The cutoff phenomenon for Ehrenfest chains

We consider families of Ehrenfest chains and provide a simple criterion on the L^p$L^p$-cutoff and the L^p$L^p$-precutoff with specified initial states for 1≤p<∞$1\le p<\infty$. For the family with an L^p$L^p$-cutoff, a cutoff time is described and a possible window is given. For the family without an L^p$L^p$-precutoff, the exact order of the L^p$L^p$-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.