Jamming and Tiling in Fragmentation of Rectangles
E. Ben-Naim and P.L. Krapivsky
We investigate a stochastic process where a rectangle breaks into
smaller rectangles through a series of horizontal and vertical
fragmentation events. We focus on the case where both the vertical
size and the horizontal size of a rectangle are discrete variables.
Because of this constraint, the system reaches a jammed state where
all rectangles are sticks, that is, rectangles with minimal width.
Sticks are frozen as they can not break any further. The average
number of sticks in the jammed state, $S$, grows as $S\simeq
A/\sqrt{2\pi\ln A}$ with rectangle area $A$ in the large-area limit,
and remarkably, this behavior is independent of the aspect ratio. The
distribution of stick length has a power-law tail, and further, its
moments are characterized by a nonlinear spectrum of scaling
exponents. We also study an asymmetric breakage process where
vertical and horizontal fragmentation events are realized with
different probabilities. In this case, there is a phase transition
between a weakly asymmetric phase where the length distribution is
independent of system size, and a strongly asymmetric phase where this
distribution depends on system size.
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