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Empirical evidence shows that the rate of irregular usage of English verbsexhibits discontinuity as a function of their frequency: the most frequentverbs tend to be totally irregular. We aim to qualitatively understand theorigin of this feature by studying simple agent--based models of languagedynamics, where each agent adopts an inflectional state for a verb and maychange it upon interaction with other agents. At the same time, agents arereplaced at some rate by new agents adopting the regular form. In models withonly two inflectional states (regular and irregular), we observe that eitherall verbs regularize irrespective of their frequency, or a continuoustransition occurs between a low frequency state where the lemma becomes fullyregular, and a high frequency one where both forms coexist. Introducing a third(mixed) state, wherein agents may use either form, we find that a third,qualitatively different behavior may emerge, namely, a discontinuous transitionin frequency. We introduce and solve analytically a very general class ofthree--state models that allows us to fully understand these behaviors in aunified framework. Realistic sets of interaction rules, including thewell-known Naming Game (NG) model, result in a discontinuous transition, inagreement with recent empirical findings. We also point out that thedistinction between speaker and hearer in the interaction has no effect on thecollective behavior. The results for the general three--state model, althoughdiscussed in terms of language dynamics, are widely applicable.

General three-state model with biased population replacement: Analytical solution and application to language dynamics