Combinatorics of the Double-Dimer Model Helen Jenne University of - PowerPoint PPT Presentation

Combinatorics of the Double-Dimer Model Helen Jenne University of Oregon Dimers in Combinatorics and Cluster Algebras 2020 August 10, 2020 This talk is being recorded 1 / 25

Outline Kuo Condensation 1 Main Result: Double-Dimer Condensation 2 Ideas of Proof 3 Non-tripartite pairings 4 2 / 25

Kuo condensation Today G = ( V 1 , V 2 , E ) is a finite bipartite planar graph. Let Z D ( G ) denote the partition function . Z D ( G ) = xyz + x + z y x z Theorem (Kuo04, Theorem 5.1) Let vertices a , b , c , and d appear in a cyclic order on a face of G . If a , c 2 V 1 and b , d 2 V 2 , then Z D ( G ) Z D ( G � { a , b , c , d } )= Z D ( G � { a , b } ) Z D ( G � { c , d } )+ Z D ( G � { a , d } ) Z D ( G � { b , c } ) a d c b a a a d d d c c c b b b 3 / 25

Kuo Condensation Theorem (Kuo04, Theorem 5.1) Let vertices a , b , c , and d appear in a cyclic order on a face of G . If a , c 2 V 1 and b , d 2 V 2 , then Z D ( G ) Z D ( G � { a , b , c , d } )= Z D ( G � { a , b } ) Z D ( G � { c , d } )+ Z D ( G � { a , d } ) Z D ( G � { b , c } ) Examples of non-bijective proofs: Fulmek, Graphical condensation, overlapping Pfa ffi ans and superpositions of Matchings Speyer, Variations on a theme of Kasteleyn, with Application to the TNN Grassmannian Theorem (Desnanot-Jacobi identity/Dodgson condensation) det( M ) det( M i , j i ) det( M j j ) � det( M j i , j ) = det( M i i ) det( M i j ) M j i is the matrix M with the i th row and the j th column removed. 4 / 25

Applications of Kuo’s work Tiling enumeration New proof of MacMahon’s product formula for the generating function for plane partitions that are subsets of an r ⇥ s ⇥ t box. Cluster algebras (LM17) Toric cluster variables for the quiver associated to the cone of the del Pezzo surface of degree 6 Main result. An analogue of Kuo’s theorem for double-dimer configs. Application: A problem in Donaldson-Thomas theory and Pandharipande-Thomas theory (joint work with Ben Young and Gautam Webb) 5 / 25

Double-dimer configurations N is a set of special vertices called nodes on the outer face of G . Definition (Double-dimer configuration on ( G , N )) 7 6 5 Configuration of ` disjoint loops 8 Doubled edges 4 Paths connecting nodes in pairs Its weight is the product of its edge weights ⇥ 2 ` 1 2 3 7 6 5 7 6 5 8 8 = + 4 4 1 1 2 3 2 3 6 / 25

Tripartite pairings Definition (Tripartite pairing) A planar pairing � of N is tripartite if the nodes can be divided into  3 sets of circularly consecutive nodes so that no node is paired with a node in the same set. 1 1 2 12 2 12 3 3 11 4 4 10 5 11 5 9 6 7 8 9 10 6 8 7 Tripartite Not tripartite We often color the nodes in the sets red, green, and blue, in which case � has no monochromatic pairs. Dividing nodes into three sets R , G , and B defines a tripartite pairing. 7 / 25

Main Result Z DD ( G , N ) denotes the weighted sum of all DD config with pairing � . � Theorem (J.) Divide N into sets R , G , and B and let � be the corr. tripartite pairing. Let x , y , w , v 2 N such that x < w 2 V 1 and y < v 2 V 2 . If { x , y , w , v } contains at least one node of each RGB color and x , y , w , v appear in cyclic order then Z DD ( G , N ) Z DD σ xywv ( G , N � { x , y , w , v } ) = σ Z DD σ xy ( G , N � { x , y } ) Z DD σ wv ( G , N � { w , v } ) + Z DD σ xv ( G , N � { x , v } ) Z DD σ wy ( G , N � { w , y } ) Example. Z DD σ ( N ) Z DD σ 1258 ( N − 1 , 2 , 5 , 8) = Z DD σ 12 ( N − 1 , 2) Z DD σ 58 ( N − 5 , 8)+ Z DD σ 18 ( N − 1 , 8) Z DD σ 25 ( N − 2 , 5) 7 6 5 7 6 7 6 5 7 6 7 6 5 7 6 8 8 8 4 4 4 4 4 4 1 1 1 2 3 3 3 2 3 2 3 3 8 / 25

