# Renewal and memory properties in the random growth of surfaces Page: 5

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recently proposed to prove the renewal character of BQD processes [15]. According to this procedure, we study first

the recursions to the origin of the fluctuation y(t), which are described by Fig. 3. We see that y(t) remains in the

positive portion of the y-axis for a given amount of time, then y moves to the negative portion of this axis, remains

there for a while, till to re-cross the origin y = 0, and so on. We evaluate the histogram and we find an inverse power

law with ,u = 1.72. According to the prescription of Ref. [15] we evaluate also the aged waiting time distribution

0(exp)(T, ta). This is done as follows. The re-crossing times generates the time series {t}. For each time ti we record

the first time of the sequence at a distance from ti equal to or larger than ti + ta. This time will be tk, with k > i.

We record the time distance T(ts, ta) = tk - (ti + ta). We repeat the procedure for all the times of the sequence {ti},

and use the sequence of these recorded time distances to generate the distribution density o(exp) (T, ta). We have also

to evaluate

(theor) ot dy (t + y)

K(ta)(11)

where K(ta) is a suitable normalization constant and 4(t) is the experimental waiting time distribution corresponding

to ta = 0. The accordance between theoryor ) (T, ta) and /(exp) (7, ta) is judged [15] to be the numerical evidence of the

renewal nature of the process under study.

Fig. 3 confirms the renewal nature of the y-process, thereby providing further support for the theoretical approach

to the stochastic growth of a single column proposed in Ref. [24] and for the adoption of the renewal theory to derive

the crucial relation of Eq. (10). Using Eq. (10) and the numerical results of Fig. 3 for p, we obtain Q = 0.28. This

same power index will be derived from the memory properties discussed in Section IV.

IV. MEMORY

In this section we adopt a single-trajectory perspective, rather than the Gibbs approach of Section III. How can we

reveal the cooperative nature of the process if we limit ourselves to studying the time evolution of only one column?

Apparently, the statistical analysis that led us to Eqs. (1) and (7) would suggest that the anomalous growth property

of the interface as a whole is annihilated by the observation of a single column.

To assess the memory properties created by cooperation, we proceed as follows. We create the diffusing variable

x(t) = (t'). (12)

Then, using the proposal made by Stanley and co-workers years ago [33, 34] we convert the single diffusing trajectory

of Eq. (12) into many diffusing trajectories. This is done as follows. We consider a window of size 1, and we move

it along the sequence. We record the space positions of the random walker at times s and s + 1. This makes it

possible for us to define the s-th random walker that in the time 1, moves from the origin x = 0 to the position

x(s, 1) x= (s + 1) - x(1). Thus we have a set of walkers, each walker denoted by the label s, moving for a time 1 and

covering the distance x(s, 1). We imagine all these walkers to depart from the origin x 0 at the same time. This

means that the set of values x(s, 1) determines a distribution, whose variance w(l) is given by

w(1) (x(s, 1) < x(s, 1)>)21/2 (13)

the averages < ... > being made on all the labels s. It is evident that if the single column reflects the complex behavior

of the whole surface, see Eq. (8), then

lim log(w(l)) =/ plog(l). (14)

l---oo

Fig. 4 shows that this prediction is fulfilled and that, as expected,/ = 0.28.

Under the stationary assumption, it is straightforward to prove (see, for instance, Ref. [35]) that

dt < (2 2)> >0< (t')dt', (15)

where Jg(t') denotes the equilibrium correlation function of the fluctuation ((t). In the time asymptotic limit, which

is the intermediate regime illustrated by both Fig. 3 and Fig. 4, with no upper bound time limit, determined by the

third regime, we getw (t)2 =< x2(t) >Ot20,

(16)

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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/m1/5/: accessed June 29, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.