The probability density function tail of the Kardar–Parisi–Zhang equation in the strongly non-linear regime

Abstract

An analytical derivation of the probability density function (PDF) tail describing the strongly correlated interface growth governed by the nonlinear Kardar-Parisi-Zhang equation is provided.The PDF tail exactly coincides with a Tracy-Widom distribution i.e. a PDF tail proportional to2 ), where w 2 is the the width of the interface. The PDF tail is computed by the instanton method in the strongly non-linear regime within the Martin-Siggia-Rose framework using a careful treatment of the non-linear interactions. In addition, the effect of spatial dimensions on the PDF tail scaling is discussed. This gives a novel approach to understand the rightmost PDF tail of the interface width distribution and the analysis suggests that there is no upper critical dimension.

Divergence of shape fluctuation for general distributions in first passage percolation

“…The scaling limits in higher dimensions are controversial issues even in physics and there are some candidates. See [3] and references therein. However, Zhang showed that if τ obeys the Bernoulli distribution, the shape fluctuation diverges [17].…”