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Existence of the upper critical dimension of the Kardar–Parisi–Zhang equation

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  • Katzav, Eytan
  • Schwartz, Moshe

Abstract

The controversy whether or not the Kardar–Parisi–Zhang (KPZ) equation has an upper critical dimension (UCD) is going on for quite a long time. Some approximate integral equations for the two-point function served as an indication for the existence of a UCD, by obtaining a dimension, above which the equation does not have a strong coupling solution. A surprising aspect of these studies, however, is that various authors who considered the same equation produced large variations in the UCD. This caused some doubts concerning the existence of a UCD. Here we revisit these calculations, describe the reason for such large variations in the results of identical calculations, show by a large-d asymptotic expansion that indeed there exists a UCD and then obtain it numerically by properly defining the integrals involved. Since many difficult problems in condensed matter physics of non-linear nature are handled with mode-coupling and self-consistent theories, this work might also contribute to other researchers working on a large class of different problems that might run into the same inconsistencies.

Suggested Citation

  • Katzav, Eytan & Schwartz, Moshe, 2002. "Existence of the upper critical dimension of the Kardar–Parisi–Zhang equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 309(1), pages 69-78.
    • Handle: RePEc:eee:phsmap:v:309:y:2002:i:1:p:69-78
    DOI: 10.1016/S0378-4371(02)00553-8
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    Keywords

    KPZ equation; Upper critical dimension;

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