Renewal and memory properties in the random growth of surfaces Page: 1
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Renewal and memory properties in the random growth of surfaces
R.Cakir, P. Grigolini,* and M. Ignaccolo
Center for Nonlinear Science, University of North Texas,
P.O. Box 31147, Denton, Texas 76203-1427, USA
(Dated: February 4, 2008)
We use the model of ballistic deposition as a simple way to establish cooperation among the
columns of a growing surface, the single individual of the same society. We show that cooperation
generates memory properties and at same time non-Poisson renewal events. The variable generating
memory can be regarded as the velocity of a particle driven by a bath with the same time scale,
and the variable generating renewal processes is the corresponding diffusional coordinate.
PACS numbers: 05.40.-a, 05.65.+b, 89.75.Da
The Science of Complexity  is an emerging field of research, which is attracting an increasing number of investiga-
tors. There is a general agreement that the deviation from the exponential behavior is a sign of complexity. However,
on the origin of this deviation is there are different proposals. There are authors emphasizing the memory properties
[2, 3, 4] and other [5, 6, 7, 8, 9, 10] stressing the non-Poisson renewal properties of complex systems.
As an attractive example of non-Poisson renewal processes we refer to the field of single-molecule spectroscopy
 and blinking quantum dots [12, 13, 14] . There is an increasing evidence that phenomena such as intermittent
fluorescence yield histograms of the "on" and "off' sojourn time distribution that depart dramatically from the
exponential condition. At the same time, a careful statistical analysis proves that these are renewal processes [14, 15].
Renewal theory  describes, for example, successive replacements of light bulbs: when a bulb fails it is immediately
replaced by a new one, which works with no memory whatsoever of the time duration of the earlier light bulb. Thus,
renewal processes are characterized by the occurrence of events that reset to zero the system's memory. In which
sense, therefore, a non-Poisson renewal process is a manifestation of organized behavior, which seems to imply [2, 3, 4]
It is tempting to express the non-exponential properties of a complex system in terms of ordinary Poisson processes
with a fluctuating Poisson parameter [17, 18, 19]. Using this perspective, it is possible to prove  that complex
networks emerge from fluctuating random graphs. However, it is worth remarking that slow fluctuations  reduce the
occurrence of renewal events and generate memory properties compatible with the adoption of stationary correlation
Another example of non-Poisson renewal process is given by the fields of seismic fluctuations , where, however,
there is no general agreement on the renewal character of the process. The authors of Ref.  argue that the
distribution of time distances between two consecutive main shocks corresponds to the properties of a non-Poisson
renewal process. Is this conclusion reliable, or does it conflict with the notion of main shocks being complex processes,
are , to some extent, predicable? Another example, of greater interest for this paper, is given by the random growth
of surfaces. The authors of a recent work  have used the concepts of non-Poisson renewal theory to describe and
derive the same properties that other authors [25, 26, 27] derive from the adoption of the Fractional Brownian Motion
(FBM) perspective . On one side, we have renewal, with jumps resetting to zero the system's memory, and on the
other we have fluctuations with infinite memory. Which is the best signature of complexity?
The purpose of this paper is to prove that both aspects are present in the processes of random growth of interfaces,
which are a natural realization of of the Renewal Cooperation (RC) perspective proposed in the earlier work of Ref.
The outline of the paper is as follows. In Section II we shall give detail on the model used for the illustration of
our approach to complexity, and we shall define the variable ((t) which is the signature of memory, and the variable
y, whose origin regression is a non-Poisson renewal process. In Section III we establish the renewal character of the
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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/m1/1/: accessed February 27, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.