(10)

where is given by

(11)

Next, the variational problem in (11) displays the dimension dependence shown in Figure 1.

In this figure has an infinite slope at when , showing that the connection with the central limit theorem is anomalous. Moreover, has a non-analyticity at when , playing the role of a

It turns out that the variational problem in (11) has a minimiser for all when , but only for when . The critical threshold is associated with `leakage' in the variational problem. In terms of the optimal strategy behind the moderate deviations, this leakage is associated with a `collapse transition': the path spends parts of its time on two different space scales.

It is not known whether the minimiser is unique when it exists. This seems to be a tough analytic problem. Very little is known about the moderate deviations in the