Dimers and embeddings Marianna Russkikh MIT Based on: [KLRR] - PowerPoint PPT Presentation

Dimers and embeddings Marianna Russkikh MIT Based on: [KLRR] “Dimers and circle patterns” joint with R. Kenyon, W. Lam, S. Ramassamy. (arXiv:1810.05616) [CLR] “Dimer model and holomorphic functions on t-embeddings” joint with D. Chelkak, B. Laslier. (arXiv:2001.11871)

Dimer model A dimer cover of a planar bipartite graph is a set of edges with the property: every vertex is contained in exactly one edge of the set. (On the square lattice / honeycomb lattice it can be viewed as a tiling of a domain on the dual lattice by dominos / lozenges.)

Height function Defined on G ⇤ , fixed reference configuration, random configuration 0 0 0 1 0 1 0 1 1 0 1 0 0 0 0 0 − 1 0 0 0 0 Note that ( h − E h ) doesn’t depend on the reference configuration.

Gaussian Free Field GFF with zero boundary conditions on a domain Ω ⊂ C is a conformally invariant random generalized function: X φ k ( z ) GFF( z ) = ξ k √ λ k [1d analog: Brownian Bridge] , k where φ k are eigenfunctions of − ∆ on Ω with zero boundary conditions, λ k is the corresp. eigenvalue, and ξ k are i.i.d. standard Gaussians. The GFF is not a random function, but a random distribution. GFF is a Gaussian process on Ω with Green’s function of the Laplacian as the covariance kernel.

� A. Borodin c [Kenyon’08], [Berestycki–Laslier–Ray’ 16]: lozenge tilings [Kenyon’00], [R.’16-18]: domino tilings (open question: domains composed of 2 ⇥ 2 blocks on Z 2 )

~ = h − E h Ambitious goal [Chelkak, Laslier, R.]: Given a big weighted bipartite planar graph to embed it so that ~ δ → GFF Q: In which metric? ( G , K ) → ( T ( G ⇤ ) , K T ) , K gauge K T ⇠ θ ( b i , v ) t-embedding θ ( w j , v ) v or circle pattern embedding ≤ δ

Results Theorem (Kenyon, Lam, Ramassamy, R.) t-embeddings exist at least in the following cases: I If G δ is a bipartite finite graph with outer face of degree 4. I If G δ is a biperiodic bipartite graph. Theorem (Chelkak, Laslier, R.) Assume G δ are perfectly t-embedded. a) Technical assumptions on faces b) The origami map is small in the bulk ⇒ convergence to π � 1 / 2 GFF D . Theorem (A ff olter; Kenyon, Lam, Ramassamy, R.) Circle pattern embeddings / t-embeddings of G ⇤ are preserved under elementary transformations of G . Application: Miquel dynamics.

Weighted dimers and gauge equivalence Weight function ν : E ( G ) → R > 0 c · ν 5 Probability measure on dimer covers: c · ν 1 Y µ ( m ) = 1 c · ν 4 ν ( e ) c · ν 2 Z e 2 m c · ν 3 Definition Two weight functions ν 1 , ν 2 are said to be gauge equivalent if there are two functions F : B → R and G : W → R such that for any edge bw , ν 1 ( bw ) = F ( b ) G ( w ) ν 2 ( bw ) . Gauge equivalent weights define the same probability measure µ .

