Simulating Simple Random Walks With a Deck of Cards
Raphael Alves, Sabrina Est\'acio, Susana Fr\'ometa, Milton Jara,, Rodrigo Marinho, Luiz F. S. Marques, Jo\~ao V. A. Pimenta

TL;DR
This paper analyzes how many cards are needed in a deck to simulate a symmetric simple random walk with minimal bias, showing convergence to a Gaussian profile and hypergeometric to multinomial distribution transition.
Contribution
It provides a quantitative analysis of deck size requirements for accurate simulation of random walks using cards, including convergence results and distribution approximations.
Findings
Minimum deck size for desired accuracy is characterized.
Total variation distance converges to a Gaussian profile as deck size grows.
Hypergeometric distribution converges to multinomial distribution with increasing deck size.
Abstract
When we want to simulate the realization of a symmetric simple random walk on , we use -side fair dice to decide to which neighbor it jumps at each step if or we simply use a fair coin when . Assume that instead of using a dice or a coin we want to do a simulation using a well shuffled deck with cards of each of the suits. In the first step the probability of jumping to each neighbor is , 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 that a deck must contain so that the total variation distance between the law of a -step simulation and the law of a -step realization of the random walk is smaller…
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Taxonomy
TopicsEvacuation and Crowd Dynamics
