The replica method originated from statistical mechanics has been successfully applied to analyzing performance of the global maximum-likelihood (GML) MIMO detector in the large-system limit. In this paper, the analysis is extended to the local maximum-likelihood (LML) detectors. A bit error rate (BER) formula for the LML detectors with a fixed neighborhood size is obtained by the replica method and interestingly by the method of Gaussian approximation as well. It is shown that the LML BER is always one of the solutions to the GML BER in any system configuration. Furthermore, the LML BER is the only solution of the GML BER in a broad range of system parameters of practical interest. In the high signal-to-noise ratio regime, both LML and GML detectors achieve the AWGN channel performance when the channel load is up to 1.51 bits/dimension with an equal-energy distribution, and the load can be higher with an unequal-energy distribution. This analytical result is verified by simulation that the sequential likelihood ascent search detector, which is a linear-complexity LML detector, can approach the BER of the NP-hard GML detector predicted by the analysis. This result might be practically useful in large MIMO systems.