Face weights For a planar bipartite graph, two weight functions are gauge equivalent if and only if their face weights are equal, where the face weight of a face with vertices w 1 , b 1 , . . . , w k , b k is X f := ν ( w 1 b 1 ) . . . ν ( w k b k ) ν ( b 1 w 2 ) . . . ν ( b k w 1 ) . b 1 w 1 w 2 b k c ν 2 f f b 2 c ν 3 c ν 1 w k c ν 4 c ν 5

Kasteleyn matrix τ 1 τ 2 τ 2 k Complex Kasteleyn signs : τ 3 τ i ∈ C , | τ i | = 1, face of degree 2 k τ 2 k − 1 τ 1 · τ 3 · . . . · τ 2 k � 1 = ( − 1) ( k +1) τ 2 τ 4 τ 2 k A (Percus–)Kasteleyn matrix K is a weighted, signed adjacency matrix whose rows index the white vertices and columns index the black vertices: K ( w , b ) = τ wb · ν ( wb ). • [Percus’69, Kasteleyn’61]: Z = | det K | = P m 2 M ν ( m ) • The local statistics for the measure µ on dimer configurations can be computed using the inverse Kasteleyn matrix .

Kasteleyn matrix as a discrete Cauchy–Riemann operator Kasteleyn C signs proposed K � 1 = K Ω × Id by Kenyon for the uniform Ω dimer model on Z 2 [flat case]: 1 0 1 1 1 = × i 1 -1 1 0 1 i − 1 1 - i 1 − i Relation for 4 values of K � 1 Ω : 1 · K � 1 Ω ( v + 1 , v 0 ) − 1 · K � 1 Ω ( v − 1 , v 0 )+ i · K � 1 Ω ( v + i , v 0 ) − i · K � 1 Ω ( v − i , v 0 ) = δ { v = v 0 } d Discrete Cauchy-Riemann: a c F ( c ) − F ( a ) = − i · ( F ( d ) − F ( b )) i b

Kasteleyn matrix as a discrete Cauchy–Riemann operator What about non-flat case / general weights / other grids? A function F • : B → C is discrete holomorphic at w ∈ W if X [¯ ∂ F • ]( w ) := F • ( b ) · K ( w , b ) = [ F • K ]( w ) = 0 . b ⇠ w For a fixed w 0 ∈ W the function K � 1 ( · , w 0 ) is a discrete holomorphic function with a simple pole at w 0 . Q: How do discrete holomorphic functions correspond to their continuous counterparts? [gauge + Kasteleyn signs + embedding] (+) [flat] uniform dimer model on Z 2 , isoradial graphs ( ? ) General weighted planar bipartite graphs [Chelkak, Laslier, R.]

Definition: circle pattern [Kenyon, Lam, Ramassamy, R.] An embedding of a bipartite graph with cyclic faces. Assume that each bounded face contains its circumcenter. The circumcenters form an embedding of the dual graph.

Definition: circle pattern [Kenyon, Lam, Ramassamy, R.] Circle pattern realisations with an embedded dual, where the dual graph is the graph of circle centres. (!) Circle patterns themselves are not necessarily embedded.

Circle pattern A circle pattern realisation with an embedded dual.

Definition: t-embedding [Chelkak, Laslier, R.] A t-embedding T : I Proper: All edges are straight segments and they don’t overlap. I Bipartite dual: The dual graph of T is bipartite. I Angle condition: For every vertex v one has θ ( b i , v ) X X θ ( w j , v ) v θ ( f , v ) = θ ( f , v ) = π , f white f black where θ ( f , v ) denotes the angle of a face f at the neighbouring vertex v .

Circle pattern = t-embedding C ( b ) C ( b ) C ( w ) C ( w ) Proposition (Kenyon, Lam, Ramassamy, R.) Suppose G is a bipartite graph and u : V ( G ⇤ ) → C is a convex embedding of the dual graph (with the outer vertex at ∞ ). Then there exists a circle pattern C : V ( G ) → C with u as centers if and only if the alternating sum of angles around every dual vertex is 0 .

Circle pattern = t-embedding C ( b ) C ( b ) T ( wb ) C ( w ) C ( w ) Proposition (Kenyon, Lam, Ramassamy, R.) Suppose G is a bipartite graph and u : V ( G ⇤ ) → C is a convex embedding of the dual graph (with the outer vertex at ∞ ). Then there exists a circle pattern C : V ( G ) → C with u as centers if and only if the alternating sum of angles around every dual vertex is 0 .