Main Result Z DD ( G , N ) denotes the weighted sum of all DD config with pairing � . � Theorem (J.) Divide N into sets R , G , and B and let � be the corr. tripartite pairing. Let x , y , w , v 2 N such that x < w 2 V 1 and y < v 2 V 2 . If { x , y , w , v } contains at least one node of each RGB color and x , y , w , v appear in cyclic order then Z DD ( G , N ) Z DD σ xywv ( G , N � { x , y , w , v } ) = σ Z DD σ xy ( G , N � { x , y } ) Z DD σ wv ( G , N � { w , v } ) + Z DD σ xv ( G , N � { x , v } ) Z DD σ wy ( G , N � { w , y } ) Example. Z DD σ ( N ) Z DD σ 1258 ( N − 1 , 2 , 5 , 8) = Z DD σ 12 ( N − 1 , 2) Z DD σ 58 ( N − 5 , 8)+ Z DD σ 18 ( N − 1 , 8) Z DD σ 25 ( N − 2 , 5) 7 6 5 7 6 7 6 5 7 6 7 6 5 7 6 8 8 8 4 4 4 4 4 4 1 1 1 2 3 3 3 2 3 2 3 3 We only need the two nodes of the same RGB color to be opposite in BW color. 8 / 25

Corollaries Theorem (Kuo04, Theorem 5.1) a d Let vertices a , b , c , and d appear in a cyclic order on a face of c G . If a , c 2 V 1 and b , d 2 V 2 , then b Z D ( G ) Z D ( G � { a , b , c , d } )= Z D ( G � { a , b } ) Z D ( G � { c , d } )+ Z D ( G � { a , d } ) Z D ( G � { b , c } ) Theorem (J.) Let x , y , w , v 2 N such that x < w 2 V 1 and 7 6 5 y < v 2 V 2 . If { x , y , w , v } contains at least one node of each RGB color and the two nodes of the same RGB 8 color are opposite in BW color then 4 Z DD ( G , N ) Z DD σ xywv ( G � { x , y , w , v } , N � { x , y , w , v } ) = σ Z DD σ xy ( G � { x , y } , N � { x , y } ) Z DD 1 2 3 σ wv ( G � { w , v } , N � { w , v } ) + Z DD σ xv ( G � { x , v } , N � { x , v } ) Z DD σ wy ( G � { w , y } , N � { w , y } ) 9 / 25

Corollaries Theorem (Kuo04, Theorem 5.1) a d Let vertices a , b , c , and d appear in a cyclic order on a face of c G . If a , c 2 V 1 and b , d 2 V 2 , then b Z D ( G ) Z D ( G � { a , b , c , d } )= Z D ( G � { a , b } ) Z D ( G � { c , d } )+ Z D ( G � { a , d } ) Z D ( G � { b , c } ) Theorem (J.) Let x , y , w , v 2 N such that x < w 2 V 1 and 7 6 5 y < v 2 V 2 . If { x , y , w , v } contains at least one node of each RGB color and the two nodes of the same RGB color are opposite in BW color then 4 Z DD ( G , N ) Z DD σ xywv ( G � { x , y , w , v } , N � { x , y , w , v } ) = σ Z DD σ xy ( G � { x , y } , N � { x , y } ) Z DD 2 3 σ wv ( G � { w , v } , N � { w , v } ) + Z DD σ xv ( G � { x , v } , N � { x , v } ) Z DD σ wy ( G � { w , y } , N � { w , y } ) 9 / 25

