In this contribution, we provide a theoretical study of two hypothesis tests allowing to detect the presence of an unknown transmitter using several sensors. Both tests are based on the analysis of the eigenvalues of the sampled covariance matrix of the received signal. The Generalized Likelihood Ratio Test (GLRT) derived in  is analyzed under the assumption that both the number K of sensors and the length N of the observation window tend to infinity at the same rate: K/N → c ∈ (0, 1). The GLRT is compared with a test based on the condition number used which is used in cognitive radio applications. Using results of random matrix theory for spiked models and tools of Large Deviations, we provide the error exponent curve associated with both test and prove that the GLRT outperforms the test based on the condition number.