Steep Dimers on Rail Yard Graphs Cdric Boutillier (UPMC) joint work - PowerPoint PPT Presentation

. . . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Steep Dimers on Rail Yard Graphs Cédric Boutillier (UPMC) joint work with J. Bouttier (CEA), G. Chapuy (LIAFA), S. Corteel (LIAFA), S. Ramassamy (Brown) . . . . . . . . . . . . . . . . . . . . . . . . . États de la recherche matrices aléatoires – 3 décembre 2014

. . . . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Dimer models planar graph 𝐻 . . . . . . . . . . . . . . . . . . . . . . . . dimer confjguration : perfect matching 5 4 6 2 3 7 1 8

. . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Several techniques to study these models Kasteleyn theory: partition function: determinant of the Kasteleyn matrix 𝐿 correlations: minors of 𝐿 −1 Non intersecting paths Lindström-Gessel-Viennot . . . . . . . . . . . . . . . . . . . . . . . . . . orthogonal polynomials

. Conclusion . . . . . . . . . Motivations and examples Rail Yard Graphs Plane partitions . Plane partitions Dimers on the hexagonal lattice: tilings with rhombi 3D interpretation: piles of cubes in the corner of a room. Partition function: McMahon’s formula ∑ 𝜌 ∞ ∏ 𝑘=1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1 − 𝑟 𝑘 ) 𝑘 𝑟 |𝜌| =

. . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Plane partitions Plane partitions: limit shape and correlations Limit shape: Cerf–Kenyon (2001) Correlations: Okounkov–Reshetikhin . . . . . . . . . . . . . . . . . . . . . . . . . . (2003)

𝜇 1 ≥ 𝜈 1 ≥ 𝜇 2 ≥ 𝜈 2 ≥ ⋯ . . . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Plane partitions Idea: cut the plane partition in vertical slices: interlacing partitions: 𝜈 ≺ 𝜇 . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Plane partitions Idea: cut the plane partition in vertical slices: interlacing partitions: 𝜈 ≺ 𝜇 . . . . . . . . . . . . . . . . . . . . . . . . 𝜇 1 ≥ 𝜈 1 ≥ 𝜇 2 ≥ 𝜈 2 ≥ ⋯

. . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Plane partitions Transfer matrices with nice algebraic properties Correlations: ℙ( particles at positions (𝑢 1 , ℎ 1 ), … (𝑢 𝑜 , ℎ 𝑜 )) = det𝐿((𝑢 𝑗 , ℎ 𝑗 ), (𝑢 𝑘 , ℎ 𝑘 )) where Φ(𝑥, 𝑢 ′ ) √𝑨𝑥 . . . . . . . . . . . . . . . . . . . . . . . . . . 𝑨 − 𝑥 𝐿((𝑢, ℎ), (𝑢 ′ , ℎ ′ )) = [𝑨 ℎ 𝑥 −ℎ ′ ] Φ(𝑨, 𝑢)

. . . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Aztec diamond Aztec diamond dimers on the square lattice: dominos Aztec diamond of size 𝑜 = 3 : . . . . . . . . . . . . . . . . . . . . . . . . . fmip accessibility:

. . . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Aztec diamond Aztec diamond dimers on the square lattice: dominos Aztec diamond of size 𝑜 = 3 : . . . . . . . . . . . . . . . . . . . . . . . . . fmip accessibility:

. . . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Aztec diamond Aztec diamond dimers on the square lattice: dominos Aztec diamond of size 𝑜 = 3 : . . . . . . . . . . . . . . . . . . . . . . . . . fmip accessibility:

. . . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Aztec diamond Aztec diamond dimers on the square lattice: dominos Aztec diamond of size 𝑜 = 3 : . . . . . . . . . . . . . . . . . . . . . . . . . fmip accessibility:

. . . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Aztec diamond Aztec diamond dimers on the square lattice: dominos Aztec diamond of size 𝑜 = 3 : . . . . . . . . . . . . . . . . . . . . . . . . . fmip accessibility:

. . . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Aztec diamond Aztec diamond dimers on the square lattice: dominos Aztec diamond of size 𝑜 = 3 : . . . . . . . . . . . . . . . . . . . . . . . . . fmip accessibility:

. to reach 𝑈 from the horizontal confjguration . . . . Motivations and examples Rail Yard Graphs Conclusion Aztec diamond Aztec diamond: partition function Number of tilings of size 𝑜 : 2 𝑜(𝑜+1) 2 Refjned partition function: if 𝑂(𝑈) miniminal number of fmips 𝑎(𝑟) = ∑ . 𝑈 𝑜 ∏ 𝑘=1 (1 + 𝑟 2𝑘−1 ) 𝑜−𝑘+1 (Elkies Kuperberg Larsen Propp) Stanley 𝑎(𝑟 𝑗 ) ∑ 𝑈 ∏ 𝑟 # fmips on diag i 𝑗 = ∏ 1≤𝑗≤𝑘≤𝑜 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝑟 𝑂(𝑈) = (1 + 𝑟 2𝑗−1 ⋯ 𝑟 2𝑘−1 )

. . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Aztec diamond Aztec diamond: limit shape encode tiling with non intersecting paths position of the highest path, Krawtchouk ensemble (Johansson) derivation of the arctic circle theorem (Jockusch Propp Shore) . . . . . . . . . . . . . . . . . . . . . . . . . . fmuctuations aroung the limit shape: Airy process (Johansson)

. . . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Aztec diamond Aztec diamond: correlations Correlations between dominos is given by determinants of In general diffjcult to compute exactly explicit expression for the inverse Kasteleyn matrix (Chhita, . . . . . . . . . . . . . . . . . . . . . . . . . Young 2013). No constructive proof. submatrices of 𝐿 −1 (inverse Kasteleyn matrix)

. . . . . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Aztec diamond . . . . . . . . . . . . . . . . . . . . . . . Pyramid partitions

. . . . . . . . . . . . Motivations and examples . Rail Yard Graphs Conclusion Aztec diamond Pyramid partitions partition function (Szendrõi, Kenyon, Young) 𝑎(𝑟) = ∏ 𝑗≥1 (1 + 𝑟 2𝑗−1 ) 2𝑗−1 (1 − 𝑟 2𝑗 ) 2𝑗 limit shape (Kenyon-Okounkov): . . . . . . . . . . . . . . . . . . . . . . . . . . . local statistics of dominos?

. Motivations and examples . . . . . . . . . . Rail Yard Graphs . Conclusion Aztec diamond Our goal: unifjed framework to study these three examples (and many more) transfer matrix approach to solve these models encode dimer confjguration as particles correlations of particles ↔ (co)interlacing partitions (Schur process) correlations of dimers explain the formula obtained by Chhita and Young . . . . . . . . . . . . . . . . . . . . . . . . . . . . study typical behaviour of such large structures

. . . . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Elementary Rail Yard Graphs 4 elementary graphs. . . Can be glued together along columns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R + R − L + L − . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion Rail Yard Graphs Rail yard graphs: sequence of glued elementary graphs. . . Structure encoded by a word in 𝑀 + /𝑀 − /𝑆 + /𝑆− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . odd vertex even vertex . . . . . . . R + L + R − R + L − R − . . . . . . . . . . . . . . . . . . . 2 r +1 . − 2 ℓ − 1

. . . . . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion vertices of degree 2: . . . . . . . . . . . . . . . . . . . . . . . square lattice If only 𝑀 ± are used, faces of degree 6: hexagonal lattice If alternate 𝑀 ± and 𝑆 ± , faces of degree 4 or degree 8 with

. . . . . . . . . . . . . . . . . Motivations and examples Rail Yard Graphs Conclusion . . . . . . . . . . . . . . . . . . . . . . . vertices of degree 2: square lattice If only 𝑀 ± are used, faces of degree 6: hexagonal lattice If alternate 𝑀 ± and 𝑆 ± , faces of degree 4 or degree 8 with