Tau-functions ` a la Dub edat and cylindrical events in the - PowerPoint PPT Presentation

Tau-functions ` a la Dub´ edat and cylindrical events in the double-dimer model Dmitry Chelkak (´ ENS, Paris, & PDMI, St.Petersburg (on leave) ) joint work w/ Mikhail Basok (PDMI & SPbSU, St.Peterburg) ICMP 2018, Montreal, 24.07.2018

Setup: double-dimer loop ensembles in Temperley discretizations on Z 2 • Temperley discretizations Ω δ on Z 2 : simply connected domains s.t. all corners are of the same type out of four: B 0 , B 1 , W 0 , W 1 . • Dimer ( = domino) model on Ω δ : perfect matchings, chosen uniformly at random. • Kasteleyn theorem: Z dimers = det K , where K : C B → C W is a weighted adjacency matrix ( = discrete ∂ operator on Ω δ ). [ Temperley domains: nice bijection with UST � Dirichlet boundary conditions for ∂ ]

Setup: double-dimer loop ensembles in Temperley discretizations on Z 2 • Temperley discretizations Ω δ on Z 2 : simply connected domains s.t. all corners are of the same type out of four: B 0 , B 1 , W 0 , W 1 . • Dimer ( = domino) model on Ω δ : perfect matchings, chosen uniformly at random. • Kasteleyn theorem: Z dimers = det K , where K : C B → C W is a weighted adjacency matrix ( = discrete ∂ operator on Ω δ ). [ Temperley domains: nice bijection with UST � Dirichlet boundary conditions for ∂ ] • Double-dimer model: two independent dimer configurations on the same domain. Configuration L dbl - d is a fully-packed collection of loops and double-edges, � 0 K ⊤ � Z dbl - d = 2 # loops ( L dbl - d ) = det K : ( C 2 ) B → ( C 2 ) W . � = det K , K 0 L dbl - d

Goal (cf. Kenyon’10, Dub´ edat’14): conformal invariance, convergence to CLE 4 • Random height functions and GFF: Choosing the orientation of loops γ ∈ L dbl - d randomly, one gets a height function h dbl - d . h dbl - d → GFF (Ω) as δ → 0. Kenyon’00: • Random loop ensembles and CLE 4 : It is a famous prediction (supported by many strong results) that L dbl - d converges to the nested conformal loop ensemble CLE 4 (Ω). [!] The convergence of h dbl - d is not strong enough for the level lines L dbl - d of h dbl - d . • Double-dimer model: two independent dimer configurations on the same domain. Configuration L dbl - d is a fully-packed collection of loops and double-edges, � 0 K ⊤ � Z dbl - d = 2 # loops ( L dbl - d ) = det K : ( C 2 ) B → ( C 2 ) W . � = det K , K 0 L dbl - d

Kenyon (2010): SL 2 ( C )-monodromies and Q-determinants for double-dimers Let ρ : π 1 (Ω \ { λ 1 , . . . , λ n } ) → SL 2 ( C ) . A 4 A 3 A 5 Down-to-earth viewpoint: draw cuts from punctures λ k to ∂ Ω and choose A k ∈ SL 2 ( C ). • Kasteleyn’s theorem generalizes as follows: = Qdet K ( ρ ) �� � γ ∈L dbl - d ( 1 2 Tr ρ ( γ )) E , det K A 1 A 2 where K ( ρ ) : ( C 2 ) B → ( C 2 ) W is obtained − 1 A 3 A 2 A 1 ρ ( γ ) = A 5 A 1 from K by putting the matrices A ± 1 on cuts. k

