44 research outputs found

    Dimers and cluster integrable systems

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    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

    Dynamic cage survey

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    Some contributions to incidence geometry and the polynomial method

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    Combinatorics

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    Planar graphs : a historical perspective.

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    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

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    We study random one-Lipschitz integer functions ff on the vertices of a finite connected graph, sampled according to the weight W(f)=v,wEcI{f(v)=f(w)}W(f) = \prod_{\langle v, w \rangle \in E} \mathbf{c}^{ \mathbb{I} \{ f(v) = f(w) \} } where c1\mathbf{c} \geq 1, 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 1c21 \leq \mathbf{ c } \leq 2, 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 p=11/cp=1-1/\mathbf{c} percolates on the same lattice (corresponding to c>2+3\mathbf{c} > 2 + \sqrt{3} 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
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