Renewal and memory properties in the random growth of surfaces Page: 4
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The authors of Ref.  studied the variable y, whose variance w(L, t), in accordance with the literature, was shown
to obey the anomalous prescription
w(t) o t,
with 3 < 0.5. Actually, for L - oc the power index 3 is expected to fit the Kardar-Parisi-Zhang (KPZ) prediction 
Q =1/3. To speed up the numerical calculation we limit our calculations to the case of L =1000, and consequently to
the value 3 = 0.28. The authors of Ref.  argued that the y-process can be reproduced with remarkable accuracy
by adopting the subordination approach [5, 6, 7, 8, 10]. It is important to stress that the variance w(t) is a Gibbs
property, obtained by making a statistical average on all the columns of the sample. The single trajectory y(t) is
derived from the stochastic trajectory of y(n), which is driven in the natural time scale by the ordinary Langevin
-Ty(n) + f(n),
and assuming that the transition from the natural time scale n to the t-time scale is realized with the prescription
t(n + 1) - t(n) = T(n), where T(n) is randomly derived from a distribution density with power law ,u < 2. This is
an approach well distinct from the ordinary memory approach [2, 3, 4], which is based on a generalized Langevin
equation, whose memory kernel is a conventional equilibrium correlation function. It is also important to point out
that there exists a connection between p and 3 stemming from It is worth remarking that all this is also a clear
evidence of the fact that the relationship
P = 2 - i,
which was derived by the authors of Ref.  from the renewal theory, rather than by using the FBM theory, as done
by earlier investigators .
The authors of Ref.  established a very good agreement between theory and numerical simulation by adopting
this subordination perspective. In this section we confirm the arguments of Ref.  by means of an aging experiment,
1000 10000 100000
FIG. 3: Probability distribution of waiting times of y(t) are shown with (+) when there is no aging. The straight solid line
is 20 x-1.72 Probability distribution of aged waiting times are dashed curve for experimental and solid curve for theoretical
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Cakir, Rasit; Grigolini, Paolo & Ignaccolo, Massimiliano. Renewal and memory properties in the random growth of surfaces, article, February 4, 2008; (digital.library.unt.edu/ark:/67531/metadc132977/m1/4/: accessed January 16, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.