dimers and integrability R. Kenyon Thursday, June 3, 2010
Dimer model on Z 2 random dimer covering = random perfect matching Thursday, June 3, 2010
Thursday, June 3, 2010
Thm (Kasteleyn 1965) For G ⊆ honeycomb, let K = adjacency matrix, � 1 i ∼ j k ij = 0 else . √ Then the number of dimer coverings is det K . Example: K is 24 × 24 and det K = 400. Thursday, June 3, 2010
Boxed plane partition Thursday, June 3, 2010
Find the cardiod Thursday, June 3, 2010
different gradients have different growth rates per unit area. Thursday, June 3, 2010
1.0 0.5 0.0 0.0 � 0.1 � 0.2 1.0 � 0.3 0.5 0.0 Honeycomb dimer surface tension (a function of the gradient of the height function). Thursday, June 3, 2010
Square-octagon lattice P ( z, w ) = 5 + z + 1 /z + w + 1 /w Ronkin function - minus the surface tension E y frozen 3 2 liquid 1 Phase space gas E x -3 -2 -1 1 2 3 -1 -2 -3 Newton polygon = allowed slopes Amoeba of P Thursday, June 3, 2010
Three phases of measures gaseous phase contours liquid phase contours Conformally invariant ([K, 2010]) height contours Thursday, June 3, 2010
Three phases of measures frozen phase contours height contours Thursday, June 3, 2010
Combinatorics The dimer world Matching theory Random walk random interfaces Gibbs measures random partitions SLE 2 , 3 , 4 Ronkin function Free field Limit shapes total positivity Cluster algebras Laplacian integrability Z 2 -actions Tropical geometry SU (2) Harnack curves Complex Burgers’ non-commutative geometry Hamiltonian dynamics Mass transport Strings Gromov-Witten Analysis Algebraic Geometry Thursday, June 3, 2010
Study the structure of dimer models on periodic planar graphs. (with A. Goncharov) Thursday, June 3, 2010
Dimers Riemann Surfaces A convex Z 2 polygon ( g, n ) bipartite graph on torus ideal triangulation mutation/ urban renewal flip move face weights cross ratios Conformal structure Harnack curve + divisor = R 6 g − 6+2 n Dimer Teichm¨ uller space Teichm¨ uller space ∼ tropical Harnack curve measured lamination cluster algebra cluster algebra commuting Hamiltonians Thursday, June 3, 2010
Start: a convex polygon with vertices in Z 2 . Thursday, June 3, 2010
Geodesics on the torus, one for each primitive edge of N . Thursday, June 3, 2010
no parallel double crossings respect circular order Isotope to a “triple-crossing diagram” [D. Thurston] Thursday, June 3, 2010
Thursday, June 3, 2010
Thursday, June 3, 2010
Obtain a bipartite graph Lemma: | white vertices | = | black vertices | = | faces | = 2Area( N ). Thursday, June 3, 2010
w 1 w 3 w 2 w 4 w 5 Label faces. Thursday, June 3, 2010
Matchings and homology Let Ω( G ) ⊂ [0 , 1] E be the matching polytope. � 1 v is white f ∈ Ω = ⇒ ∂f ( v ) = − 1 v is black. Lemma: Vertices of Ω are dimer covers of G . If f 0 , f 1 ∈ Ω then ∂ ( f 1 − f 0 ) = 0 so defines a homology class [ f 1 − f 0 ] in R 2 = H 1 ( T 2 , R ). Lemma: The image of Ω under f �→ [ f − f 0 ] is (a translate of) N . Thursday, June 3, 2010
Let M ( G ) be the set of dimer covers of G . Fix m 0 ∈ M ( G ). For any m ∈ M ( G ), [ m − m 0 ] ∈ H 1 ( G, R ). But H 1 ( G ) is generated by the [ w i ] and [ z 1 ] , [ z 2 ]. So [ m − m 0 ] = � α i [ w i ] + h x [ z 1 ] + h y [ z 2 ]. Define the weight of m to be ν ( m ) = � w α i 1 z h y i z h x 2 . and the “partition function” � ν ( m )( − 1) h x h y . P ( z 1 , z 2 ; w ) = m ∈ M ( G ) Thursday, June 3, 2010
w 2 1 w 3 w 4 z 2 2 C 0 , 1 z 2 z − 1 1 C 0 , 0 w 1 w 2 w 3 z 1 z − 1 2 C 0 , 1 = w 1 (1 + w 3 + w 3 w 4 + w 1 w 3 w 4 + w 1 w 2 w 3 w 4 ) C 0 , 0 = 1 + w 1 + w 1 w 3 + w 1 w 2 w 3 + w 1 w 2 w 3 w 4 Thursday, June 3, 2010
Define a Poisson structure on ( C ∗ ) n +2 (a Poisson bracket on C [ w ± 1 n , z ± 1 1 z ± 1 1 , . . . , w ± 1 2 ]) { w i , w j } = ε ij w i w j where ε is a skew-symmetric form ε ij = − 1 if ε ij = 1 if w j w i w j w i ε ij = 0 else. A similar rule for { w i , z j } and { z i , z j } . Thursday, June 3, 2010
Theorem [Goncharov-K] This Poisson bracket defines a completely integrable system of dimension 2 + 2Area( N ), with symplectic leaves of dimension 2int( N ), (twice the number of interior vertices). A basis for the Casimir elements is given by (ratios of) boundary coe ffi cients of P . The commuting Hamiltonians are the ‘interior’ coefficients of P . A quantum integrable system can be defined using q -commuting variables: w i w j = q 2 ε ij w j w i . Thursday, June 3, 2010
w 2 1 w 3 w 4 z 2 2 C 0 , 1 z 2 z − 1 1 C 0 , 0 w 1 w 2 w 3 z 1 z − 1 2 C 0 , 1 = w 1 (1 + w 3 + w 3 w 4 + w 1 w 3 w 4 + w 1 w 2 w 3 w 4 ) C 0 , 0 = 1 + w 1 + w 1 w 3 + w 1 w 2 w 3 + w 1 w 2 w 3 w 4 Casimirs: w 1 w 3 w 4 z 2 2 z 1 (commute with everything) 1 z − 2 w 1 w 2 w 3 z 2 2 Hamiltonians: H 0 = C 0 , 0 z 1 (commute with each other) H 1 = C 0 , 1 z 1 Complete integrability 6 = 2 + 2 ∗ 2 Thursday, June 3, 2010
∂ z i ∂ t = { z i , H } ∂ w i ∂ t = { w i , H } Thursday, June 3, 2010
Triangle flip Λ 4 Λ 4 � λ − 1 λ � = 0 0 λ � = λ 1 (1 + λ 0 ) 1 � Λ 0 Λ 0 � � λ 2 (1 + λ − 1 Λ 1 Λ 3 Λ 1 Λ 3 0 ) − 1 λ � = 2 λ � = λ 3 (1 + λ 0 ) 3 λ 4 (1 + λ − 1 0 ) − 1 λ � = 4 � Λ 2 Λ 2 Urban renewal w 4 w 4 � w − 1 w � = 0 0 w � = w 1 (1 + w 0 ) 1 w 2 (1 + w − 1 0 ) − 1 w � = w 1 w 0 w 3 w 1 w 0 w 3 � � � 2 w � = w 3 (1 + w 0 ) 3 w 4 (1 + w − 1 0 ) − 1 w � = 4 w 2 w 2 � Thursday, June 3, 2010
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