The stochastic process under consideration is as follows: the basic object is the finite set

of sites, where

for some positive integer

. The state space of the process is the powerset

, and the initial state is

(

), where

is the state for which every site is occupied

*. *The allowed state transitions are in one-to-one correspondence with the sites. They are of the form

, that is, they just remove

from the set

. Of course, this emptying of site

is the trivial action for states that do not contain

(states for which site

is empty). The stochastics of the process is defined by stipulating that state transitions (steps) occur in a fixed time raster and with equal probability for all transitions. The cardinality

of a state

is referred to as its population size. Obviously the first step reduces the population size from

to

. The smaller the population becomes in a sequence of steps, the smaller the probability that the next step will reduce the population even more: the sites proposed for emptying may be empty already. This stochastic law is simple enough that all pertinent problems can be worked out analytically. The most important property is that the expectation value of the population size after

steps is

. A

* *realization

* *of the stochastic process asks for actually deciding which site should be selected for emptying in all the steps under consideration. As in all modern applications of the Monte Carlo method, this is done by calling a random number generator. If this function is set up to return integer numbers in the range

it can be directly used for this decision. Any realization of the process gives a curve representing the actual population sizes as they decrease with growing step number. The behavior of this curve can be compared with that of the mean value curve as given above. If there is a significant discrepancy between the mean value and the particular realization, we may have to blame the random generator for failing to give all its outputs with equal probability. Such a failure is very unlikely, however, for any reasonably designed random generator, and so we make the test more selective by making it sensitive for statistical dependencies of successive results: for realizing one step we call three times in succession a random number generator that returns numbers in the smaller set

. Let

be the triplet of numbers obtained in this way. The site proposed for emptying, then, is

. This procedure suggests itself if we consider the sites as arranged in a finite cubic lattice and let the random generator output three discrete Cartesian coordinates of the site to be selected. The "name emptying a cube" refers to this situation [1].

Snapshot 2: the gray curves carry essentially the same information as the green and red lines. Whereas the gray curves can always be obtained by the Monte-Carlo method, the red and green curves can only be obtained by analytical methods. For the present simple process, these are in place, however. The blue curve remains always within the area spanned by the 10 gray curves. Thus, there is no reason to blame the sine-floor random generator.

[1] D. Stauffer, "Random Number Generation,"

*Computational Physics* (K. H. Hoffmann and M. Schreiber, eds.), New York: Springer, 1996.