Confluence in labeled chip-firing

Lattices in Chip-Firing

“…Note the diamonds that appear at the bottom of the diagrams for n = 10 and n = 20. These are indeed known to exist for any even n [6].…”

“…When n is even, this process is still globally confluent, as the final positions of the n chips must be in sorted order. The structure of the move poset guarantees that a relatively small collection of locally confluent moves at the end of the process guarantee that all inversions between chips will be removed by the end of the process [6].…”

Labeled Chip-firing on Binary Trees

“…In this section, we will consider the labeled chip-firing process with 2 n − 1 chips (labeled from 1 to 2 n − 1) initially placed at the root. Let (i, j) be the jth-to-last fire of node i, we will adapt the relation used by Klivans in [2] and denote the relation (i 1 , j 1 ) > (i 2 , j 2 ) if (i 1 , j 1 ) must occur before (i 2 , j 2 ). The relation may seem counter-intuitive at first, but the intention will be clearer when we study the poset of firing moves.…”

“…It was proved in [1] and [2] that the labeled chip-firing process on an infinite one-dimensional line with an even number of chips initially placed at the origin terminates in a unique configuration regardless of the order in which nodes fire and regardless of the choice of chips made at each node. Moreover, in the unique terminal configuration, the chips are in sorted order.…”

Toppleable Permutations, Excedances and Acyclic Orientations

“…But for n odd, they conjectured based on simulations that the probability of being sorted tends to 1/3 as n → ∞. This conjecture has motivated a lot of work since then [GHMP18,GHMP19,HP19,KL20b,FK21,KL20a].…”