Space-time dispersion of graphene conductivity

We present an analytic calculation of the conductivity of pure graphene as a function of frequency, wave-vector k, and temperature for the range where the energies related to all these parameters are small in comparison with the band parameter, but much larger than the collision rate. The simple asymptotic expressions are given in various limiting cases. For instance, the conductivity for high frequencies turns out to be independent of the band structure parameters. Our results are also used to explain the known dependence of the graphite conductivity on temperature and pressure.