We investigate the equilibrium dynamics of our recently proposed toy model of dense fluid in the infinite damping limit. Contrary to naive expectation, the correlators involving the velocity-like variables do not quickly relax away. Instead, after a very fast transient relaxation, they exhibit rather slow relaxations due to the coupling to the density-like variable. Hence, the so-called "hopping" processes are not suppressed even in the large damping limit. These hopping processes can only be controlled by tuning the parameter delta, which is the ratio of the numbers of the components of the velocity-like and the density-like variables in the model. We analytically prove that there must exist an ergodic-to-nonergodic phase transition for delta such that 0 < delta < 1. The slow dynamics and the dynamic transition in the model are distinct from those in the idealized mode coupling theory.
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