We consider families of Ehrenfest chains and provide a simple criterion on the ~(Lp)-cutoff and the ~(Lp)-precutoff with specified initial states for 1≤p<∞. For the family with an ~(Lp)-cutoff, a cutoff time is described and a possible window is given. For the family without an ~(Lp)-precutoff, the exact order of the ~(Lp)-mixing time is determined. The result is consistent with the well-known conjecture on cutoffs of Markov chains proposed by Peres in 2004, which says that a cutoff exists if and only if the multiplication of the spectral gap and the mixing time tends to infinity.
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