Renewal and memory properties in the random growth of surfaces Page: 6
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FIG. 4: Variance of x is shown with dashed line. The solid line is 0.256 x0.288
namely, we reiterate that the diffusion process has anomalous scaling defined by 3, with P < 1/2. By differentiating
Eq. (15) with respect to time we establish a connection with the time asymptotic properties of #(t). Due to the fact
that 3 < 1/2 we conclude that the correlation function (I(t) must have a negative tail, namely that
lim (t)= - (17)
= - 23. (18)
This property establishes a closer connection with the earlier work of Ref. , where the variable responsible for
memory yields a correlation function with a negative tail. In the case of Ref.  the correlation function is known
theoretically, so that it is possible to move from the correlation function to the variance time evolution using Eq. (15).
In the present case, we do not have available any analytical approach to the equilibrium correlation function of I(t).
Its numerical derivation, as pointed out by Eq. (15), would be equivalent to differentiating twice the variance w(t),
which is numerically a source of big errors. In, fact, the numerical approach to the equilibrium correlation function,
not shown here, yields a negative tail, but the assessment of the correct power requires rich statistics and excessive
Thus, to do double check our conclusion, that the variable ((t) is the source of memory, with no renewal events,
we prefer to accurately examine the physically plausible conjecture made by the authors of Ref.  that the non-
Poisson renewal events may be hidden among the pseudo-Poissonian fluctuations of the variable ((t). According to
this conjecture, either the fluctuations of ((t) above a given threshold R or the time distances between two consecutive
fluctuations of ((t), exceeding the threshold R, may deviate from the exponential distribution. Fig. 5 and Fig. 6
illustrate these two conditions, and prove that in both cases the asymptotic time behavior is exponential.
We therefore conclude that there is no evidence of renewal events in the dynamics of f(t). The stationary assumption
made to deal with the correlation function of ((t) does not conflict with the non-stationary nature of y. This is so
because ((t) and y(t) are the signatures of cooperation and renewal, respectively. Cooperation generates a slow
decaying correlation function, but the diffusion process generated by these fluctuations is renewal, in accordance with
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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/m1/6/: accessed September 22, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.