Renewal and memory properties in the random growth of surfaces Page: 3
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The height of the column at a give time t is given by
with o = 0. To produce a renewal process we use the variable y(t) defined by
where < h(t) > denotes the average over all the sample columns. In the earlier work of Ref.  it was argued
that the regressions of the variable y to the origin y 0 is a renewal process. We shall prove this property with a
compelling numerical experiment.
Let us now define the variable that is the signature of the memory properties generated by the BD model. This is
the variable ((t) defined by
- (t) - f(t),
where the symbol ( denotes a time average. We generate a very long, but finite, sequence f(t), and we define the time
1 10 100 1000
FIG. 2: Probability distribution of jumps, (, (+), fitting to dashed line 0.22
fitting to solid line 0.001 exp(-0.001 t). L = 1000
exp(-0.35 t); and waiting times of jumps, (x),
Fig. 2 illustrates the distribution density 2/(T) and 4(), and prove that the latter distribution, as well as the
former, is an exponential,
0() = a exp(- be).
On the basis of these results one would be tempted to conclude that the single column process is Poisson. We
shall prove that it is not so because the fluctuation ((t) has memory, in spite of its exponential distribution. This
memory is a consequence of the process of transverse transport of information, and it can be considered a signature
y(t)- h(t)- < h(t) >,
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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/m1/3/: accessed April 27, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.