Limit shapes in the Schur process Dan Betea LPMA (UPMC Paris VI), - PowerPoint PPT Presentation

Limit shapes in the Schur process Dan Betea LPMA (UPMC Paris VI), CNRS (Collaboration with C. Boutillier, M. Vuleti´ c ) Aprilis XIII, MMXV

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Outline ◮ Pyramid partitions ◮ Interlude into partitions and the Schur process ◮ Asymptotics of pyramid partitions ◮ Asymptotics of non–uniform Aztec diamonds ◮ Some related phenomena

Pyramid partitions Figure : Piles of 2 × 2 × 1 boxes, each viewed as a pair of dominoes in the 2D projection looking downwards. On the left, the empty pyramid partition.

Flips and the volume ◮ pyramid partition = what’s left after a finite number of box removals from the empty configuration (introduced by Kenyon and Szendr¨ oi) ◮ removal = flip (adjacent vertical dominoes ↔ adjacent horizontal dominoes) ◮ Volume = Number of flips Theorem (Young 2010) (1 + q 2 n − 1 ) 2 n − 1 q Volume (Λ) = � � . (1 − q 2 n ) 2 n Λ n ≥ 1

How do large pyramid partitions look like?

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Partitions Figure : Partition (2 , 2 , 2 , 1 , 1) in English, French and Russian notation, with associated Maya diagram (particle-hole representation).

Horizontal and vertical strips Given partitions µ ⊆ λ , we can form skew diagram λ/µ , which we call a ◮ horizontal strip, and write µ ≺ λ if λ 1 ≥ µ 1 ≥ λ 2 ≥ µ 2 ≥ λ 3 . . . ******** / ***** -----*** ***** / *** ---** *** / *** = --- * / * ◮ vertical strip, and write µ ≺ ′ λ , if λ ′ ≺ µ ′ ( ′ = conjugate) or λ i − µ i ∈ { 0 , 1 } ******** / ******* -------* ***** / ***** ----- ***** / **** ----* ***** / **** = ----* * / * - * / * * / *

The Schur process Let ω = ( ω 1 , ω 2 , . . . , ω n ) ∈ {≺ , ≻ , ≺ ′ , ≻ ′ } n be a word. We say a sequence of partitions Λ = ( ∅ = λ (0) , λ (1) , . . . , λ ( n ) = ∅ ) is ω -interlaced if λ ( i − 1) ω i λ ( i ), for i = 1 , . . . , n . The Schur process of word ω with parameters Z = ( z 1 , . . . , z n ) is the measure on the set of ω -interlaced sequences of partitions Λ = ( ∅ = λ (0) , λ (1) , . . . , λ ( n ) = ∅ ) given by n z || λ ( i ) |−| λ ( i − 1) || � Prob (Λ) ∝ . i i =1 Remark For a more general definition, see the original work of Okounkov–Reshetikhin 2003, or Borodin–Rains 2006.

The Schur process is a determinantal point process Theorem (OR 2003; BR 2006) Prob ( λ ( i s ) contains a particle at position k s , 1 ≤ s ≤ n ) = 1 ≤ u , v ≤ n K ( i u , k u ; i v , k v ) det where √ zw  � z k � Φ( z ; Z ,ω ; i ) i ≤ i ′ , z − w ,  w k ′ Φ( w ; Z ,ω ; i ′ ) K ( i , k ; i ′ , k ′ ) = � Φ( z ; Z ,ω ; i ′ ) √ zw z k � i > i ′ − w − z ,  w k ′ Φ( w ; Z ,ω ; i ) with z j � − ǫ j � � (1 + ǫ j z j z ) ǫ j � Φ( z ; Z , ω ; i ) = 1 + ǫ j z j : j ≤ i , ω j ∈{≺ , ≺ ′ } j : j > i , ω j ∈{≻ , ≻ ′ } ω j = ≺′ , ω j = ≻′ , � � 1 , 1 , ǫ j = ǫ j = − 1 , ω j = ≺ . − 1 , ω j = ≻ .

