Communications in Mathematical Physics

, Volume 341, Issue 1, pp 219–261

# Weakly Asymmetric Non-Simple Exclusion Process and the Kardar–Parisi–Zhang Equation

Article

## Abstract

We analyze a class of non-simple exclusion processes and the corresponding growth models by generalizing the discrete Cole–Hopf transformation of Gärtner (Stoch Process Appl, 27:233–260, 1987). We identify the main non-linearity and eliminate it by imposing a gradient type condition. For hopping range at most 3, using the generalized transformation, we prove the convergence of the exclusion process toward the Kardar–Parisi–Zhang (kpz) equation. This is the first universality result under the weak asymmetry concerning interacting particle systems. While this class of exclusion processes are not explicitly solvable, under the weak asymmetry we obtain the exact one-point limiting distribution for the step initial condition by using the previous result of Amir et al. (Commun Pure Appl Math, 64(4): 466–537, 2011) and our convergence result.

## Preview

### References

1. 1.
Alberts T., Khanin K., Quastel J.: The intermediate disorder regime for directed polymers in dimension 1 + 1. Ann. Probab. 42(3), 1212–1256 (2014)
2. 2.
Amir G., Corwin I., Quastel J.: Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions. Commun. Pure Appl. Math. 64(4), 466–537 (2011)
3. 3.
Assing S.: A rigorous equation for the Cole–Hopf solution of the conservative KPZ equation. Stoch Partial Differ Equ Anal Comput 1(2), 365–388 (2013)
4. 4.
Balázs, M., Komjáthy, J., Seppäläinen, T.: Microscopic concavity and fluctuation bounds in a class of deposition processes. In: Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, vol. 48, pp. 151–187. Institut Henri Poincaré, Paris (2012)Google Scholar
5. 5.
Bertini L., Cancrini N.: The stochastic heat equation: Feynman–Kac formula and intermittence. J. Stat. Phys. 78(5-6), 1377–1401 (1995)
6. 6.
Bertini L., Bertini L.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183(3), 571–607 (1997)
7. 7.
Borodin A., Borodin A.: Macdonald processes. Probab. Theory Relat. Fields 158(1-2), 225–400 (2014)
8. 8.
Borodin A., Corwin I., Ferrari P.: Free energy fluctuations for directed polymers in random media in 1 + 1 dimension. Commun. Pure Appl. Math. 67(7), 1129–1214 (2014)
9. 9.
Corwin I.: The Karder–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 01(01), 1130001 (2012)
10. 10.
Forster D., Nelson D.R., Stephen M.J.: Large-distance and long-time properties of a randomly stirred fluid. Phys. Rev. A 16(2), 732–749 (1977)
11. 11.
Gärtner J.: Convergence towards Burgers equation and propagation of chaos for weakly asymmetric exclusion processes. Stoch. Process. Appl. 27, 233–260 (1987)
12. 12.
Gonçalves, P., Jara, M., Sethuraman, S.: A stochastic Burgers equation from a class of microscopic interactions. Ann. Probab. 43(1), 286–338 (2015)Google Scholar
13. 13.
Hairer M.: Solving the KPZ equation. Ann. Math. 178(2), 559–664 (2013)
14. 14.
Johansson K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209(2), 437–476 (2000)
15. 15.
Johansson K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242(1–2), 277–329 (2003)
16. 16.
Kardar M., Parisi G., Zhang Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889–892 (1986)
17. 17.
Kipnis C., Landim C.: Scaling Limits of Interacting Particle Systems, vol. 320. Springer, New York (1999)
18. 18.
Krug J., Meakin P., Halpin-Healy T.: Amplitude universality for driven interfaces and directed polymers in random media. Phys. Rev. A 45(2), 638 (1992)
19. 19.
Liggett T.M.: Interacting Particle Systems. Springer, New York (2005)
20. 20.
Mueller C.: On the support of solutions to the heat equation with noise. Stochastics 37(4), 225–245 (1991)
21. 21.
Prähofer M., Spohn H.: Scale invariance of the png droplet and the airy process. J. Stat. Phys. 108(5–6), 1071–1106 (2002)
22. 22.
Quastel, J.: Introduction to KPZ. http://math.arizona.edu/~mathphys/school_2012/IntroKPZ-Arizona.pdf (2012) (unpublished)
23. 23.
Sasamoto T., Spohn H.: Exact height distributions for the KPZ equation with narrow wedge initial condition. Nucl. Phys. B 834(3), 523–542 (2010)
24. 24.
Spohn, H.: KPZ scaling theory and the semi-discrete directed polymer model. In: Random Matrix Theory, Interacting Particle Systems and Integrable Systems, vol. 65. Cambridge University Press, Cambridge (2014)Google Scholar
25. 25.
Tracy C.A., Widom H.: A Fredholm determinant representation in ASEP. J. Stat. Phys. 132(2), 291–300 (2008)
26. 26.
Tracy C.A., Widom H.: Integral formulas for the asymmetric simple exclusion process. Commun. Math. Phys. 279(3), 815–844 (2008)
27. 27.
Tracy C.A., Widom H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290(1), 129–154 (2009)
28. 28.
Tracy C.A., Widom H.: Erratum to: Integral formulas for the asymmetric simple exclusion process. Commun. Math. Phys. 304(3), 875–878 (2011)
29. 29.
Walsh, J.B.: An introduction to stochastic partial differential equations. In: École d’Été de Probabilités de Saint Flour XIV—1984. Lecture Notes in Mathematics, vol. 1180, pp. 265–439. Springer, Berlin (1986)Google Scholar