Plane partitions with two-periodic weights Sevak Mkrtchyan - PowerPoint PPT Presentation

The model Homogeneous weights Inhomogeneous weights Other regimes Plane partitions with two-periodic weights Sevak Mkrtchyan University of Rochester GGI June 15, 2015

The model Homogeneous weights Plane partitions Inhomogeneous weights Skew Plane partitions Other regimes The Model The Model

The model Homogeneous weights Plane partitions Inhomogeneous weights Skew Plane partitions Other regimes Plane partitions Definition 8 7 7 6 7 5 5 5 5 3 2 4 4 3 2 0 1 2 2 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 Notation: π = { π i , j } i > 0 , j > 0 , confined to an M × N rectangle (here 6 × 6).

The model Homogeneous weights Plane partitions Inhomogeneous weights Skew Plane partitions Other regimes Plane partitions as stacks of cubes 

The model Homogeneous weights Plane partitions Inhomogeneous weights Skew Plane partitions Other regimes Skew plane partitions Definition 6 8 12 4 6 5 11 12 3 2 3 3 4 5 0 1 0 2 2 0 0 0 0 0 0 0 0 0 0 Notation: π λ = { π i , j } , defined for all pairs ( i , j ) not in λ .

The model Homogeneous weights Plane partitions Inhomogeneous weights Skew Plane partitions Other regimes Skew plane partitions as stacks of cubes  

The model Homogeneous weights Plane partitions Inhomogeneous weights Skew Plane partitions Other regimes Plane partitions as dimer configurations

The model Homogeneous weights Plane partitions Inhomogeneous weights Skew Plane partitions Other regimes Plane partitions as dimer configurations

The model Homogeneous weights Plane partitions Inhomogeneous weights Skew Plane partitions Other regimes Plane partitions as dimer configurations

The model Homogeneous weights Plane partitions Inhomogeneous weights Skew Plane partitions Other regimes Plane partitions as dimer configurations

The model The probability measure Homogeneous weights The discrete correlation kernel Inhomogeneous weights The scaling limit Other regimes The Model Homogeneous weights

The model The probability measure Homogeneous weights The discrete correlation kernel Inhomogeneous weights The scaling limit Other regimes The probability measure 6 8 12 4 6 5 11 12 3 2 3 3 4 5 0 1 0 2 2 0 0 0 0 0 0 0 0 0 0 Consider the system consisting of all skew plane partitions with boundary λ , confined to the N × M box, with the distribution Prob ( π ) ∝ q | π | = q volume , for some q ∈ (0 , 1), where | π | = � π i , j is the total volume. q volume ↔ homogeneous weights.

The model The probability measure Homogeneous weights The discrete correlation kernel Inhomogeneous weights The scaling limit Other regimes An instance of the Schur-process Recall the Schur-process introduced by Okounkov-Reshetikhin (2003): ◮ A measure on sequences of Young diagrams { λ ( i ) } i . ◮ Position-dependent transition weights between two Young diagrams: S ( t ) ( λ ( t ) , λ ( t − 1)) . ◮ Schur-process: � S ( t ) ( λ ( t ) , λ ( t − 1)) . Prob ( { λ ( i ) } i ) ∝ t q volume on skew plane partitions is a special case of the Schur-process. Okounkov-Reshetikhin showed that this is a determinantal process and computed the correlation kernel.

The model The probability measure Homogeneous weights The discrete correlation kernel Inhomogeneous weights The scaling limit Other regimes The correlation functions The positions of the horizontal tiles completely determine the (skew) plane partition. To understand the fluctuations, study the local correlation functions of the positions of the horizontal tiles. Denote by ρ (( t 1 , h 1 ) , . . . , ( t n , h n )) the probability that there are horizontal tiles at positions ( t i , h i ) , i = 1 , . . . , n .

The model The probability measure Homogeneous weights The discrete correlation kernel Inhomogeneous weights The scaling limit Other regimes The discrete correlation kernel Theorem (Okounkov-Reshetikhin 2003) ρ (( t 1 , h 1 ) , . . . , ( t n , h n )) = det( K (( t i , h i ) , ( t j , h j )) n i , j =1 . The correlation kernel K (( t 1 , h 1 ) , ( t 2 , h 2 )) is given by: �� Φ − ( z , t 1 )Φ + ( w , t 2 ) √ zw 1 z − w z − h 1 + b λ q ( t 1 ) − 1 / 2 w h 2 − b λ q ( t 2 ) − 1 / 2 dzdw zw , (2 π i ) 2 Φ + ( z , t 1 )Φ − ( w , t 2 ) where � (1 ∓ z ± 1 q ± m ) , Φ ± ( z , t ) = t , m ∈ Z + 1 m > 2 < slope at m is ∓ 1 and b λ q encodes the ”back wall”.

