  We examine several characteristics of conformal maps that resemble the
variance of a Gaussian: asymptotic variance, the constant in Makarov's law of
iterated logarithm and the second derivative of the integral means spectrum at
the origin. While these quantities need not be equal in general, they agree for
domains whose boundaries are regular fractals such as Julia sets or limit sets
of quasi-Fuchsian groups. We give a new proof of these dynamical equalities. We
also show that these characteristics have the same universal bounds and prove a
central limit theorem for extremals. Our method is based on analyzing the local
variance of dyadic martingales associated to Bloch functions.
