- Title Information
- Title
- Confluence in Chip-Firing and Related Combinatorial Processes
- Type of Resource (primo)
- dissertations
- Name:
Personal
- Name Part
- Liscio, Patrick Hirsch
- Role
- Role Term:
Text
- creator
- Name:
Personal
- Name Part
- Klivans, Caroline
- Role
- Role Term:
Text
- Advisor
- Name:
Personal
- Name Part
- Guzman, Johnny
- Role
- Role Term:
Text
- Reader
- Name:
Personal
- Name Part
- Chan, Melody
- Role
- Role Term:
Text
- Reader
- Name:
Corporate
- Name Part
- Brown University. Department of Applied Mathematics
- Role
- Role Term:
Text
- sponsor
- Origin Information
- Copyright Date
- 2022
- Physical Description
- Extent
- xxii, 229 p.
- digitalOrigin
- born digital
- Note:
thesis
- Thesis (Ph. D.)--Brown University, 2022
- Genre (aat)
- theses
- Abstract
- Confluence is an important property in many combinatorial processes. A globally confluent process is one in which a fixed initial state always leads to a fixed final state, even if different choices of moves are made throughout the process. In many processes, including chip-firing, global confluence is proven using local confluence: if two moves are available at the same time, they may be performed in either order.
We consider processes that exhibit global confluence without local confluence. We begin with labeled chip-firing, a modified chip-firing process in which chips with numerical labels are fired according to certain rules. Under some conditions, the chips are guaranteed to end the process in sorted order. We provide a new proof that labeled chip-firing from an initial configuration with $2m$ chips at the origin always leads to a sorted final configuration. The proof involves analyzing a poset of firing moves, which contains a diamond of locally confluent moves at the end of the process.
We use similar methods to prove confluence in related processes, including other initial configurations, variations of the line graph, and classical root system types. We also analyze the class of move posets. We show that locally confluent diamonds are generated in move posets on a much larger class of bipartite graphs. We also show that the move poset is isomorphic to the poest of join-irreducible chip configurations. We prove this by introducing edge colorings, called $S$-colorings, of lower locally distributive lattices. We show that $S$-colorings describe moves in many other combinatorial processes, and we use $S$-colorings to provide a new proof of a theorem stating that all shortest paths between flip-connected domino tilings use the same flip moves in a possibly different order.
- Subject
- Topic
- Combinatorics
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/01400410")
- Topic
- Applied mathematics
- Language
- Language Term (ISO639-2B)
- English
- Record Information
- Record Content Source (marcorg)
- RPB
- Record Creation Date
(encoding="iso8601")
- 20220706