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

1 3 8 7 2 4 5 6

dimer confjguration: perfect matching

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

• rthogonal polynomials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion 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

𝑟|𝜌| =

∞

∏

𝑘=1

1 (1 − 𝑟𝑘)𝑘

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion Plane partitions

Plane partitions: limit shape and correlations

Limit shape: Cerf–Kenyon (2001) Correlations: Okounkov–Reshetikhin (2003)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion Plane partitions

Idea: cut the plane partition in vertical slices: interlacing partitions: 𝜈 ≺ 𝜇 𝜇1 ≥ 𝜈1 ≥ 𝜇2 ≥ 𝜈2 ≥ ⋯

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion Plane partitions

Idea: cut the plane partition in vertical slices: interlacing partitions: 𝜈 ≺ 𝜇 𝜇1 ≥ 𝜈1 ≥ 𝜇2 ≥ 𝜈2 ≥ ⋯

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion Plane partitions

Transfer matrices with nice algebraic properties Correlations: ℙ(particles at positions (𝑢1, ℎ1), … (𝑢𝑜, ℎ𝑜)) = det𝐿((𝑢𝑗, ℎ𝑗), (𝑢𝑘, ℎ𝑘)) where 𝐿((𝑢, ℎ), (𝑢′, ℎ′)) = [𝑨ℎ𝑥−ℎ′] Φ(𝑨, 𝑢) Φ(𝑥, 𝑢′) √𝑨𝑥 𝑨 − 𝑥

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 to reach 𝑈 from the horizontal confjguration 𝑎(𝑟) = ∑

𝑈

𝑟𝑂(𝑈) =

𝑜

∏

𝑘=1

(1 + 𝑟2𝑘−1)𝑜−𝑘+1 (Elkies Kuperberg Larsen Propp) Stanley 𝑎(𝑟𝑗) ∑

𝑈

∏ 𝑟#fmips on diag i

𝑗

= ∏

1≤𝑗≤𝑘≤𝑜

(1 + 𝑟2𝑗−1 ⋯ 𝑟2𝑘−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 submatrices of 𝐿−1 (inverse Kasteleyn matrix) In general diffjcult to compute exactly explicit expression for the inverse Kasteleyn matrix (Chhita, Young 2013). No constructive proof.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 + 𝑟2𝑗−1)2𝑗−1 (1 − 𝑟2𝑗)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.

. . . . . .

R+

. . . . . . . . . . . .

R−

. . . . . . . . . . . .

L+

. . . . . . . . . . . .

L−

. . . . . .

Can be glued together along columns.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion

Rail Yard Graphs

Rail yard graphs: sequence of glued elementary graphs.

−2ℓ − 1 2r+1

• dd vertex

even vertex

. . . . . . . . . . . .

R+ L+ R− R+ L− R−

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Structure encoded by a word in 𝑀 + /𝑀 − /𝑆 + /𝑆−.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion

If only 𝑀± are used, faces of degree 6: hexagonal lattice If alternate 𝑀± and 𝑆±, faces of degree 4 or degree 8 with vertices of degree 2: square lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion

If only 𝑀± are used, faces of degree 6: hexagonal lattice If alternate 𝑀± and 𝑆±, faces of degree 4 or degree 8 with vertices of degree 2: square lattice

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion

Steep dimers on Rail Yard Graphs

boundary conditions: vacuum vertices with negative ordinate on the left, and positive

• rdinate of the right are left unmatched.

the other vertices on the boundary are covered by a dimer.

. . .

y = 0

}

all covered

}

none covered

}

all covered

}

none covered y = 0

−2ℓ − 1 2r+1

. . . . . . . . .

R+ L+ R− R+ L− R−

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

steep confjgurations: on each column, fjnite number of diagonal dimers.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion

Connection to tilings

Only 𝑀 + /𝑀−: plane partitions / skew plane partitions (Okounkov-Reshetikhin, Borodin,…) Alternance 𝑀 ± /𝑆±: steep domino tilings (considered by Bouttier, Chapuy, Corteel) ex: L+/R-/L+/R- corresponds to the Aztec diamond

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion

From dimers to Maya diagrams and partitions

From dimers, construct particle confjgurations {•, ∘} (Maya diagrams) on columns of odd vertices: Put • if vertex matched to the left. Put ∘ if vertex matched to the right. For graphs 𝑀+, 𝑀−: plane partitions dimer confjgurations ↔ interlacing • particles. number of diagonal edges: total displacement of • particules given two Maya diagrams, number of compatible dimer confjgurations is 1 if • particles interlaced, 0 otherwise.

. . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion

Transfer matrix

State of odd columns encoded by vectors |𝜇⟩ of a Hilbert space. Transfer operators: Γ𝑀−(𝑦)|𝜇⟩ = ∑

𝜈≻𝜇

𝑦|𝜈|−|𝜇||𝜈⟩, Γ𝑀+(𝑧)|𝜇⟩ = ∑

𝜈≺𝜇

𝑧|𝜇|−|𝜈||𝜈⟩ Localisation operators: 𝜔𝑙, 𝜔∗

𝑙 create, annihilate particles at

position 𝑙. 𝜔𝑙𝜔∗

𝑙 projector on diagrams with a particle at site 𝑙.

