Limit shapes ICERM, April 13-17, 2015 Limit shapes for interacting particle systems and their universal fluctuations P.L. Ferrari in collaboration with A. Borodin http://wt.iam.uni-bonn.de/ ∼ ferrari
An example of random tilings 1 An Aztec diamond of size N = 240 Introduction Scaling theory 2d seq dynamics 2d par dynamics Random tilings Interacting particles
An example of random tilings 2 The border of the four regular facets, as the size N → ∞ : has a circular limit shape (aka arctic circle) Jockush, Propp, Shor’98 the fluctuations of border of the four facets are O ( N 1 / 3 ) and (GUE) Tracy-Widom distributed As a process, it converges to the Airy 2 process on the ( N 2 / 3 , N 1 / 3 ) scale Johansson’03 Introduction Scaling theory 2d seq dynamics 2d par dynamics Random tilings Interacting particles
An example of interacting particle system 3 TASEP: Totally Asymmetric Simple Exclusion Process Configurations � 1 , if j is occupied , η = { η j } j ∈ Z , η j = 0 , if j is empty . Dynamics Independently, particles jump on the right site with rate 1 , provided the right is empty. Introduction Scaling theory 2d seq dynamics 2d par dynamics Random tilings Interacting particles
An example of interacting particle system 4 ⇒ Particles are ordered: position of particle n is x n ( t ) Step initial condition is x n (0) = − n , n ≥ 1 . Introduction Scaling theory 2d seq dynamics 2d par dynamics Random tilings Interacting particles
An example of interacting particle system 4 ⇒ Particles are ordered: position of particle n is x n ( t ) Step initial condition is x n (0) = − n , n ≥ 1 . Introduction Scaling theory 2d seq dynamics 2d par dynamics Random tilings Interacting particles
An example of interacting particle system 4 ⇒ Particles are ordered: position of particle n is x n ( t ) Step initial condition is x n (0) = − n , n ≥ 1 . Introduction Scaling theory 2d seq dynamics 2d par dynamics Random tilings Interacting particles
An example of interacting particle system 4 ⇒ Particles are ordered: position of particle n is x n ( t ) Step initial condition is x n (0) = − n , n ≥ 1 . Introduction Scaling theory 2d seq dynamics 2d par dynamics Random tilings Interacting particles
An example of interacting particle system 5 ⇒ Particles are ordered: position of particle n is x n ( t ) Step initial condition is x n (0) = − n , n ≥ 1 . Some known asymptotic results: law of large number: lim t →∞ x ηt ( t ) /t = 1 − 2 √ η , η ∈ [ − 1 , 1] Rost’81 the fluctuations of particles are O ( t 1 / 3 ) and (GUE) Tracy-Widom distributed As a process in n , it converges to the Airy 2 process on the ( t 2 / 3 , t 1 / 3 ) scale Johansson’03 (LPP); Borodin,Ferrari’07 (TASEP) Introduction Scaling theory 2d seq dynamics 2d par dynamics Random tilings Interacting particles
KPZ scaling theory 6 Given a height function of a model in the Kardar-Parisi-Zhang universality class in one-dimension: x �→ h ( x, t ) (example: n �→ x n ( t ) ) Deterministic limit shape h ma ( ξ ) = lim t →∞ h ( ξt, t ) /t Stationary spatial diffusivity lim t →∞ Var( h ( ξt, t ) − h ( ξt + x, t )) A = lim | x | x →∞ Define further Γ = | λ | A 2 λ = h ′′ ma ( ξ ) and Rescaled process ( u ) := h ( ξt + ut 2 / 3 , t ) − th ma ( ξ + ut − 1 / 3 ) h resc . t t 1 / 3 Introduction Scaling theory 2d seq dynamics 2d par dynamics
KPZ scaling theory 7 Rescaled process ( u ) := h ( ξt + ut 2 / 3 , t ) − th ma ( ξ + ut − 1 / 3 ) h resc . t t 1 / 3 If h ′′ ma ( ξ ) � = 0 , one expects the following: � Au � t →∞ h resc ( u ) = sgn( λ )(Γ / 2) 1 / 3 A 2 lim t 2Γ 2 / 3 where A 2 is the Airy 2 process Pr¨ ahofer,Spohn’02 For flat interfaces (i.e., if h ′′ ( ξ ) = 0 ) one has similar formulas but with either the Airy 1 process Sasamoto’05; Borodin,Ferrari,Pr¨ ahofer,Sasamoto’06 or the Airy stat depending on the initial conditions Baik,Ferrari,P´ ech´ e’09 Introduction Scaling theory 2d seq dynamics 2d par dynamics
Interacting particles and random tilings 8 With Alexei Borodin, in Anisotropic growth of random surfaces in 2+1 dimensions (arXiv:0804.3035), we introduced and studied a model of interacting particles in 2 + 1 -dimensions In discrete time, we have either parallel update or sequential update A discrete time parallel update includes (as different space-time projections) the Aztec diamond and the discrete time TASEP simultaneously Introduction Scaling theory 2d seq dynamics 2d par dynamics Example Basics MC on GT N Properties
