Fluctuation exponent of the KPZ/stochastic Burgers equation

Authors:
M. Balázs, J. Quastel and T. Seppäläinen

Journal:
J. Amer. Math. Soc. **24** (2011), 683-708

MSC (2010):
Primary 60H15, 82C22; Secondary 35R60, 60K35

Published electronically:
January 19, 2011

MathSciNet review:
2784327

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the stochastic heat equation

**1.**T. Alberts, K. Khanin, and J. Quastel.

The intermediate disorder regime for directed polymers in dimension 1 + 1.*Phys. Rev. Lett.*, 105, 2010.**2.**Jinho Baik, Percy Deift, and Kurt Johansson,*On the distribution of the length of the longest increasing subsequence of random permutations*, J. Amer. Math. Soc.**12**(1999), no. 4, 1119–1178. MR**1682248**, 10.1090/S0894-0347-99-00307-0**3.**Márton Balázs and Timo Seppäläinen,*Exact connections between current fluctuations and the second class particle in a class of deposition models*, J. Stat. Phys.**127**(2007), no. 2, 431–455. MR**2314355**, 10.1007/s10955-007-9291-3**4.**Márton Balázs and Timo Seppäläinen,*Fluctuation bounds for the asymmetric simple exclusion process*, ALEA Lat. Am. J. Probab. Math. Stat.**6**(2009), 1–24. MR**2485877****5.**Albert-László Barabási and H. Eugene Stanley,*Fractal concepts in surface growth*, Cambridge University Press, Cambridge, 1995. MR**1600794****6.**Lorenzo Bertini and Giambattista Giacomin,*Stochastic Burgers and KPZ equations from particle systems*, Comm. Math. Phys.**183**(1997), no. 3, 571–607. MR**1462228**, 10.1007/s002200050044**7.**Sérgio Bezerra, Samy Tindel, and Frederi Viens,*Superdiffusivity for a Brownian polymer in a continuous Gaussian environment*, Ann. Probab.**36**(2008), no. 5, 1642–1675. MR**2440919**, 10.1214/07-AOP363**8.**Patrick Billingsley,*Convergence of probability measures*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0233396****9.**Terence Chan,*Scaling limits of Wick ordered KPZ equation*, Comm. Math. Phys.**209**(2000), no. 3, 671–690. MR**1743612**, 10.1007/PL00020963**10.**Patrik L. Ferrari and Herbert Spohn,*Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process*, Comm. Math. Phys.**265**(2006), no. 1, 1–44. MR**2217295**, 10.1007/s00220-006-1549-0**11.**Dieter Forster, David R. Nelson, and Michael J. Stephen,*Large-distance and long-time properties of a randomly stirred fluid*, Phys. Rev. A (3)**16**(1977), no. 2, 732–749. MR**0459274****12.**Helge Holden, Bernt Øksendal, Jan Ubøe, and Tusheng Zhang,*Stochastic partial differential equations*, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1996. A modeling, white noise functional approach. MR**1408433****13.**Kurt Johansson,*Transversal fluctuations for increasing subsequences on the plane*, Probab. Theory Related Fields**116**(2000), no. 4, 445–456. MR**1757595**, 10.1007/s004400050258**14.**K. Kardar, G. Parisi, and Y.Z. Zhang.

Dynamic scaling of growing interfaces.*Phys. Rev. Lett.*, 56:889-892, 1986.**15.**T. Kriecherbauer and J. Krug.

A pedestrian's view on interacting particle systems, KPZ universality, and random matrices.*J. Phys. A: Math. Theor.*, 43, 2001.**16.**H. Krug and H. Spohn.*Kinetic roughening of growing surfaces*, pages 412-525.

Cambridge Univ. Press., 1991.**17.**C. Licea, C. M. Newman, and M. S. T. Piza,*Superdiffusivity in first-passage percolation*, Probab. Theory Related Fields**106**(1996), no. 4, 559–591. MR**1421992**, 10.1007/s004400050075**18.**Olivier Mejane,*Upper bound of a volume exponent for directed polymers in a random environment*, Ann. Inst. H. Poincaré Probab. Statist.**40**(2004), no. 3, 299–308 (English, with English and French summaries). MR**2060455**, 10.1016/S0246-0203(03)00072-4**19.**Carl Mueller,*On the support of solutions to the heat equation with noise*, Stochastics Stochastics Rep.**37**(1991), no. 4, 225–245. MR**1149348****20.**M. Petermann.*Superdiffusivity of directed polymers in random environment*.

Ph.D. thesis, University of Zürich, 2000.**21.**M. S. T. Piza,*Directed polymers in a random environment: some results on fluctuations*, J. Statist. Phys.**89**(1997), no. 3-4, 581–603. MR**1484057**, 10.1007/BF02765537**22.**Michael Prähofer and Herbert Spohn,*Current fluctuations for the totally asymmetric simple exclusion process*, In and out of equilibrium (Mambucaba, 2000) Progr. Probab., vol. 51, Birkhäuser Boston, Boston, MA, 2002, pp. 185–204. MR**1901953****23.**Giuseppe Da Prato and Jerzy Zabczyk,*Stochastic equations in infinite dimensions*, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. MR**1207136****24.**Jeremy Quastel and Benedek Valko,*𝑡^{1/3} Superdiffusivity of finite-range asymmetric exclusion processes on ℤ*, Comm. Math. Phys.**273**(2007), no. 2, 379–394. MR**2318311**, 10.1007/s00220-007-0242-2**25.**T. Seppäläinen.

Scaling for a one-dimensional directed polymer with boundary conditions.*To appear in Ann. Probab.,*, 2009.`arXiv:0911.2446`**26.**T. Seppäläinen and B. Valkó.

Bounds for scaling exponents for a 1+1 dimensional directed polymer in a Brownian environment.*To appear in Alea,*, 2010.`arXiv:1006.4864`**27.**John B. Walsh,*An introduction to stochastic partial differential equations*, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 265–439. MR**876085**, 10.1007/BFb0074920

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Additional Information

**M. Balázs**

Affiliation:
Department of Stochastics, Budapest University of Technology and Economics, 1 Egry Jozsef u, H ep V 7, Budapest, 1111 Hungary

Email:
balazs@math.bme.hu

**J. Quastel**

Affiliation:
Departments of Mathematics and Statistics, University of Toronto, 40 St. George Street, Room 6290, Toronto, ON M5S 1L2 Canada

Email:
quastel@math.toronto.edu

**T. Seppäläinen**

Affiliation:
Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53706-1388

Email:
seppalai@math.wisc.edu

DOI:
http://dx.doi.org/10.1090/S0894-0347-2011-00692-9

Keywords:
Kardar-Parisi-Zhang equation,
stochastic heat equation,
stochastic Burgers equation,
random growth,
asymmetric exclusion process,
anomalous fluctuations,
directed polymers.

Received by editor(s):
October 16, 2009

Received by editor(s) in revised form:
October 28, 2010

Published electronically:
January 19, 2011

Additional Notes:
The first author is supported by the Hungarian Scientific Research Fund (OTKA) grants K-60708 and F-67729, by the Bolyai Scholarship of the Hungarian Academy of Sciences, and by the Morgan Stanley Mathematical Modeling Center.

The second author is supported by the Natural Sciences and Engineering Research Council of Canada.

The third author is supported by the National Science Foundation grant DMS-0701091 and by the Wisconsin Alumni Research Foundation.

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.