for asymmetric exclusion processes with open boundaries. J Stat Phys, 123(2):277–300, 2006.
• [BS10] M. Balázs and T. Seppäläinen. Order of current variance and diffusivity in the asymmetric simple exclusion process. Annals of Mathematics, pages 1237–1265, 2010.
• [CKP01] H. Cohn, R. Kenyon, and J. Propp. A variational principle for domino tilings. J Amer Math Soc, 14(2):297–346, 2001.
• [dGKW18] J. de Gier, R. Kenyon, and S. S. Watson. Limit shapes for the asymmetric five vertex model. arXiv:1812.11934, 2018.
• [DL98] B. Derrida and Lebowitz. Exact large deviation function in the asymmetric exclusion process. Phys Rev Lett, 80(2):209—212, Jan 1998.
• [DLS03] B. Derrida, J. L. Lebowitz, and E. Speer. Exact large deviation functional of a stationary open driven diffusive system: the asymmetric exclusion process. J Stat Phys, 110(3-6):775—810, Jan 2003.
• [DZ99] J.-D. Deuschel and O. Zeitouni. On increasing subsequences of iid samples. Comb Probab Comput, 8(3):247–263, 1999.
• [EK09] S. N. Ethier and T. G. Kurtz. Markov processes: characterization and convergence, volume 282. John Wiley & Sons, 2009.
• [GKS10] N. Georgiou, R. Kumar, and T. Seppalainen. TASEP with discontinuous jump rates. Alea, 7:293–318, 2010.
• [GS92] L.-H. Gwa and H. Spohn. Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian. Phys Rev Lett, 68(6):725, 1992.
• [Jen00] L. Jensen. The asymmetric exclusion process in one dimension. PhD thesis, New York Univ., New York, 2000.
• [Joh00] K. Johansson. Shape fluctuations and random matrices. Comm Math Phys, 209(2):437–476, 2000.
• [KL13] C. Kipnis and C. Landim. Scaling limits of interacting particle systems, volume 320. Springer Science & Business Media, 2013.
• [KOV89] C. Kipnis, S. Olla, and S. Varadhan. Hydrodynamics and large deviation for simple exclusion processes. Comm Pure Appl Math, 42(2):115–137, 1989.
• [Lig05] T. M. Liggett. Interacting Particle Systems. Springer, 2005.
• [Lig13] T. M. Liggett. Stochastic interacting systems: contact, voter and exclusion processes, volume 324. springer science & Business Media, 2013.
• [Mar10] M. Mariani. Large deviations principles for stochastic scalar conservation laws. Probab Theory Related Fields, 147:607—-648, Jan 2010.
• [Rez91] F. Rezakhanlou. Hydrodynamic limit for attractive particle systems on $\mathbb{Z} ^d$. Comm Math Phys, 140(3):417–448, 1991.
• [Ros81] H. Rost. Non-equilibrium behaviour of a many particle process: Density profile and local equilibria. Probab Theory Related Fields, 58(1):41–53, 1981.
• [Rud87] W. Rudin. Real and complex analysis. Tata McGraw-Hill Education, 1987.
• [Sch97] G. M. Schütz. Exact solution of the master equation for the asymmetric exclusion process. J Stat Phys, 88(1):427–445, 1997.
• [Sep98a] T. Seppäläinen. Coupling the totally asymmetric simple exclusion process with a moving interface. Markov Process Related Fields, 4(4):593–628, 1998.
• [Sep98b] T. Seppäläinen. Large deviations for increasing sequences on the plane. Probab Theory Related Fields, 112(2):221–244, 1998.
• [TW94] C. A. Tracy and H. Widom. Level-spacing distributions and the Airy kernel. Commun Math Phys, 159(1):151–174, 1994.
• [Var04] S. Varadhan. Large deviations for the asymmetric simple exclusion process. In Stochastic analysis on large scale interacting systems, volume 39 of Adv Stud Pure Math, pages 1—-27, Tokyo, 2004. Math Soc Japan.