Simulating Simple Random Walks With a Deck of Cards

When we want to simulate the realization of a symmetric simple random walk on $\mathbb Z^d$, we use $(2d)$-side fair dice to decide to which neighbor it jumps at each step if $d\geq 2$ or we simply use a fair coin when $d=1$. Assume that instead of using a dice or a coin we want to do a simulation using a well shuffled deck with $K$ cards of each of the $2d$ suits. In the first step the probability of jumping to each neighbor is $(2d)^{-1}$, but from the second step it becomes biased. Of course if we continue performing this simulation, the total variation distance between its law and the law of the random walk will increase until all cards are used. In this paper we investigate the minimum number of cards $N=2d K$ that a deck must contain so that the total variation distance between the law of a $n$-step simulation and the law of a $n$-step realization of the random walk is smaller than a chosen threshold $\varepsilon \in (0,1)$. More generally, we prove that when $N=cn$ this distance converges, as $n \to \infty$, to a Gaussian profile which depends on $c\geq 2d$. Furthermore, our analysis shows that this Gaussian profile vanishes as $c \to \infty$, proving the convergence of a multivariate hypergeometric distribution to a multinomial distribution in total variation.