Exact Analysis of Level-Crossing Statistics for (d+1)-Dimensional Fluctuating Surfaces

الأصل · EN

We carry out an exact analysis of the average frequency ναₓᵢ+ in the direction xᵢ of positive-slope crossing of a given level α such that, h(x,t)-h=α, of growing surfaces in spatial dimension d. Here, h(x,t) is the surface height at time t, and h is its mean value. We analyze the problem when the surface growth dynamics is governed by the Kardar-Parisi-Zhang (KPZ) equation without surface tension, in the time regime prior to appearance of cusp singularities (sharp valleys), as well as in the random deposition (RD) model. The total number N+ of such level-crossings with positive slope in all the directions is then shown to scale with time as tᵈ/² for both the KPZ equation and the RD model.