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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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IV.25 Probabilistic Models of Critical Phenomena

Gordon Slade

1 Critical Phenomena

1.1 Examples

A population can explode if its birth rate exceeds its death rate, but otherwise it becomes extinct. The nature of the population’s evolution depends critically on which way the balance tips between adding new members and losing old ones.

A porous rock with randomly arranged microscopic pores has water spilled on top. If there are few pores, the water will not percolate through the rock, but if there are many pores, it will. Surprisingly, there is a critical degree of porosity that exactly separates these behaviors. If the rock’s porosity is below the critical value, then water cannot flow completely through the rock, but if its porosity exceeds ...

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