The performance of an orthogonal frequency-division multiplexing (OFDM)-based spectrum sharing communication system that assumes random access and no information about the primary user's (PU) channel occupation information is studied. Also, no cooperation, information exchange overhead, or subcarrier allocation mechanism between the secondary users (SUs) is assumed. SUs randomly access the subcarriers of the primary network and collide with the PUs' subcarriers with a certain probability. To maintain the quality of service (QoS) requirement of PUs, the interference that SUs causes onto PUs is controlled by adjusting SUs transmit power below a predefined threshold, termed interference temperature. Herein paper, the average capacity of SU with subcarrier collisions is employed as performance measure to investigate the proposed random access scheme. Bounds and scaling laws of average capacity with respect to the number of SUs', PUs' and available subcarriers are derived, and corroborated by computer simulations.