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This paper considers joint active user detection (AUD) and channel estimation (CE) for massive connectivity scenarios with sporadic traffic. The state-of-art method under a Bayesian framework to perform joint AUD and CE in such scenarios is approximate message passing (AMP). However, the existing theoretical analysis of AMP-based joint AUD and CE can only be performed with a given fixed point of the AMP state evolution function, lacking the analysis of AMP phase transition and Bayes-optimality. In this paper, we propose a novel theoretical framework to analyze the performance of the joint AUD and CE problem by adopting the replica method in the Bayes-optimal condition. Specifically, our analysis is based on a general channel model, which reduces to particular channel models in multiple typical MIMO communication scenarios. Our theoretical framework allows ones to measure the optimality and phase transition of AMP-based joint AUD and CE as well as to predict the corresponding performance metrics under our model. To reify our proposed theoretical framework, we analyze two typical scenarios from the massive random access literature, i.e., the isotropic channel scenario and the spatially correlated channel scenario. Accordingly, our performance analysis produces some novel results for both the isotropic Raleigh channel and spatially correlated channel case.