  We study the superconducting correlations induced in graphene when it is
placed between two superconductors, focusing in particular on the supercurrents
supported by the 2D system. For this purpose we make use of a formalism placing
the emphasis on the many-body aspects of the problem, with the aim of
investigating the dependence of the critical currents on relevant variables
like the distance L between the superconducting contacts, the temperature, and
the doping level. Thus we show that, despite the vanishing density of states at
the Fermi level in undoped graphene, supercurrents may exist at zero
temperature with a natural 1/L^3 dependence at large L. When temperature
effects are taken into account, the supercurrents are further suppressed beyond
the thermal length L_T (~ v_F / k_B T, in terms of the Fermi velocity v_F of
graphene), entering a regime where the decay is given by a 1/L^5 dependence. On
the other hand, the supercurrents can be enhanced upon doping, as the Fermi
level is shifted by a chemical potential \mu from the charge neutrality point.
This introduces a new crossover length L* ~ v_F / \mu, at which the effects of
the finite charge density start being felt, marking the transition from the
short-distance 1/L^3 behavior to a softer 1/L^2 decay of the supercurrents at
large L. It turns out that the decay of the critical currents is given in
general by a power-law behavior, which can be seen as a consequence of the
perfect scaling of the Dirac theory applied to the low-energy description of
graphene.
