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Random matrices over the integers, finite fields, and p-adic integers have been studied since the late 1980s as natural models for random groups appearing in number theory and combinatorics. I will discuss some exact results relating them to Hall-Littlewood processes, explain their parallels with classical real/complex random matrix theory, and give probabilistic results on local limits which are the most difficult application of these tools. The latter results find a new limit process, the reflecting Poisson sea, which is a discrete-space local interacting particle system and appears as a universal limit of Hall-Littlewood processes.