- Title Information
- Title
- Cluster integrable systems and statistical mechanics
- Name:
Personal
- Name Part
- George, Terrence
- Role
- Role Term:
Text
- creator
- Name:
Personal
- Name Part
- Kenyon, Richard
- Role
- Role Term:
Text
- Advisor
- Name:
Personal
- Name Part
- Abramovich, Dan
- Role
- Role Term:
Text
- Reader
- Name:
Personal
- Name Part
- Chan, Melody
- Role
- Role Term:
Text
- Reader
- Name:
Corporate
- Name Part
- Brown University. Department of Mathematics
- Role
- Role Term:
Text
- sponsor
- Origin Information
- Copyright Date
- 2020
- Physical Description
- Extent
- VIII, 162 p.
- digitalOrigin
- born digital
- Note:
thesis
- Thesis (Ph. D.)--Brown University, 2020
- Genre (aat)
- theses
- Abstract
- The work in this thesis concerns three projects at the interface of statistical mechanics
and cluster integrable systems. We describe each of these in the paragraphs to
follow.
First, we compute the group of automorphisms of the dimer integrable systems, proving
a conjecture of Fock and Marshakov. Probabilistically, non-torsion elements of
the group are ways of shuffling the underlying bipartite graph, generalizing dominoshuffling.
Algebro-geometrically, this group is identified with the Picard group of an
algebraic surface associated to the integrable system.
Next, we study the spectral transform of biperiodic resistor networks associated to
the discrete Laplacian. We give a complete classification of networks (modulo a natural
equivalence) in terms of the spectral transform. The space of networks has a
large group of cluster automorphisms arising from the star-triangle transformation.
We show that the spectrum provides action-angle coordinates for the discrete cluster
integrable systems defined by these automorphisms.
Lastly, we study groves, which are spanning forests of a finite region of the triangular
lattice that are in bijection with Laurent monomials that arise in solutions of the
cube recurrence. We introduce a large class of probability measures on groves for
which we can compute exact generating functions for edge probabilities. Using the
machinery of asymptotics of multivariate generating functions, this lets us explicitly
compute arctic curves, generalizing the arctic circle theorem of Petersen and Speyer.
- Subject
- Topic
- Dimer model
- Subject
- Topic
- Integrable systems
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/01132070")
- Topic
- Statistical mechanics
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/01763173")
- Topic
- Cluster algebras
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/01738506")
- Topic
- Spanning trees (Graph theory)
- Language
- Language Term (ISO639-2B)
- English
- Record Information
- Record Content Source (marcorg)
- RPB
- Record Creation Date
(encoding="iso8601")
- 20200720
- Access Condition:
rights statement
(href="http://rightsstatements.org/vocab/InC/1.0/")
- In Copyright
- Access Condition:
restriction on access
- Collection is open for research.
- Identifier:
DOI
- 10.26300/bk3a-ps57
- Type of Resource (primo)
- dissertations