  We study the thermal distribution of intervalley phonons in a graphene sheet.
These phonons have two components with the same frequency. The degeneracy of
the two modes is preserved by weak electron-phonon coupling. A sufficiently
strong electron-phonon coupling, however, can result in a splitting into an
optical and an acoustic phonon branch, which creates a fluctuating gap in the
electronic spectrum. We describe these effects by treating the phonon
distribution within a saddle-point approximation. Fluctuations around the
saddle point indicate a Berezinskii-Kosterlitz-Thouless transition of the
acoustic branch. This transition might be observable in the polarization of
Raman scattered light.
