Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 215 / 2 / c

Take $d=1$ and start from the lower endpoint. The stationary distribution of this birth-death chain satisfies detailed balance withso it is concentrated within $O(1)$ of the upper endpoint $n$. Before reaching that region the walk has drift $1/6$. The weak law of large numbers and exponential concentration therefore imply that its hitting time of $n-O(1)$ isAt time $(6-\epsilon )n$ the chain is still macroscopically below the stationary region with probability tending to one, so its total-variation distance tends to one. Under the monotone coupling from part (b), by time $(6+\epsilon )n$ the extremal copies have coalesced with probability tending to one, so the distance tends to zero. Therefore the family has