dimers and integrability R. Kenyon Thursday, June 3, 2010 Dimer - PowerPoint PPT Presentation

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