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.
George, Terrence,
"Cluster integrable systems and statistical mechanics"
(2020).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/bk3a-ps57
