Chip-Firing Game as Probability Abacus by Characterization of ULD Lattice

Namrata Kaushal, Madhu Tiwari, C. L. Parihar

Abstract


Based on the graph and distribution of chips on its vertices three types of dynamic models are available. Out of these three models, CFGs (Chip-Firing Game) have received extensive attention on account of their  application in many areas of mathematics such as algebra, combinatory, dynamical systems, statistics, algorithms, and computational complexity. In this paper, we use characterization properties of set of configurations called configuration space which are ordered by predecessor relations. We use CFG as a procedure called “Probability abacus” to determine absorbing probability in the form of vector addition language. Furthermore, by using characterization property of ULD lattice we show that termination state of any CFG on absorbing Markova chain with rational transition matrix does not depend on order of firing and at termination state critical loading re-occurred.

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