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## Condensation of electron-hole pairs in a two-layer graphene system: Correlation effects

Condensation of pairs formed by spatially separated electrons and holes in a system of two isolated graphene layers is studied beyond the mean-field approximation. Suppression of the screening of the pairing interaction at large distances, caused by the appearance of the gap, is considered self-consistently. A mutual positive feedback between the appearance of the gap and the enlargement of the interaction leads to a sharp transition to a correlated state with a greatly increased gap above some critical value of the coupling strength. At a coupling strength below the critical value, this correlation effect increases the gap approximately by a factor of 2. The maximal coupling strength achievable in experiments is close to the critical value. This indicates the importance of correlation effects in closely spaced graphene bilayers at weak substrate dielectric screening. Another effect beyond the mean-field approximation considered is the influence of vertex corrections on the pairing, which is shown to be very weak.