Jamming and tiling in fragmentation of rectangles

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 ≃ A/ √ 2π 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.

On multidimensional record patterns

“…is nothing but the length of the stick adjacent to the origin in the symmetric fragmentation process considered in [21,22].…”

“…Their statistics therefore involve some universal combinatorics, generalizing that involved in the case of usual records. The present work was originally inspired by recent studies on the fragmentation of rectangles drawn on the square lattice [21,22], where an initial rectangle of size L × M may break either vertically or horizontally into two smaller rectangles. In a symmetric variant of the model it breaks simultaneously in both directions, giving birth to four rectangles.…”

“…In a symmetric variant of the model it breaks simultaneously in both directions, giving birth to four rectangles. The process is repeated until it stops when the entire system is filled with frozen rectangles, called sticks, of size K × 1 or 1 × K. Several quantities of interest have been studied in [21,22], including the mean number of sticks and the distribution of the stick length K. The outcomes provide interesting examples of the non-trivial scaling laws which are observed in fragmentation processes (see e.g. [23,24,25,26,27,28,29,30,31], and [32] for a review).…”

“…In section 2 we present a selfcontained reminder on the statistics of usual records and describe a backward recursive construction of those records for finite signals of length L. The multidimensional record patterns obtained by extending the latter recursive construction to higher-dimensional lattices are investigated in later sections. Section 3 is devoted to a detailed analysis of the two-dimensional case, which is related to the fragmentation process studied in [21,22]. We obtain exact combinatorial results on the statistics of the number N of records on finite rectangular samples of size L × M , as well as asymptotic results for large samples.…”

Non-extensive super-cluster states in aggregation with fragmentation

“…4b). The nonextensive growth has been detected in a few aggregationfragmentation processes with standard spontaneous fragmentation [66,67], pure aggregation [68] and pure fragmentation [69]. Neither monomers nor fluctuations play any special role there.…”