Central Limit Theorem for discrete loggases Vadim Gorin MIT - PowerPoint PPT Presentation

Central Limit Theorem for discrete log–gases Vadim Gorin MIT (Cambridge) and IITP (Moscow) (based on joint work with Alexei Borodin and Alice Guionnet) April, 2015

Setup and overview λ 1 ≤ λ 2 ≤ · · · ≤ λ N , ℓ i = λ i + θ i Probability distributions on discrete N –tuples of the form. N 1 Γ( ℓ j − ℓ i + 1)Γ( ℓ j − ℓ i + θ ) � � w ( ℓ i ; N ) , Z Γ( ℓ j − ℓ i )Γ( ℓ j − ℓ i + 1 − θ ) 1 ≤ i < j ≤ N i =1 Discrete log–gas. We go beyond specific integrable weights.

Setup and overview λ 1 ≤ λ 2 ≤ · · · ≤ λ N , ℓ i = λ i + θ i Probability distributions on discrete N –tuples of the form. N 1 Γ( ℓ j − ℓ i + 1)Γ( ℓ j − ℓ i + θ ) � � w ( ℓ i ; N ) , Z Γ( ℓ j − ℓ i )Γ( ℓ j − ℓ i + 1 − θ ) 1 ≤ i < j ≤ N i =1 Discrete log–gas. We go beyond specific integrable weights. • Appearance in probabilistic models of statistical mechanics. • Law of Large Numbers and Central Limit Theorem for global fluctuations as N → ∞ under mild assumptions on w ( x ; N ) . • Our main tool: discrete loop equations .

Appearance of discrete log–gases N 1 Γ( ℓ j − ℓ i + 1)Γ( ℓ j − ℓ i + θ ) � � w ( ℓ i ; N ) , Z Γ( ℓ j − ℓ i )Γ( ℓ j − ℓ i + 1 − θ ) 1 ≤ i < j ≤ N i =1 At θ = 1 becomes...

Appearance of discrete log–gases N 1 Γ( ℓ j − ℓ i + 1)Γ( ℓ j − ℓ i + θ ) � � w ( ℓ i ; N ) , Z Γ( ℓ j − ℓ i )Γ( ℓ j − ℓ i + 1 − θ ) 1 ≤ i < j ≤ N i =1 At θ = 1 becomes... N 1 � ( ℓ j − ℓ i ) 2 � w ( ℓ i ; N ) , Z 1 ≤ i < j ≤ N i =1 which frequently appears in natural stochastic systems. E.g.

Krawtchouk ensemble empty • N independent simple 7 random walks 6 • probability of jump p 5 4 • started at adjacent lattice 3 points 2 • conditioned never to 1 collide 0 time

Krawtchouk ensemble empty • N independent simple 7 random walks 6 • probability of jump p 5 4 • started at adjacent lattice 3 points 2 • conditioned never to 1 collide 0 t = 7 Claim. (Konig–O’Connel–Roch) Distribution of N walkers at time t N 1 � � M �� � ( ℓ j − ℓ i ) 2 � p ℓ i (1 − p ) M − ℓ i , M = N + t − 1 . ℓ i Z 1 ≤ i < j ≤ N i =1

Krawtchouk ensemble 7 6 5 4 3 2 1 0 t = 7 Claim. (Konig–O’Connel–Roch) Distribution of N walkers at time t N 1 � � M �� � ( ℓ j − ℓ i ) 2 � p ℓ i (1 − p ) M − ℓ i , M = N + t − 1 . ℓ i Z 1 ≤ i < j ≤ N i =1 Claim. (Johansson) In random domino tilings of Aztec diamond.

Krawtchouk ensemble 7 6 5 4 3 2 1 0 t = 7 Claim. (Konig–O’Connel–Roch) Distribution of N walkers at time t N 1 � � M �� � ( ℓ j − ℓ i ) 2 � p ℓ i (1 − p ) M − ℓ i , M = N + t − 1 . ℓ i Z 1 ≤ i < j ≤ N i =1 Claim. (Johansson) In random domino tilings of Aztec diamond.

Hahn ensemble • Regular A × B × C hexagon B • 3 types of lozenges C A

Hahn ensemble • Regular A × B × C hexagon B • 3 types of lozenges C A • uniformly random tiling

Hahn ensemble • Regular A × B × C hexagon B • uniformly random tiling C • Distribution of N horizontal lozenges on t –th vertical? A t

Hahn ensemble • Regular A × B × C hexagon B • uniformly random tiling C • Distribution of N horizontal lozenges on t –th vertical? A N = B + C − t t > max( B , C ) ( a ) n = a ( a +1) . . . ( a + n − 1) t Claim. (Cohn–Larsen–Propp) N 1 � � � ( ℓ i − ℓ j ) 2 � ( A + B + C + 1 − t − ℓ i ) t − B ( ℓ i ) t − C Z i < j i =1

Two–interval support • Regular A × B × C hexagon • Rhombic hole of size D at vertical position H .

Two–interval support • Regular A × B × C hexagon • Rhombic hole of size D at vertical position H . • uniformly random tiling

Two–interval support • Regular A × B × C hexagon • Rhombic hole of size D at vertical position H . • uniformly random tiling • Distribution of N horizontal lozenges on the vertical going through the axis of the hole?

