  In a disordered mesoscopic system, the typical spacing between the peaks and
the valleys of the conductance as a function of Fermi energy $E_F$ is called
the conductance energy correlation range $E_c$. Under the ergodic hypothesis,
the latter is determined by the half-width of the ensemble averaged conductance
correlation function: $F= < \delta g(E_F) \delta g(E_F + \Delta E) >$. In
ordinary diffusive metals, $E_c\sim D/L^2$, where $D$ is the diffusion constant
and $L$ is the linear dimension of the phase-coherent sample. However, near a
quantum phase transition driven by the location of the Fermi energy $E_F$, the
above picture breaks down. As an example of the latter, we study, for the first
time, the conductance correlations near the integer quantum Hall transitions of
which $E_F$ is a critical coupling constant. We point out that the behavior of
$F$ is determined by the interplay between the static and the dynamic
properties of the critical phenomena.