The model The probability measure Homogeneous weights The discrete correlation kernel Inhomogeneous weights The scaling limit Other regimes The scaling limit The thermodynamic limit of the system is when q → 1 − . Write q as q = e − r , and study the limit r → 0 + . Question: How should the parameters N , M and λ change in the limit? Answer for N , M : The typical size of a plane partition not restricted to a finite box is 1 r , so one should study the limit when N and M grow at the rate 1 r , and scale the system by r in all directions. Answer for back wall: Let b λ q ( t ) be the functions giving the back walls. h t 1 2 b Λ � t � Study the limit when after rescaling b λ q ( τ ) converges to a 1-Lipschitz function.

The model The probability measure Homogeneous weights The discrete correlation kernel Inhomogeneous weights The scaling limit Other regimes M. - Skew plane partitions with arbitrary piecewise linear back walls Most general case studied: back wall is a piecewise linear curve of slopes in [ − 1 , 1]. Earlier works on limit shapes by Nienhuis (plane partitions), Kenyon (plane partitions), Okounkov and Reshetikhin (slopes ± 1), Boutilier, M., Reshetikhin and Tingley (slopes in ( − 1 , 1)).

The model The probability measure Homogeneous weights The discrete correlation kernel Inhomogeneous weights The scaling limit Other regimes M. - Skew plane partitions with arbitrary piecewise linear back walls

The model The probability distribution Homogeneous weights Periodic weights Inhomogeneous weights Intermediate regime Other regimes Almost periodic weights The Model Inhomogeneous weights

The model The probability distribution Homogeneous weights Periodic weights Inhomogeneous weights Intermediate regime Other regimes Almost periodic weights The probability distribution Recall the Schur-Process representation: write a skew plane partition π as a sequence { π i } i of its diagonal slices. Given { q i } i ∈ Z , q i > 0, consider the system consisting of all skew plane partitions with boundary λ , with the distribution q | π i | � Prob ( π ) ∝ , i i ∈ Z where | π i | is the total volume of the i -th slice of π .

The model The probability distribution Homogeneous weights Periodic weights Inhomogeneous weights Intermediate regime Other regimes Almost periodic weights Discrete correlation kernel for non-homogeneous weights Theorem (Okounkov-Reshetikhin 2003) ρ (( t 1 , h 1 ) , . . . , ( t n , h n )) = det( K (( t i , h i ) , ( t j , h j )) n i , j =1 . The correlation kernel K (( t 1 , h 1 ) , ( t 2 , h 2 )) is given by: √ zw �� Φ − ( z , t 1 )Φ + ( w , t 2 ) 1 z − w z − h 1 + b λ q ( t 1 ) − 1 / 2 w h 2 − b λ q ( t 2 ) − 1 / 2 dzdw zw , (2 π i ) 2 Φ + ( z , t 1 )Φ − ( w , t 2 ) where � (1 ∓ z ± 1 a ± 1 q ± 1 . . . q ± 1 Φ ± ( z , t ) = 2 ) , 0 m − 1 t , m ∈ Z + 1 m > 2 < slope at m is ∓ 1 and b λ q encodes the ”back wall”. Only change is q m is replaced with aq 0 . . . q m − 1 2 .

The model The probability distribution Homogeneous weights Periodic weights Inhomogeneous weights Intermediate regime Other regimes Almost periodic weights Periodic weights Consider weights with q 0 = q 2 k and q 1 = q 2 k +1 ∀ k ∈ Z . What scaling limit should we study? Nothing new, if you take q 0 → 1 − and q 1 → 1 − . More interesting: α ≥ 1, q 0 = α q , q 1 = α − 1 q and q → 1 − . Obstacle: partition function may be infinite.

The model The probability distribution Homogeneous weights Periodic weights Inhomogeneous weights Intermediate regime Other regimes Almost periodic weights Periodic weights q q q q q Α Α Α Α Α Α q Α q Α q Α q Α q Α q q q q q q q q Α Α Α Α Α Α Α Α q Α q Α q Α q Α q Α q q q q q q Α Α Α Α Α Α q Α q Α q Α q q q q Α Α Α Α q Α q q Α Note, that we have q = e − r and we are scaling by r , thus there are ( V 2 − V 1 ) / r microscopic linear sections between V 1 and V 2 . Hence in order for the measure to be well defined, we must have q ( V 2 − V 1 ) / r α < 1 , or equivalently e − ( V 2 − V 1 ) α < 1 .

The model The probability distribution Homogeneous weights Periodic weights Inhomogeneous weights Intermediate regime Other regimes Almost periodic weights Unbounded floor: Frozen boundary

The model The probability distribution Homogeneous weights Periodic weights Inhomogeneous weights Intermediate regime Other regimes Almost periodic weights Unbounded floor: A sample