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In this paper we study the rate of convergence of the iterates of \iid randompiecewise constant monotone maps to the time-$1$ transport map for the processof coalescing Brownian motions. We prove that the rate of convergence is givenby a power law. The time-1 map for the coalescing Brownian motions can beviewed as a fixed point for a natural renormalization transformation acting inthe space of probability laws for random piecewise constant monotone maps. Ourresult implies that this fixed point is exponentially stable.

On asymptotic behavior of iterates of piecewise constant monotone maps