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On dimers and spanning trees Introduction Temperley’s bijection T-graphs
On dimers and spanning trees Introduction Temperley’s bijection T-graphs
On dimers and spanning trees February 13, 2017 On dimers and - - PowerPoint PPT Presentation
On dimers and spanning trees February 13, 2017 On dimers and - - PowerPoint PPT Presentation
On dimers and spanning trees Introduction Temperleys bijection T-graphs On dimers and spanning trees February 13, 2017 On dimers and spanning trees Introduction Temperleys bijection 1 Introduction T-graphs 2 Temperleys
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On dimers and spanning trees Introduction Temperley’s bijection T-graphs
Motivation
We want to study random surfaces embedded in space, for example arising as interfaces in some statistical physics model. Questions :
- Law of large number
- Fluctuation
- Universality
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On dimers and spanning trees Introduction Temperley’s bijection T-graphs
Dimer covering and height function
- In a bipartite planar graph, a dimer configuration is
associated to a height function.
- The construction uses an arbitrary choice of reference flow.
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On dimers and spanning trees Introduction Temperley’s bijection T-graphs
The dimer measure
For a (planar bipartite) graph G with some positive edge weights w we want to consider the measure µ(M) ∝
- e∈M
w(e). The reason to study this fairly wide class is twofold:
- We want to understand universality.
- Weights can naturally appear due to gauge invariance.
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On dimers and spanning trees Introduction Temperley’s bijection T-graphs
1 Introduction 2 Temperley’s bijection 3 T-graphs
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On dimers and spanning trees Introduction Temperley’s bijection T-graphs
Square grid
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On dimers and spanning trees Introduction Temperley’s bijection T-graphs
General graph for the tree
Limitations :
- All black vertices have degree 4.
- Dual edges have weight 1.
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On dimers and spanning trees Introduction Temperley’s bijection T-graphs
Boundary condition
Natural dimer boundary condition can correspond to very degenerate conditioning on the tree.
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On dimers and spanning trees Introduction Temperley’s bijection T-graphs
1 Introduction 2 Temperley’s bijection 3 T-graphs
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On dimers and spanning trees Introduction Temperley’s bijection T-graphs
T-graph example
8 segments and 5 points.
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On dimers and spanning trees Introduction Temperley’s bijection T-graphs
Associated bipartite graph
- A white vertex in each face
- A black vertex in each segment
- Edge for adjacency relation, weight = length.
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On dimers and spanning trees Introduction Temperley’s bijection T-graphs
From tree to dimers
Match a white vertex with the segment crossed by the outgoing dual tree edge.
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On dimers and spanning trees Introduction Temperley’s bijection T-graphs