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Pyramid partitions as Schur processes, pictorially ◦ ◦ ◦ ◦ ◦ ◦ ◦ • ◦ • • • • • ◦ • ◦ ◦ ◦ ◦ • • ◦ • ◦ • • ◦ • • ◦ − 5 • ◦ ◦ • • • • ◦ − 4 • ◦ ◦ ◦ ◦ • • ◦ − 3 • ◦ ◦ ◦ ◦ ◦ • ◦ − 2 • • • • • • • − 1 0 1 2 3 4 5 Figure : A pyramid partition of width 5 corresponding to the sequence ∅ ≺ (1) ≺ ′ (2) ≺ (2 , 2) ≺ ′ (3 , 3) ≺ (3 , 3 , 2) ≻ ′ (2 , 2 , 1) ≻ (2 , 1) ≻ ′ (1 , 1) ≻ (1) ≻ ′ ∅ .

Pyramid partitions as Schur processes II Let n = 2 n 0 be an even integer. A pyramid partition is (bijectively) a sequence of 2 n + 1 partitions Λ = ( ∅ = λ ( − n ) ≺ λ ( − n +1) ≺ ′ λ ( − n +2) ≺ · · · ≺ ′ λ (0) ≻ λ (1) ≻ ′ λ (2) ≻ · · · ≻ ′ λ ( n ) = ∅ ) . It is this a Schur process for the word ω pyr = ( ≺ , ≺ ′ ) n 0 ( ≻ , ≻ ′ ) n 0 and parameters Z = ( z − n , . . . , z − 1 , z 1 , . . . , z n ). Remark For volume weighting weighting, z − i = z i = q i − 1 2 , 1 ≤ i ≤ n .

A simple word on asymptotics Everything we’d like to know about asymptotics of large pyramid partitions can be translated into asymptotics of large particle–hole systems associated to the corresponding Schur process.

How to compute the limit shape Let t = 2 t 0 < n , k ∈ Z + 1 2 . A weak Wick lemma shows that: Lemma (db–Boutillier–Vuleti´ c 2015) Prob ( λ ( − t ) contains a particle at position k ) = � z k � J ( z ; t 0 ) √ zw = w k J ( w ; t 0 ) z − w � � J ( z ; t 0 ) 1 1 dz dw = z k − 1 2 w − k − 1 z − w J ( w ; t 0 ) 2 π iz 2 π iw 2 where (with ( u ; q ) m = � m − 1 i =0 (1 − q i u ) ) 1 J ( z ; t 0 ) = ( − q 2 t 0 + 1 2 z ; q 2 ) n 0 − t 0 ( q 2 z ; q 2 ) n 0 . 3 ( q 2 t 0 + 3 2 z ; q 2 ) n 0 − t 0 ( − q 2 z ; q 2 ) n 0

Asymptotics regime We let the size of the partition grow with q → 1 as ǫ → 0 like so: q ( ǫ ) = exp( − γǫ ) , n 0 ( ǫ ) = a 0 /ǫ, t 0 ( ǫ ) = x 0 /ǫ, k ( ǫ ) = y /ǫ.

A few limit formulas If q = exp( − r ) and r → 0+, we have log( z ; q ) ∞ ∼ − Li 2 ( z ) r and furthermore, r ∼ 1 r ( Li 2 ( e − A z ) − Li 2 ( z )) log( z ; q ) A where z 2 � n 2 , | z | < 1 Li 2 ( z ) = n ≥ 1 with analytic continuation given by � z log(1 − u ) Li 2 ( z ) = − du , z ∈ C \ [1 , ∞ ) . u 0

Asymptotics of the kernel Lemma (db–Boutillier–Vuleti´ c 2015) In the limit (x = 2 x 0 is rescaled t, y is rescaled k), d T � � e S ( z ; x , y ) − S ( w ; x , y )) Prob ( λ ( − t ) contains a particle at position k ) ∼ z − w where S ( z ; x , y ) = 1 � Li 2 ( − Az ) − Li 2 ( − Xz ) + Li 2 ( A z ) − Li 2 ( 1 z )+ 2 γ + Li 2 ( Xz ) − Li 2 ( Az ) + Li 2 ( − 1 z ) − Li 2 ( − A � z ) − y log z and X = exp( − γ x ) , A = exp( − 2 γ a 0 ) .