Kasteleyn weights X X T → ( G , K T ) , where K T ( w , b ) = K T ( w , b ) = 0 b w u 1 u 2 K T ( w 1 , b 1 ) w 1 w u 2 k b k K T ( w, b ) b 1 w k u w 2 b b 2 Then K T is a Kasteleyn matrix. Kasteleyn sign condition angle condition P white = π mod 2 π Q K T ( w i , b i ) K T ( w i +1 , b i ) ∈ ( − 1) k +1 R +

Circle patterns and elementary transformations b 1 w x + y b 4 b b 1 | u 1 � u 0 | | u 2 � u 0 | = x w y b 2 b b 3 b u 2 w 1 x b 2 w b u 1 b 1 u 0 w 2 b 4 b 4 y w w 1 a a w 2 b 3 b 2 b b 3 a u 2 u 2 u 1 u 1 b d u 3 u 3 c u 4 u 4 1 1 c/ ∆ d/ ∆ b/ ∆ a/ ∆ 1 1 [A ff olter; Kenyon, Lam, Ramassamy, R.]: ∆ = ac + bd T-embeddings of G ⇤ are preserved under elementary transformations of G .

Circle patterns and elementary transformations Miquel theorem: u 1 u 2 Central move u u 3 ( u 2 � u )( u 4 � u ) ( u 1 � u )( u 3 � u ) = ( u 2 � ˜ u )( u 4 � ˜ u ) ( u 1 � ˜ u )( u 3 � ˜ u ) u 4

Circle patterns and elementary transformations a b d u 2 u 1 u 2 u 1 c u 3 u 3 1 1 u 4 u 4 c/ ∆ d/ ∆ b/ ∆ a/ ∆ 1 1 ∆ = ac + bd [A ff olter; Kenyon, Lam, Ramassamy, R.]: The Miquel move for circle centers corresponds to the urban renewal for dimer model.

Miquel dynamics on the square lattice • Miquel dynamics defined as a discrete-time dynamics on the space of square-grid circle patterns: alternate Miquel moves on all the green faces then on all the orange faces. • Its integrability follows from the identification with the Goncharov-Kenyon dimer dynamics. • The evolution is governed by cluster algebras mutations. green move

Miquel dynamics on the square lattice [Goncharov, Kenyon]: Green move: Step 1: Apply an urban renuval move to the green faces. Step 2: Contract all the degree-2 vertices. X ] (1 + X ] )(1 + X [ ) X → X � 1 X → X (1 + X � 1 + )(1 + X � 1 X − X + X � ) X [ green move orange move

Existence of t-embeddings ( G , K ) → ( G ⇤ , K ) → ( T ( G ⇤ ) , K T ) , where K gauge K T . ⇠ Theorem (Kenyon, Lam, Ramassamy, R.) t-embeddings of the dual graph G ⇤ exist at least in the following cases: I If G is a bipartite finite graph with outer face of degree 4, with an equivalence class of real Kasteleyn edge weights under gauge equivalence. I If G is a biperiodic bipartite graph, with an equivalence class of biperiodic real Kasteleyn edge weights under gauge equivalence. K gauge K T ⇠ K T ( wb ) = G ( w ) K ( wb ) F ( b ) ← →

Coulomb gauge for finite planar graphs Def: Functions G : W → C and f 11 w 1 b 1 F : B → C are said to give Coulomb gauge for G if for all internal white vertices w X G ( w ) K wb F ( b ) = 0 , f 21 f 12 b and for all internal black vertices b X G ( w ) K wb F ( b ) = 0 . w 2 b 2 f 22 w P 12 P 11 X W 1 G ( w ) K wb i F ( b i ) = B i w X B 2 B 1 Boundary conditions: a convex quadrilateral P G ( w i ) K w i b F ( b ) = − W i . b P 21 W 2 P 22