This thesis investigates a generalisation of the coupon collector’s problem, and its implications for cover times of random walks. The generalisation allows one to adjust the rate at which coupons are drawn, and therefore dynamically slow down or speed up coupon collection. The motivation for studying a dynamic coupon collector is to probe the robustness of coupon collecting statistics. An old result of Erdős and Rényi established that the coupon collection time of the original model follows Gumbel statistics with a mean collection time scaling as N ln N , for N coupons. By solving our generalised model exactly, we show that the Gumbel distribution is but one of a family of distributions, following the statistics of sums of weighted exponentials, and that the mean collection time is, generally, a power-law in N with an adjustable exponent.
Building on the work of Aldous and Belius, we transfer the results of our generalised coupon collector to transient random walks on regular lattices. The cover times, closely analogous to coupon collection times, essentially follow the same statistics, as confirmed by simulations. Curiously, we identify one anomalous result for an accelerated cover time in three dimensions, with possible connections to random matrix theory.
We also closely investigate one-dimensional cover, since it can be treated analytically in detail via the diffusion equation. Thus we consider covering a one-dimensional interval with Poissonian resetting. We derive the position at which cover completes, and describe the effect of resetting.
Finally, we make initial steps towards a field-theory of cover processes. Ultimately, these powerful methods will, we hope, provide additional insight into cover processes that are either dynamically adjusted or reset.