The rate function

A closer analysis of (7-8) reveals a number of surprises. First, the $a$-dependence can be removed by Brownian scaling. This gives (recall that $\kappa_a \equiv 2\pi$ for $d=2$)

$I_d^{\kappa_a}(b) = \frac{1}{2\kappa_a^{2/d}} \chi_d(b/\kappa_a),$ (10)

where $\chi_d \colon (0,\infty) \to [0,\infty)$ is given by

$\chi_d(u) = \inf \{ \Vert\nabla \psi \Vert _2^2 \colon ~\psi \in H^1(\mathbb{R... ...si\Vert _2 = 1, ~\textstyle\int_{\mathbb{R}^d} (1-e^{-\psi^2}) \leqslant u \}.$ (11)

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

In this figure $\chi_d$ has an infinite slope at $u=1$ when $d \geqslant 3$, showing that the connection with the central limit theorem is anomalous. Moreover, $\chi_d$ has a non-analyticity at $u=u_d \in (0,1)$ when $d \geqslant 5$, playing the role of a critical threshold.

It turns out that the variational problem in (11) has a minimiser for all $u \in (0,1)$ when $2 \leqslant d \leqslant 4$, but only for $u \in (0,u_d]$ when $d \geqslant 5$. The critical threshold $u_d$ 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 upward direction, i.e., events $\{\vert W^a(t)\vert \geqslant bt\}$ with $b \geqslant 1$. These are expected to behave completely differently.