The arctic curve To compute the arctic curve, one solves for ( x , y ) (or X = exp( − γ x ) , Y = exp(2 γ y )) corresponding to double critial points of S ( z ; x , y ). That is, Theorem (db–Boutillier–Vuleti´ c 2015) The arctic curve is the locus ( x , y ) satisfying: f ( z ; X ) = Y , f ′ ( z ; X ) = 0 where f ( z ; X ) = ( z +1)( z − A )( z − 1 / A )( z +1 / X ) ( z − 1)( z + A )( z +1 / A )( z − 1 / X ) . Remark Alternatively, it can be seen as given by the algebraic equation ∆ [( z + 1)( z − A )( z − 1 / A )( z + 1 / X ) − Y ( z − 1)( z + A )( z + 1 / A )( z − 1 / X )] = 0 where ∆ represents taking the discriminant.

The arctic curve, pictorially Notice the cusps (which correspond to the triple critical point of S at z = 0). This cusp phenomenon has appeared in the case of (skew) plane partitions with two different q ’s, Mkrtchyan 2013. Intuitively, we have replaced “two different q ’s, word ω = ≺ 2 n 0 ≻ 2 n 0 ” with “one single q , word ω = ( ≺ , ≺ ′ ) n 0 ( ≻ , ≻ ′ ) n 0 ”. If this makes no sense, it’s probably because it doesn’t make much sense.

Arctic curve in the infinite regime What happens when a 0 → ∞ , or equivalently, A → 0? The cusps move to ∞ and the arctic curve becomes (1 + Z + W − ZW )(1 + Z − W + ZW )(1 − Z + W + ZW )(1 − Z − W − ZW ) = 0 √ √ where ( Z , W ) = ( X , Y ) which is the boundary of the amoeba of the (square lattice determined) polynomial P ( Z , W ) = 1 + Z + W − ZW .

Arctic curve in the infinite regime, pictorially 4 2 0 � 2 � 4 � 4 � 2 0 2 4

A large sample in the infinite regime, up to affine transformations

A word on fluctuations around the arctic curve Everywhere but at the cusps, fluctuations are of Airy type (cf., for example, Okounkov–Reshetikhin 2006). At the cusps, because of the appearence of the triple critical point, one would conjecture Pearcey process fluctuations, but this has not yet been rigurously established.

A slide on details: vertex operators 1 Γ + ( x )Γ − ( y ) = 1 − xy Γ − ( y )Γ + ( x ) , 1 ˜ Γ + ( x )˜ Γ − ( y )˜ ˜ Γ − ( y ) = Γ + ( x ) , 1 − xy ˜ Γ + ( x )Γ − ( y ) = (1 + xy )Γ − ( y )˜ Γ + ( x ) , Γ + ( x )˜ Γ − ( y ) = (1 + xy )˜ Γ − ( y )Γ + ( x ) , 1 Γ + ( x ) ψ ( z ) = 1 − xz ψ ( z )Γ + ( x ) , Γ + ( x ) ψ ∗ ( w ) = (1 − xw ) ψ ∗ ( w )Γ + ( x ) , 1 Γ − ( y ) ψ ( z ) = ψ ( z )Γ − ( y ) , 1 − y z Γ − ( y ) ψ ∗ ( w ) = (1 − y w ) ψ ∗ ( w )Γ − ( y ) , Γ + ( x ) ψ ( z ) = (1 + xz ) ψ ( z )˜ ˜ Γ + ( x ) , 1 ˜ Γ + ( x ) ψ ∗ ( w ) = 1 + xw ψ ∗ ( w )˜ Γ + ( x ) , Γ − ( y ) ψ ( z ) = (1 + y ˜ z ) ψ ( z )˜ Γ − ( y ) , 1 ˜ Γ − ( y ) ψ ∗ ( w ) = ψ ∗ ( w )˜ Γ − ( y ) . 1 + y w

Other stuff: “skew pyramid partitions” Figure : Skew pyramid partitions: word ( ≺ , ≺ ′ ) 50 , ( ≻ , ≻ ′ ) 50 , ( ≺ , ≺ ′ ) 50 , ( ≻ , ≻ ′ ) 50 , q = 0 . 99. The analogue in pyramid partition land of OR 2006’s skew plane partitions. Vertical cusps should have Pearcey fluctuations.

Other stuff: symmetric “pyramid partitions”

Symmetric “pyramid partitions” as plane overpartitions This limit shape seems to be the same that Vuleti´ c 2009 analyzed in the context of strict plane partitions.