Two-periodic Aztec diamond Arno Kuijlaars (KU Leuven) joint work - PowerPoint PPT Presentation

Two-periodic Aztec diamond Arno Kuijlaars (KU Leuven) joint work with Maurice Duits (KTH Stockholm) Optimal and Random Point Configurations ICERM, Providence, RI, U.S.A., 27 February 2018

Outline 1. Aztec diamond 2. The model and main result 3. Non-intersecting paths 4. Matrix Valued Orthogonal Polynomials (MVOP) 5. Analysis of RH problem 6. Saddle point analysis 7. Periodic tilings of a hexagon

1. Aztec diamond

Aztec diamond North West East South

Tiling of an Aztec diamond North West East South Tiling with 2 × 1 and 1 × 2 rectangles (dominos) Four types of dominos

Large random tiling Deterministic pattern near corners Solid region or Frozen region Disorder in the middle Liquid region Boundary curve Arctic circle

Recent development Two-periodic weighting Chhita, Johansson (2016) Beffara, Chhita, Johansson (2018 to appear)

Two-periodic weights A new phase within the liquid region: gas region

Phase diagram solid solid liquid gas solid solid

2. The model and main result

Two periodic weights Weight w ( T ) of a tiling T is a the product of b the weights of b b dominos a a a a b Partition function b b b � Z N = w ( T ) a a a a T b b b Probability for T a Prob ( T ) = w ( T ) Z N Aztec diamond of size 2 N

Equivalent weights α = a 2 and β = b 2 North and East dominos have β weight 1 α α Without loss of β β generality α α αβ = 1 β β and α ≥ 1 α Since North dominos have weight 1, we can transfer the weights to non-intersecting paths.

Particles in West and South dominos Particles along diagonal lines are interlacing Positions of particles are random in the two-periodic Aztec diamond. Structure of determinantal point process We found explicit formula for kernel K N using matrix valued orthogonal polynomials (MVOP).

Coordinates n runs m runs from 0 to 2 N − 1 from 0 to 2 N

Formula for correlation kernel THEOREM 1 Assume N is even and m + n and m ′ + n ′ are even. � � K N ( m , n ; m ′ , n ′ ) K N ( m , n + 1; m ′ , n ′ ) K N ( m , n ; m ′ , n ′ + 1) K N ( m , n + 1; m ′ , n ′ + 1) = − χ m > m ′ � dz m ′− m + n ′− n A m − m ′ ( z ) z z + 2 2 π i γ 0 , 1 N − m − n ( z − 1) N 1 � dz � dw z 2 ( w − 1) N A N − m ′ ( w ) F ( w ) A − N + m ( z ) (2 π i ) 2 z − w N − m ′− n ′ z w γ 0 , 1 γ 1 2 where � � 1 2 α z α ( z + 1) A ( z ) = β z ( z + 1) 2 β z z − 1 � ( α − β ) z F ( z ) = 1 1 � α ( z + 1) 2 I 2 + β z ( z + 1) − ( α − β ) z � 2 z ( z + α 2 )( z + β 2 )

3. Non-intersecting paths

Non-intersecting paths Line segments on West, East and South dominos North West East South

Double Aztec diamond 2 N particles along each diagonal line

Non-intersecting paths on a graph Paths are transformed to fit on a graph 6 5 4 3 2 1 0 -1 -2 0 0.5 1 1.5 2 2.5 3 3.5 4

Weights on the graph 6 β β β β β β β β β β 5 α α α α α α α α α α 4 β β β β β β β β β β 3 α α α α α α α α α α 2 β β β β β β β β β β 1 α α α α α α α α α α 0 β β β β β β β β β β -1 α α α α α α α α α α -2 0 0.5 1 1.5 2 2.5 3 3.5 4

Weights on non-intersecting paths Any tiling of double Aztec diamond is equivalent to system ( P 0 , . . . , P 2 N − 1 ) of 2 N non-intersecting paths P j is path on the graph from (0 , j ) to (2 N , j ) , P i is vertex disjoint from P j if i � = j .

Transitions and LGV theorem There are 2 N + 1 levels, 0 , 1 , . . . , 2 N . Transition from level m to level m ′ > m � T m , m ′ ( x , y ) = w ( P ) , x , y ∈ Z P :( m , x ) → ( m ′ , y )

Transitions and LGV theorem There are 2 N + 1 levels, 0 , 1 , . . . , 2 N . Transition from level m to level m ′ > m � T m , m ′ ( x , y ) = w ( P ) , x , y ∈ Z P :( m , x ) → ( m ′ , y ) Lindstr¨ om-Gessel-Viennot theorem Probability that paths at level m are at positions x ( m ) < x ( m ) < · · · < x ( m ) 2 N − 1 : 0 1 1 � 2 N − 1 � 2 N − 1 � � T 0 , m ( i , x ( m ) T m , 2 N ( x ( m ) det ) i , k =0 · det , j ) k k Z N k , j =0 Lindstr¨ om (1973) Gessel-Viennot (1985)

Determinantal point process Corollary: The positions at level m are determinantal with kernel 2 N − 1 � � G − t � K N , m ( x , y ) = T 0 , m ( i , x ) i , j T m , 2 N ( y , j ) i , j =0 G = [ T 0 , 2 N ( i , j )] 2 N − 1 where i , j =0 Multi-level extension is known as Eynard-Mehta theorem.

Block Toeplitz matrices In our case: Transition matrices are 2 periodic T ( x + 2 , y + 2) = T ( x , y ) Block Toeplitz matrices, infinite in both directions, ∞ � B j z j with block symbol A ( z ) = j = −∞  ... ... ...  ... ...   B 0 B 1     ... ...   if T = B − 1 B 0 B 1     ... ...   B − 1 B 0     ... ... ...