A 2 + 1 -dimensional model - building bricks 9 The state space of our model is the Gelfand-Tsetlin pattern GT N = { X N = ( x 1 , . . . , x N ); x n = ( x n n ) | x n ≺ x n +1 , ∀ n } 1 , . . . , x n where x n ≺ x n +1 ⇔ x n +1 < x n 1 ≤ x n +1 < x n 2 ≤ . . . < x n n ≤ x n +1 1 2 n +1 means that x n and x n +1 interlace. x n is the called configuration at level n Introduction Scaling theory 2d seq dynamics 2d par dynamics Example Basics MC on GT N Properties
A 2 + 1 -dimensional model - building bricks 10 The Markov chain at level n (discrete time) is given by x n 1 , . . . , x n n being one-sided random walk conditioned to stay forever in W n = { x n ∈ Z n | x n 1 < x n 2 < . . . < x n n } . It is the Doob h-transform of the free walk with h function the Vandermonde determinant ∆ n ( x n ) = � ( x n j − x n i ) , 1 ≤ i<j ≤ n i.e., it has the one-time transition probability given by P n ( x n , y n ) = ∆ n ( y n ) ∆ n ( x n ) det( P ( x n i , y n j )) n i,j =1 with P ( x, y ) = pδ y,x +1 + (1 − p ) δ y,x . Introduction Scaling theory 2d seq dynamics 2d par dynamics Example Basics MC on GT N Properties
A 2 + 1 -dimensional model - building bricks 11 The chain at fixed time t is the one that, given x N , it generates the uniform measure on the interlacing configurations: Λ n n − 1 ( x n , x n − 1 ) : = P ( x n − 1 | x n ) = #GT n − 1 with given x n − 1 ✶ x n − 1 ≺ x n #GT n with given x n = ( n − 1)!∆ n − 1 ( x n − 1 ) ✶ x n − 1 ≺ x n ∆ n ( x n ) Introduction Scaling theory 2d seq dynamics 2d par dynamics Example Basics MC on GT N Properties
A 2 + 1 -dimensional model - building bricks 12 The key property used below is the intertwining property of the chains: Diaconis, Fill ’90 ∆ n n − 1 := P n Λ n n − 1 = Λ n n − 1 P n − 1 Introduction Scaling theory 2d seq dynamics 2d par dynamics Example Basics MC on GT N Properties
A 2 + 1 -dimensional model - sequential update 13 The sequential update is the following: x 1 ( t ) → x 1 ( t + 1) according to P 1 ( x 1 ( t ) , x 1 ( t + 1)) , 1 x 2 ( t ) → x 2 ( t + 1) to be the middle point of the chain 2 ( P 2 ◦ Λ 2 1 )( x 2 ( t ) , x 1 ( t + 1)) and so on 3 Projection on { x 1 1 , x 2 1 , . . . , x N 1 } is TASEP in discrete time with sequential update Introduction Scaling theory 2d seq dynamics 2d par dynamics Example Basics MC on GT N Properties
A 2 + 1 -dimensional model - conserved measures 14 There is a class of measure which form is invariant under P N Λ . Let µ N ( x N ) be a probability measure on W N and define M N ( X N ) := µ N ( x N )Λ N N − 1 ( x N , x N − 1 ) · · · Λ 2 1 ( x 2 , x 1 ) . Then, applying t times P N Λ we have ( M N ( P N Λ ) t )( Y N ) = ( µ N ( P N ) t )( y N )Λ N N − 1 ( y N , y N − 1 ) · · · Λ 2 1 ( y 2 , y 1 ) This is a consequence of the intertwining properties of the Markov chains! Introduction Scaling theory 2d seq dynamics 2d par dynamics Example Basics MC on GT N Properties
✶ A 2 + 1 -dimensional model - conserved measures 15 Consider further the ”packed” initial condition: x n k (0) = − n + k , 1 ≤ k ≤ n ≤ N . One can see that it can be written as above with µ N of the form µ N ( x N ) = ∆ N ( x N ) det(Ψ j ( x N i , 0)) N i,j =1 . Introduction Scaling theory 2d seq dynamics 2d par dynamics Example Basics MC on GT N Properties
A 2 + 1 -dimensional model - conserved measures 15 Consider further the ”packed” initial condition: x n k (0) = − n + k , 1 ≤ k ≤ n ≤ N . One can see that it can be written as above with µ N of the form µ N ( x N ) = ∆ N ( x N ) det(Ψ j ( x N i , 0)) N i,j =1 . ⇒ The measure at time t has the form N − 1 � ✶ [ x n ≺ x n +1 ] det(Ψ j ( x N i , t )) N i,j =1 n =1 ⇒ The measure at fixed level N and times t 1 < . . . < t m has the form m − 1 � det(Ψ j ( x N i ( t 1 ) , t 1 )) N det( P t k ,t k +1 ( x N ( t k ) , x N ( t k +1 ))∆ N ( x N ( t m )) i,j =1 k =1 Measure of this form have determinantal correlations as they are conditional L -ensembles Borodin, Rains’06 Introduction Scaling theory 2d seq dynamics 2d par dynamics Example Basics MC on GT N Properties
A 2 + 1 -dimensional model - correlations 16 Correlation structure of the blue lozenges / particles Theorem (arXiv:0804.3035) Consider any N triples ( x j , n j , t j ) such that t 1 ≤ t 2 ≤ . . . ≤ t N , n 1 ≥ n 2 ≥ . . . ≥ n N . Then, � at each ( x j , n j , t j ) , j = 1 , . . . , N, P � there exists a blue lozenge / particle = det[ K ( x i , n i , t i ; x j , n j , t j )] 1 ≤ i,j ≤ N for an explicit kernel K . Introduction Scaling theory 2d seq dynamics 2d par dynamics Example Basics MC on GT N Properties
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