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We analyze asynchronous carrier sense multiple access (CSMA) policies for scheduling packet transmissions in multihop wireless networks subject to collisions under primary interference constraints. While the (asymptotic) achievable rate region of CSMA policies for single-hop networks has been well-known, their analysis for general multihop networks has been an open problem due to the complexity of complex interactions among coupled interference constraints. Our work resolves this problem for networks with primary interference constraints by introducing a novel fixed-point formulation that approximates the link service rates of CSMA policies. This formulation allows us to derive an explicit characterization of the achievable rate region of CSMA policies for a limiting regime of large networks with a small sensing period. Our analysis also reveals the rate at which CSMA achievable rate region approaches the asymptotic capacity region of such networks. Moreover, our approach enables the computation of approximate CSMA link transmission attempt probabilities to support any given arrival vector within the achievable rate region. As part of our analysis, we show that both of these approximations become (asymptotically) accurate for large networks with a small sensing period. Our numerical case studies further suggest that these approximations are accurate even for moderately sized networks.