# Jamming and Tiling in Fragmentation of Rectangles

## TL;DR

This paper studies a stochastic fragmentation process of rectangles with discrete dimensions, revealing a jammed state with sticks, a size-independent growth of sticks, and phase transitions in asymmetric cases.

## Contribution

It introduces a model of rectangle fragmentation with discrete sizes, analyzes the jammed state, and uncovers phase transitions based on asymmetry in breakage probabilities.

## Key findings

- Average number of sticks grows as $A/\sqrt{2\pi\ln A}$ in large areas.
- Stick length distribution has a power-law tail with nonlinear scaling exponents.
- Phase transition occurs between size-independent and size-dependent distributions in asymmetric breakage.

## Abstract

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.