Dimer model

introduction

• relation to Bethe ansatz http://staff.science.uva.nl/~nienhuis/tiles.pdf
• domino tiling

 

 

basic notions

• dimer configurations
• set of dimer configurations
• partition function
• Kasteleyn matrix
• height function
• spectral curve
• surface tension

 

 

Termperley equivalence

• spanning trees on \gamma rooted at x
• dimers on D(\gamma)

 

 

Domino tiling and height function

• bipartite graph

 

 

energy and weight systems

• define a weight function on the edges of the graph \gamma
  \(w:E(\Gamma)\to \mathbb{R}_{\geq 0}\)
  
• For a dimer configuration D,
  \(w(D)=\prod_{e\in D} w(e)\)
  
• energy function
  \(\epsilon:E(\Gamma)\to \mathbb{R}\)
  
• For a dimer configuration D,
  \(\epsilon(D)=\sum_{e\in D} \epsilon(e)\)
  
• energy and weight function
  \(w(e)=\exp (-\frac{\epsilon(e)}{T})\)
  
• partition function
  \(Z_{\Gamma}=\sum_{D\subset \Gamma} \prod_{e\in D} w(e)\)
  

 

 

fH

 

P(z_1,z_2,w) if weights are positive real., then P=0 is a Harnack curve of genus

g=|int(N)|

P(z_0,z_2)=0 is harnack if the amoeba map is at most 2-to-1.

 

 

 

 

 

하위페이지

• dimer model
  
  • 2010 Fall Reshetikhin dimer seminar
    
  • 2012 Spring James Propp Topics in Random Spatial Processes
    
    • fully packed loops models
      
  • dimers on triangular and hexagonal lattice
    
  • discrete conformal transform
    
  • domino tiling
    
  • height function
    
  • Lozenge tilings (rhombus tiling)
    
  • non-intersecting paths
    
  • plane partitions
    
  • random surface
    

 

 

 

 

memo

• http://www.math.brown.edu/~rkenyon/talks/
• http://www.umich.edu/~mctp/SciPrgPgs/events/2006/2006glsc/talks/hanany.pdf
• http://www.lif.univ-mrs.fr/~fernique/info/slides_csr.pdf

 

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related itemsSchramm–Loewner evolution

• basic thermodynamics & statistical mechanics
• Schramm–Loewner evolution (SLE)
• 픽의 정리(Pick's Theorem)
• Gaussian free field theory