44 research outputs found
Dimers and cluster integrable systems
We show that the dimer model on a bipartite graph on a torus gives rise to a
quantum integrable system of special type - a cluster integrable system. The
phase space of the classical system contains, as an open dense subset, the
moduli space of line bundles with connections on the graph. The sum of
Hamiltonians is essentially the partition function of the dimer model. Any
graph on a torus gives rise to a bipartite graph on the torus. We show that the
phase space of the latter has a Lagrangian subvariety. We identify it with the
space parametrizing resistor networks on the original graph.We construct
several discrete quantum integrable systems.Comment: This is an updated version, 75 pages, which will appear in Ann. Sci.
EN
Planar graphs : a historical perspective.
The field of graph theory has been indubitably influenced by the study of planar graphs. This thesis, consisting of five chapters, is a historical account of the origins and development of concepts pertaining to planar graphs and their applications. The first chapter serves as an introduction to the history of graph theory, including early studies of graph theory tools such as paths, circuits, and trees. The second chapter pertains to the relationship between polyhedra and planar graphs, specifically the result of Euler concerning the number of vertices, edges, and faces of a polyhedron. Counterexamples and generalizations of Euler\u27s formula are also discussed. Chapter III describes the background in recreational mathematics of the graphs of K5 and K3,3 and their importance to the first characterization of planar graphs by Kuratowski. Further characterizations of planar graphs by Whitney, Wagner, and MacLane are also addressed. The focus of Chapter IV is the history and eventual proof of the four-color theorem, although it also includes a discussion of generalizations involving coloring maps on surfaces of higher genus. The final chapter gives a number of measurements of a graph\u27s closeness to planarity, including the concepts of crossing number, thickness, splitting number, and coarseness. The chapter conclused with a discussion of two other coloring problems - Heawood\u27s empire problem and Ringel\u27s earth-moon problem
Logarithmic delocalization of random Lipschitz functions on honeycomb and other lattices
We study random one-Lipschitz integer functions on the vertices of a
finite connected graph, sampled according to the weight where
, and restricted by a boundary condition. For planar graphs,
this is arguably the simplest ``2D random walk model'', and proving the
convergence of such models to the Gaussian free field (GFF) is a major open
question. Our main result is that for subgraphs of the honeycomb lattice (and
some other cubic planar lattices), with flat boundary conditions and , such functions exhibit logarithmic variations. This is in
line with the GFF prediction and improves a non-quantitative delocalization
result by P. Lammers. The proof goes via level-set percolation arguments,
including a renormalization inequality and a dichotomy theorem for level-set
loops. In another direction, we show that random Lipschitz functions have
bounded variance whenever the wired FK-Ising model with
percolates on the same lattice (corresponding to on
the honeycomb lattice). Via a simple coupling, this also implies, perhaps
surprisingly, that random homomorphisms are localized on the rhombille lattice.Comment: 70 pages; 10 figures with 20 illustration