Contents
** Next:** The expected size
**Up:** A probabilistic model
** Previous:** A random walk

Let us explore the consequences of the plausible assumption . By (6.3), we
expect that , where the **O** constant is uniform in
. Thus the region , being the product of two intervals of lengths **1** and
and disks of radius , has volume . Since the points of are distributed with density
, and assuming is in general
position, there are points of
in . If is sufficiently large, it is thus possible to
push through the argument of .2 for **d = 4** and the birthday paradox argument of
.3 for **d = 6** to conclude that the orbit of should be finite.

More precisely, , so
, where

If is large, then the largest terms in are the **d-1** terms
involving the conjugate ; their product is about . If the
conjugates on the unit circle are nicely distributed, then we would expect the product of
terms involving them not to be too small and so ``typically'' and from (6.4). Values of which
are untypically small will lead to large values of .

Contents
** Next:** The expected size
**Up:** A probabilistic model
** Previous:** A random walk