Random maximal independent sets and the unfriendly theater seating arrangement problem

People arrive one at a time to a theater consisting of m rows of length n . Being unfriendly they choose seats at random so that no one is in front of them, behind them or to either side. What is the expected number of people in the theater when it becomes full, i.e., it cannot accommodate any more unfriendly people? This is equivalent to the random process of generating a maximal independent set of an m × n grid by randomly choosing a node, removing it and its neighbors, and repeating until there are no nodes remaining. The case of m = 1 was posed by Freedman and Shepp [D. Freedman, L. Shepp, An unfriendly seating arrangement (problem 62-3), SIAM Rev. 4 (2) (1962) 150] and solved independently by Friedman, Rothman and MacKenzie [H.D. Friedman, D. Rothman, Solution to: An unfriendly seating arrangement (problem 62-3), SIAM Rev. 6 (2) (1964) 180–182; J.K. MacKenzie, Sequential filling of a line by intervals placed at random and its application to linear adsorption, J. Chem. Phys. 37 (4) (1962) 723–728] by proving the asymptotic limit 1 2 − 1 2 e 2 . In this paper we solve the case m = 2 and prove the asymptotic limit 1 2 − 1 4 e . In addition, we consider the more general case of m × n grids, m ≥ 1 , and prove the existence of asymptotic limits in this general setting. We also make several conjectures based upon Monte Carlo simulations.