The most fun paper in this field (in my opinion) is by Last and Thouless in 1971. In this era of Higgs Boson-sized experimental budget, it is a study in elegance. They punched carefully randomized holes in carbon paper (2D conducting sheet) and measured the resistivity as a function of hole density. The experiment trumped the theory at the time and the apparatus (for once) actually cost less than the pencil/paper theory... --R Physical Review Letters, Vol 27, Number 25, 20 December 1971, pp1719-1722. -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Allan Wechsler Sent: Monday, February 10, 2014 10:16 PM To: math-fun Subject: Re: [math-fun] Connected components in random Cartesian mazes It is percolation theory; there is an enormous literature on it. Start with the Wikipedia article. I hope you have a spare lifetime to spend. On Mon, Feb 10, 2014 at 10:00 PM, Thane Plambeck <tplambeck@gmail.com>wrote:
Draw a maze on a two-dimensional n x n grid by erecting a wall between each pair of adjacent (ie, distance one) lattice points independently with probability p.
Eyeballing these things in Mathematica, it looks to me like if p < 1/2, there tends to be one "large component" that connects almost all the cells that are not walled off into "locally small" (say 1x1 or 1x2) walled gardens.
I'm sure I can't be the first to have considered something like this.
I'd welcome information about prior work.
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
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