A drawback of the Propp–Wilson algorithm introduced in the previous two chapters is the need to reuse old random numbers: Recall that Markov chains are started at times -N1, -N2, … (where N1 < N2 < …) and so on until j is large enough so that starting from time -Nj gives coalescence at time 0. A crucial ingredient in the algorithm is that when the Markov chains start at time -Ni, the same random numbers as in previous runs should be used from time -Ni-1 and onwards. The typical implementation of the algorithm is therefore to store all new random numbers, and to read them again when needed in later runs. This may of course be costly in terms of computer memory, and the worst-case scenario is that one suddenly is forced to abort a simulation when the computer has run out of memory.

Various approaches to coping with this problem have been tried. For instance, some practitioners of the algorithm have circumvented the need for storage of random numbers by certain manipulations of (the seeds of) the random number generator. Such manipulations may, however, lead to all kinds of unexpected and unpleasant problems, and we therefore advise the reader to avoid them.

There have also been various attempts to modify the Propp–Wilson algorithm in such a way that each random number only needs to be used once. For instance, one could modify the algorithm by using new random variables each time that old ones are supposed to be used.