On dimers and spanning trees February 13, 2017 On dimers and - PowerPoint PPT Presentation

On dimers and spanning trees Introduction Temperley’s bijection T-graphs On dimers and spanning trees February 13, 2017

On dimers and spanning trees Introduction Temperley’s bijection 1 Introduction T-graphs 2 Temperley’s bijection 3 T-graphs

On dimers and spanning trees Motivation Introduction We want to study random surfaces embedded in space, for Temperley’s example arising as interfaces in some statistical physics model. bijection T-graphs Questions : • Law of large number • Fluctuation • Universality

On dimers and spanning trees Dimer covering and height function Introduction Temperley’s bijection T-graphs • In a bipartite planar graph, a dimer configuration is associated to a height function. • The construction uses an arbitrary choice of reference flow.

On dimers and spanning trees The dimer measure Introduction Temperley’s bijection T-graphs For a (planar bipartite) graph G with some positive edge weights w we want to consider the measure � µ ( M ) ∝ w ( e ) . e ∈ M The reason to study this fairly wide class is twofold: • We want to understand universality. • Weights can naturally appear due to gauge invariance.

On dimers and spanning trees Introduction Temperley’s bijection 1 Introduction T-graphs 2 Temperley’s bijection 3 T-graphs

On dimers and spanning trees Square grid Introduction Temperley’s bijection T-graphs

On dimers and spanning trees General graph for the tree Introduction Temperley’s bijection T-graphs Limitations : • All black vertices have degree 4. • Dual edges have weight 1.

On dimers and spanning trees Boundary condition Introduction Temperley’s bijection T-graphs Natural dimer boundary condition can correspond to very degenerate conditioning on the tree.

On dimers and spanning trees Introduction Temperley’s bijection 1 Introduction T-graphs 2 Temperley’s bijection 3 T-graphs

On dimers and spanning trees T-graph example Introduction Temperley’s bijection T-graphs 8 segments and 5 points.

On dimers and spanning trees Associated bipartite graph Introduction Temperley’s bijection T-graphs • A white vertex in each face • A black vertex in each segment • Edge for adjacency relation, weight = length.

On dimers and spanning trees From tree to dimers Introduction Temperley’s bijection T-graphs Match a white vertex with the segment crossed by the outgoing dual tree edge.

On dimers and spanning trees Real life example Introduction Temperley’s bijection T-graphs