Kenyon (2010): SL 2 ( C )-monodromies and Q-determinants for double-dimers Let ρ : π 1 (Ω \ { λ 1 , . . . , λ n } ) → SL 2 ( C ) . A 4 A 3 A 5 Down-to-earth viewpoint: draw cuts from punctures λ k to ∂ Ω and choose A k ∈ SL 2 ( C ). • Kasteleyn’s theorem generalizes as follows: = Qdet K ( ρ ) �� � γ ∈L dbl - d ( 1 2 Tr ρ ( γ )) E , det K A 1 A 2 where K ( ρ ) : ( C 2 ) B → ( C 2 ) W is obtained n ( L ) = (2 , 2 , 2 , 1 , 1 , 1 , 2 , 0 , 1 , 3 , 3 , 1 , 2) e ∈E from K by putting the matrices A ± 1 on cuts. k Remark: A better viewpoint is to fix a triangulation of Ω \ { λ 1 , . . . , λ n } and to consider discrete C 2 -vector bundles and flat SL 2 ( C )-connections on them: ( Fun ( π 1 (Ω \ { λ 1 , . . . , λ n } ) → SL 2 ( C ))) SL 2 ( C ) ≃ ( Fun ( SL 2 ( C ) E )) SL 2 ( C ) F .

Dub´ edat (2014): locally unipotent monodromies and convergence to Dub´ edat (2014): the Jimbo–Miwa–Ueno isomonodronic τ -function Let Ω δ , δ → 0, be a sequence of Temperley approximations to a simply connected domain Ω ⊂ C . Fix a collection of (pairwise distinct) punctures λ 1 , . . . , λ n ∈ Ω. Theorem (Dub´ edat, 2014): Let ρ : π 1 (Ω \ { λ 1 , . . . , λ n } ) → SL 2 ( C ) be such that Tr ρ ([ γ ❦ ]) = 2 for each of the loops [ λ k ] surrounding a single puncture λ k . γ ∈L dbl - d ( 1 � � � =: τ δ ( ρ ) → τ JMU ( ρ ) as δ → 0. (i) Then 2 Tr ρ ( γ )) E Remark: In fact, this convergence is uniform on compact subsets of ❳ unip ⊂ ❳ := { ρ : π 1 (Ω \ { λ 1 , . . . , λ n } ) → SL 2 ( C ) } . (ii) Moreover, provided that ρ ∈ X unip is close enough to Id , one has γ ∈L CLE 4 ( 1 τ JMU ( ρ ) = τ CLE 4 ( ρ ) := E � � � 2 Tr ρ ( γ )) .

Dub´ edat (2014): locally unipotent monodromies and convergence to Dub´ edat (2014): the Jimbo–Miwa–Ueno isomonodronic τ -function Notation: Lamination L = collection of loops in Ω \ { λ 1 , . . . , λ n } up to homotopies. ▲ := 2 − # loops ( L ) · P [ L dbl - d ≃ macro L] , ♣ δ f L ( ρ ) := � γ ∈ L Tr ρ ( γ ) . edat give τ δ ( ρ ) = � ▲ − macro ♣ δ ▲ ❢ ▲ ( ρ ) → τ JMU ( ρ ) , ρ ∈ ❳ unip . � � The results of Dub´ The goal is to deduce the convergence of ♣ δ ▲ for each macroscopic lamination L . ( ρ ) Remark: The isomonodronic τ -function can be thought of as : det ∂ [Ω; λ 1 ,...,λ ♥ ] : , ( ρ ) stands for the ∂ operator acting on functions Ω → C 2 with monodromy ρ . where ∂ • The function τ JMU ( ρ ) is defined for all ρ ∈ X unip and is conformally invariant. • The identity τ JMU = τ CLE 4 is a separate statement (also due to Dub´ edat’14).