Corollaries Theorem (Kuo04, Theorem 5.2) a d Let vertices a , c , b , and d appear in a cyclic order on a face of G . If a , c 2 V 1 and b , d 2 V 2 , then c b Z D ( G ) Z D ( G � { a , b , c , d } )= Z D ( G � { a , d } ) Z D ( G � { b , c } ) � Z D ( G � { a , b } ) Z D ( G � { c , d } ) Theorem (J.) Let x , y , w , v 2 N such that x < w 2 V 1 and y < v 2 V 2 . If { x , y , w , v } contains at least one node of each RGB color and the two nodes of the same RGB color are the same in BW color then Z DD ( G , N ) Z DD σ xywv ( G � { x , y , w , v } , N � { x , y , w , v } ) = σ Z DD σ xy ( G � { x , y } , N � { x , y } ) Z DD σ wv ( G � { w , v } , N � { w , v } ) � Z DD σ xv ( G � { x , v } , N � { x , v } ) Z DD σ wy ( G � { w , y } , N � { w , y } ) 10 / 25

Corollaries Theorem (Kuo04, Theorem 5.2) a d Let vertices a , c , b , and d appear in a cyclic order on a face of G . If a , c 2 V 1 and b , d 2 V 2 , then c b Z D ( G ) Z D ( G � { a , b , c , d } )= Z D ( G � { a , d } ) Z D ( G � { b , c } ) � Z D ( G � { a , b } ) Z D ( G � { c , d } ) Theorem (J.) Let x , y , w , v 2 N such that x < w 2 V 1 and y < v 2 V 2 . If { x , y , w , v } contains at least one node of each RGB color and the two nodes of the same RGB color are the same in BW color then Z DD ( G , N ) Z DD σ xywv ( G � { x , y , w , v } , N � { x , y , w , v } ) = σ Z DD σ xy ( G � { x , y } , N � { x , y } ) Z DD σ wv ( G � { w , v } , N � { w , v } ) � Z DD σ xv ( G � { x , v } , N � { x , v } ) Z DD σ wy ( G � { w , y } , N � { w , y } ) 10 / 25

Background: Double-dimer pairing probabilities 1 2 6 ⇣ 1 3 5 ⌘ b Pr = X 1 , 4 X 2 , 5 X 3 , 6 + X 1 , 2 X 3 , 4 X 5 , 6 2 4 6 3 5 4 ⇣ 1 3 5 7 ⌘ b Pr = X 1 , 8 X 3 , 4 X 5 , 2 X 7 , 6 � X 1 , 4 X 3 , 8 X 5 , 2 X 7 , 6 + X 1 , 6 X 3 , 4 X 5 , 8 X 7 , 2 8 4 2 6 � X 1 , 8 X 3 , 6 X 5 , 2 X 7 , 4 � X 1 , 4 X 3 , 6 X 5 , 8 X 7 , 2 + X 1 , 6 X 3 , 8 X 5 , 2 X 7 , 4 Definition (KW11a) Z D ( G BW ) Z D ( G BW ) , where G BW ✓ G only contains nodes that are black and X i , j = i , j odd or white and even. 4 1 1 4 1 4 1 4 3 2 2 2 3 2 G BW G BW G = G BW G BW G 1 , 2 2 , 4 11 / 25

X i , j = 0 if i and j have the same parity 1 1 1 2 8 2 8 2 8 3 7 3 7 3 7 4 6 4 6 4 6 ⇣ 1 3 5 7 ⌘ 5 5 5 b Pr = X 1 , 8 X 3 , 4 X 5 , 2 X 7 , 6 � X 1 , 4 X 3 , 8 X 5 , 2 X 7 , 6 + X 1 , 6 X 3 , 4 X 5 , 8 X 7 , 2 8 4 2 6 � X 1 , 8 X 3 , 6 X 5 , 2 X 7 , 4 � X 1 , 4 X 3 , 6 X 5 , 8 X 7 , 2 + X 1 , 6 X 3 , 8 X 5 , 2 X 7 , 4 Each term in b Pr( � ) is of the form Q X ⌧ := X i , j , where ⌧ is an odd-even pairing. ( i , j ) 2 ⌧ Kenyon and Wilson made a simplifying assumption that all nodes are black and odd or white and even. Theorem (KW11a, Theorem 1.3) b Pr ( � ) is an integer-coe ff homogeneous polynomial in the quantities X i , j 12 / 25