Double contour integral formula THEOREM 2: Suppose transition matrices are 2 -periodic. Then � � K N , m (2 x , 2 y ) K N , m (2 x + 1 , 2 y ) K N , m (2 x , 2 y + 1) K N , m (2 x + 1 , 2 y + 1) 1 w y � � = A m , 2 N ( w ) R N ( w , z ) A 0 , m ( z ) z x +1 w N dzdw (2 π i ) 2 γ γ A m , 2 N and A 0 , m are block symbols for the transition matrices T m , 2 N and T 0 , m . R N ( w , z ) is a reproducing kernel for matrix valued polynomials.

4. Matrix Valued Orthogonal Polynomials (MVOP)

MVOP Matrix valued polynomial of degree j , j � C i z i P j ( z ) = i =0 each C i is d × d matrix, det C j � = 0 W ( z ) is d × d matrix valued weight Orthogonality 1 � P j ( z ) W ( z ) P t k ( z ) dz = H j δ j , k 2 π i γ

Reproducing kernel N − 1 � j ( w ) H − 1 P t R N ( w , z ) = P j ( z ) j j =0 is reproducing kernel for matrix polynomials of degree ≤ N − 1 If Q has degree ≤ N − 1 , then 1 � Q ( w ) W ( w ) R N ( w , z ) dw = Q ( z ) 2 π i γ There is a Christoffel-Darboux formula for R N and a Riemann Hilbert problem

Riemann-Hilbert problem Y : C \ γ → C 2 d × 2 d satisfies Y is analytic, � I d � W Y + = Y − on γ , 0 d I d � z N I d � 0 d Y ( z ) = ( I 2 d + O ( z − 1 )) as z → ∞ . z − N I d 0 d Gr¨ unbaum, de la Iglesia, Mart´ ınez-Finkelshtein (2011)

Solution of RH problem Unique solution (provided P N uniquely exists) is 1 � P N ( s ) W ( s )   P N ( z ) ds 2 π i s − z γ   Y ( z ) =   1 � Q N − 1 ( s ) W ( s )   Q N − 1 ( z ) ds 2 π i s − z γ where P N is monic MVOP of degree N and Q N − 1 = − H − 1 N − 1 P N − 1 has degree N − 1

Solution of RH problem Unique solution (provided P N uniquely exists) is 1 � P N ( s ) W ( s )   P N ( z ) ds 2 π i s − z γ   Y ( z ) =   1 � Q N − 1 ( s ) W ( s )   Q N − 1 ( z ) ds 2 π i s − z γ where P N is monic MVOP of degree N and Q N − 1 = − H − 1 N − 1 P N − 1 has degree N − 1 Christoffel Darboux formula � I d � 1 Y − 1 ( w ) Y ( z ) � � R N ( w , z ) = 0 d I d 0 d z − w Delvaux (2010)

Our case of interest Weight matrix in special case of two periodic Aztec diamond is W N ( z ) , with � ( z + 1) 2 + 4 α 2 z 1 � 2 α ( α + β )( z + 1) W ( z ) = ( z + 1) 2 + 4 β 2 z ( z − 1) 2 2 β ( α + β ) z ( z + 1) No symmetry in W . Existence and uniqueness of MVOP are not immediate.

Our case of interest Weight matrix in special case of two periodic Aztec diamond is W N ( z ) , with � ( z + 1) 2 + 4 α 2 z 1 � 2 α ( α + β )( z + 1) W ( z ) = ( z + 1) 2 + 4 β 2 z ( z − 1) 2 2 β ( α + β ) z ( z + 1) No symmetry in W . Existence and uniqueness of MVOP are not immediate. Scalar valued analogue � N on circle around z = 1 and OPs are � z +1 Weight z − 1 Jacobi polynomials P ( − N , N ) ( z ) with nonstandard j parameters

5. Analysis of RH problem

Surprise Steepest descent analysis of RH problem leads to explicit formula RH problem is solved in terms of contour integrals. For example: MVOP is P N ( z ) = ( z − 1) N W N / 2 ∞ W − N / 2 ( z ) , if N is even.

Surprise Steepest descent analysis of RH problem leads to explicit formula RH problem is solved in terms of contour integrals. For example: MVOP is P N ( z ) = ( z − 1) N W N / 2 ∞ W − N / 2 ( z ) , if N is even. It leads to proof of THEOREM 1

6. Saddle point analysis

Asymptotic analysis Saddle point analysis on the double contour integral N − 2 x 2 ( z − 1) N 1 � dz � dw z 2 ( w − 1) N A N − m ( w ) F ( w ) A − N + m ( z ) (2 π i ) 2 N − 2 y z z − w w γ 0 , 1 γ 1 when N → ∞ m , x , y scale with N in such a way that x , y ≈ (1 + ξ 1 + ξ 2 m ≈ (1 + ξ 1 ) N , ) N 2 Saddle points are critical points of 2 log( z − 1) − (1 + ξ 2 ) log z + ξ 1 log λ ( z ) where λ ( z ) is an eigenvalue of W ( z ) = A 2 ( z ) . z

Saddle point analysis Let − 1 < ξ 1 , ξ 2 < 1 . There are always four saddle points, depending on ξ 1 , ξ 2 , and they lie on the Riemann surface for y 2 = z ( z + α 2 )( z + β 2 ) (genus one) with branch points − α 2 < − β 2 < 0 and infinity. At least two saddles are in z ∈ [ − α 2 , − β 2 ] .