Main result (joint w/ Mikhail Basok, 2018) Let D r denote the “ball of radius R ” in X = { ρ : π 1 (Ω \ { λ 1 , . . . , λ n } ) → SL 2 ( C ) } . √ [ normalization: � A � := Tr ( AA ∗ ), in particular X ∩ D r = ∅ if r � 2 ] Theorem: There exists an absolute constant k 0 > 1 such that the following holds: √ 2, R := k 0 r and F : X unip ∩ D R → C be a holomorphic function. (i) Let r > r −| n ( L ) | · � F � L ∞ ( D R ) � � (i) Then there exist coefficients p L = O such that � F ( ρ ) = L − macro p L f L ( ρ ) , ρ ∈ X unip ∩ D r . √ p L = O ( r −| n ( L ) | ) be such that (ii) Let r > k 0 2 and two sets of coefficients p L , ˜ � � ρ ∈ X unip ∩ D r . L − macro p L f L ( ρ ) = L − macro ˜ p L f L ( ρ ) , (ii) Then p L = ˜ p L for all macroscopic laminations L .

Main result (joint w/ Mikhail Basok, 2018) Corollary: Since the isomonodromic tau-function is holomoprhic on the whole X unip , there exist unique coefficients p JMU s.t. τ JMU ( ρ ) = � L − macro p JMU f L ( ρ ), ρ ∈ X unip . L L Theorem: There exists an absolute constant k 0 > 1 such that the following holds: √ 2, R := k 0 r and F : X unip ∩ D R → C be a holomorphic function. (i) Let r > r −| n ( L ) | · � F � L ∞ ( D R ) � � (i) Then there exist coefficients p L = O such that � F ( ρ ) = L − macro p L f L ( ρ ) , ρ ∈ X unip ∩ D r . √ p L = O ( r −| n ( L ) | ) be such that (ii) Let r > k 0 2 and two sets of coefficients p L , ˜ � � L − macro p L f L ( ρ ) = L − macro ˜ p L f L ( ρ ) , ρ ∈ X unip ∩ D r . (ii) Then p L = ˜ p L for all macroscopic laminations L .

Main result (joint w/ Mikhail Basok, 2018) Corollary: Since the isomonodromic tau-function is holomoprhic on the whole X unip , there exist unique coefficients p JMU s.t. τ JMU ( ρ ) = � L − macro p JMU f L ( ρ ), ρ ∈ X unip . L L Theorem: There exists an absolute constant k 0 > 1 such that the following holds: √ 2, R := k 0 r and F : X unip ∩ D R → C be a holomorphic function. (i) Let r > r −| n ( L ) | · � F � L ∞ ( D R ) � � (i) Then there exist coefficients p L = O such that � F ( ρ ) = L − macro p L f L ( ρ ) , ρ ∈ X unip ∩ D r . Corollary: (a) Uniform boundedness of topological correlators τ δ on D R for all R > 0 Corollary: (a) implies the uniform (in δ ) estimate p δ L = O ( r −| n ( L ) | ) for all r > 0. (b) Convergence (as δ → 0) of topological correlators τ δ → τ JMU on D R implies (b) convergence of coefficients: p δ L → p JMU for all macroscopic laminations L . L

Main result (joint w/ Mikhail Basok, 2018) Corollary: Since the isomonodromic tau-function is holomoprhic on the whole X unip , there exist unique coefficients p JMU s.t. τ JMU ( ρ ) = � L − macro p JMU f L ( ρ ), ρ ∈ X unip . L L √ = O ( r −| n ( L ) | Warning: It is easy to see that p CLE 4 ) for some r 0 > 2 and L 0 edat proved that τ CLE 4 ( ρ ) = τ JMU ( ρ ) for ρ ∈ X unip ∩ D r 0 (= near Id ). Warning: Dub´ Unfortunately, this does not directly imply p CLE 4 = p JMU for all laminations L : L √ L 2 k 0 is enough) decay of p CLE 4 we also need a superexponential (in fact, r 0 > . L Corollary: (a) Uniform boundedness of topological correlators τ δ on D R for all R > 0 L = O ( r −| n ( L ) | ) for all r > 0. Corollary: (a) implies the uniform (in δ ) estimate p δ (b) Convergence (as δ → 0) of topological correlators τ δ → τ JMU on D R implies (b) convergence of coefficients: p δ L → p JMU for all macroscopic laminations L . L