Convergence to equilibrium for density dependent Markov jump processes

We investigate the convergence to (quasi-)equilibrium of a density dependent Markov chain in Zd, whose drift satisfies a system of ordinary differential equations having an attractive fixed point. For a sequence of such processes XN, indexed by a size parameter N, the time taken until the distribution of XN, started in some given state, approaches its (quasi-)equilibrium distribution πN typically increases with N. To first order, it corresponds to the time tN at which the solution to the drift equations reaches a distance of N from their fixed point. However, the length of the time interval over which the total variation distance between L(XN(t)) and its (quasi-)equilibrium distribution πN changes from being close to 1 to being close to zero is asymptotically of smaller order than tN. In this sense, the chains exhibit ‘cutoff’, and we are able to prove that the cutoff window is of (optimal) constant size.