Commutation relations: Γ𝑀+(𝑦), Γ𝑀−(𝑧), Ψ(𝑨) = ∑𝑙 𝜔𝑙𝑨𝑙 satisfy nice commutation relations: Γ𝑀+(𝑧)Γ𝑀−(𝑦) = 1 1 − 𝑦𝑧Γ𝑀−(𝑦)Γ𝑀+(𝑧) Γ𝑀+(𝑧)Ψ(𝑨) = 1 1 − 𝑦𝑨Ψ(𝑨)Γ𝑀+(𝑧)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion Partition function

Case of plane partitions: 𝑎(𝑟) = ⟨∅| Γ𝑀+(𝑟𝑛−1/2) ⋯ Γ𝑀+(𝑟1/2) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝑛

Γ𝑀−(𝑟1/2) ⋯ Γ𝑀+(𝑟𝑜−1/2) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝑜

|∅⟩ =

𝑛

∏

𝑘=1 𝑜

∏

𝑙=1

1 1 − 𝑟𝑗+𝑘−1 Apply as many times as necessary the commutation relation Γ𝑀+/Γ𝑀−.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion Partition function

Graphs R+ and R-

Exchange the role of white/black, left/right. Now ∘ particles are interlacing (the corresponding partitions are cointerlacing). Two new operators Γ𝑆−(𝑦), Γ𝑆+(𝑧). Γ𝑆+(𝑧)Γ𝑀−(𝑦) = (1 + 𝑦𝑧)Γ𝑀−(𝑦)Γ𝑆+(𝑧)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion Partition function

Theorem Let 𝐻 is a rail yard graph, encoded by 𝑏 = 𝑏1 ⋯ 𝑏𝑜 and 𝑐 = 𝑐1 ⋯ 𝑐𝑜, 𝑏𝑗 ∈ {𝑀, 𝑆}, 𝑐𝑗 ∈ {+, −}. Let 𝑦 = (𝑦1, … , 𝑦𝑜) the weights per diagonal dimer on each elementary graph. The partition function of the steep dimer confjgurations on 𝐻 is 𝑎(𝑏, 𝑐, 𝑦) = ∏

1≤𝑗<𝑘≤𝑜 𝑐𝑗=+,𝑐𝑘=−

𝑨𝑗𝑘; 𝑨𝑗𝑘 = {1 + 𝑦𝑗𝑦𝑘, if 𝑏𝑗 ≠ 𝑏𝑘 (1 − 𝑦𝑗𝑦𝑘)−1, if 𝑏𝑗 = 𝑏𝑘.

bi = + bj = − ai = aj: zij = (1 + xixj) ai = aj: zij = 1 1 − xixj

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion Correlations for particles

Computing particle probabilities: ℙ(• particles at (𝑢1, ℎ1), … , (𝑢𝑙, ℎ𝑙)) = 1 𝑎 × ⟨∅| Γ(𝑦1) ⋯ Γ(𝑦𝑢1) ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟

𝑢1

𝜔ℎ1𝜔∗

ℎ1

⋯ ⏟

𝑢2−𝑢1

𝜔ℎ2𝜔∗

ℎ2 ⋯ |∅⟩

View 𝜔ℎ𝑘 as some coeffjcient extraction from Ψ(𝑨𝑘) and again make use of commutation relations.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion Correlations for dimers

Map from dimers to particles is local Reconstructing the dimer confjguration from the Maya diagrams not local. Easy case: dimers in the simple columns Equivalent to localisation of particles.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion Correlations for dimers

Dimers in double column: position not (locally) related to presence

• f particles. But:

Bijection between confjgurations inside a “slice” by rerouting dimers around central vertices. Conclusion: localisation operators in terms of terms of creation/annihilation operators.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion Correlations for dimers

Let 𝐺𝑗(𝑨) = ∏𝑛<𝑗/2∶𝑆+(1 + 𝑦𝑛𝑨) ∏𝑛>𝑗/2∶𝑀−(1 − 𝑦𝑛𝑨−1) ∏𝑛<𝑗/2∶𝑀+(1 − 𝑦𝑛𝑨) ∏𝑛<𝑗/2∶𝑆−(1 + 𝑦𝑛𝑨−1) Defjne the matrix 𝐷𝛽𝛾 indexed by vertices of 𝐻 (rows are odd vertices/columns are even vertices) 𝐷𝛽𝛾 = [𝑨𝑙𝛽𝑥−𝑙′

𝛾]

𝐺𝑗𝛽(𝑨) 𝐺𝑗′𝛾(𝑥) √𝑨𝑥 𝑨 − 𝑥 Theorem The probability that edges (𝑓1, … , 𝑓𝑜), with 𝑓𝑗 = (𝑥𝑗, 𝑐𝑗) belong to the random confjguration, is (product of the weights) × det𝐷𝑐𝑗,𝑥𝑘 𝐷 is an inverse of the Kasteleyn matrix on 𝐻.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion

Applications

In the particular case of the Aztec diamond:

gives a constructive derivation of the formula for the inverse Kasteleyn matrix found by Chhita and Young yet another derivation of the arctic circle theorem, fmuctuations…

Mixtures of hexagonal/square lattice Special case of interest: pyramid partitions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Motivations and examples Rail Yard Graphs Conclusion

Applications

In the particular case of the Aztec diamond:

gives a constructive derivation of the formula for the inverse Kasteleyn matrix found by Chhita and Young yet another derivation of the arctic circle theorem, fmuctuations…

Mixtures of hexagonal/square lattice Special case of interest: pyramid partitions