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The ensemble of regular low-definition parity-check (LDPC) codes is considered. Using concentration results on the weight distribution, lower bounds on the error rate of a random code in the ensemble are derived. These bounds hold with some confidence level. Combining these results with known lower bounds on the error exponent, confidence intervals on the error exponent, under maximum-likelihood (ML) decoding, are obtained. Over a large range of channel parameter and transmission rate values, when the graph connectivity is sufficiently large, the upper bound of the interval approaches the lower bound, and the probability that the error exponent is within the interval can be arbitrarily close to one. In fact, in this case the true error exponent approaches the maximum between the random coding and the expurgated random coding exponents, with probability that approaches one.