Two–interval support • Regular A × B × C hexagon • Rhombic hole of size D at vertical position H . • uniformly random tiling • Distribution of N horizontal lozenges on the vertical going through the axis of the hole? Claim. It is: (and similarly for k holes) N � � � � ( ℓ i − ℓ j ) 2 ( A + B + C +1 − t − ℓ i ) t − B ( ℓ i ) t − C ( H − ℓ i ) D ( H − ℓ i ) D i < j i =1

General θ case N 1 Γ( ℓ j − ℓ i + 1)Γ( ℓ j − ℓ i + θ ) � � w ( ℓ i ; N ) , Γ( ℓ j − ℓ i )Γ( ℓ j − ℓ i + 1 − θ ) Z 1 ≤ i < j ≤ N i =1 • ℓ i = L · x i , L → ∞ , β = 2 θ . N 1 � � ( x j − x i ) β w ( ℓ i ; N ) . Z 1 ≤ i < j ≤ N i =1 Eigenvalue ensembles of random matrix theory . β = 1 , 2 , 4 corresponds to real/complex/quaternion matrices.

General θ case N 1 Γ( ℓ j − ℓ i + 1)Γ( ℓ j − ℓ i + θ ) � � w ( ℓ i ; N ) , Γ( ℓ j − ℓ i )Γ( ℓ j − ℓ i + 1 − θ ) Z 1 ≤ i < j ≤ N i =1 • ℓ i = L · x i , L → ∞ , β = 2 θ . N 1 � � ( x j − x i ) β w ( ℓ i ; N ) . Z 1 ≤ i < j ≤ N i =1 Eigenvalue ensembles of random matrix theory . β = 1 , 2 , 4 corresponds to real/complex/quaternion matrices. • Another appearance — asymptotic representation theory (Olshanski: (z,w)-measures). Factor Γ( ℓ j − ℓ i +1)Γ( ℓ j − ℓ i + θ ) Γ( ℓ j − ℓ i )Γ( ℓ j − ℓ i +1 − θ ) links to evaluation formulas for Jack symmetric polynomials.

Large N setup N 1 Γ( ℓ j − ℓ i + 1)Γ( ℓ j − ℓ i + θ ) � � w ( ℓ i ; N ) , Z Γ( ℓ j − ℓ i )Γ( ℓ j − ℓ i + 1 − θ ) 1 ≤ i < j ≤ N i =1 k regions with prescribed filling fractions a 1 b 1 a k b k . . . n 1 particles n k particles 1. w ( · ; N ) vanishes at the boundaries of the regions. 2. All data regularly depends on N → ∞

Large N setup N 1 Γ( ℓ j − ℓ i + 1)Γ( ℓ j − ℓ i + θ ) � � w ( ℓ i ; N ) , Z Γ( ℓ j − ℓ i )Γ( ℓ j − ℓ i + 1 − θ ) 1 ≤ i < j ≤ N i =1 k regions with prescribed filling fractions a 1 b 1 a k b k . . . n 1 particles n k particles 1. w ( · ; N ) vanishes at the boundaries of the regions. 2. All data regularly depends on N → ∞ a i = α i N + . . . , b i = β i N + . . . , n i = ˆ n i N + . . . � x � �� w ( x ; N ) = exp , NV N ( z ) = NV ( z ) + . . . NV N N Potential V ( z ) should have bounded derivative (except at end–points, where we allow V ( z ) ≈ c · z ln( z ) ).

Law of Large Numbers N 1 Γ( ℓ j − ℓ i + 1)Γ( ℓ j − ℓ i + θ ) � � w ( ℓ i ; N ) , Z Γ( ℓ j − ℓ i )Γ( ℓ j − ℓ i + 1 − θ ) 1 ≤ i < j ≤ N i =1 Theorem. Suppose that all data regularly depends on N → ∞ , then the LLN holds: There exists µ ( x ) dx with 0 ≤ µ ( x ) ≤ θ − 1 , such that for any Lipshitz f and any ε > 0 � N � 1 � ℓ i � � � � N →∞ N 1 / 2 − ε � lim − f ( x ) µ ( x ) dx � = 0 f � � N N � � � i =1 In fact the difference is O (1 / N ) .

Law of Large Numbers N 1 Γ( ℓ j − ℓ i + 1)Γ( ℓ j − ℓ i + θ ) � � w ( ℓ i ; N ) , Z Γ( ℓ j − ℓ i )Γ( ℓ j − ℓ i + 1 − θ ) 1 ≤ i < j ≤ N i =1 Theorem. Suppose that all data regularly depends on N → ∞ , then the LLN holds: There exists µ ( x ) dx with 0 ≤ µ ( x ) ≤ θ − 1 , such that for any Lipshitz f and any ε > 0 � N � � ℓ i � 1 � � � � N →∞ N 1 / 2 − ε lim f − f ( x ) µ ( x ) dx � = 0 � � � N N � � i =1 µ ( x ) dx is the unique maximizer of the functional I V � ∞ �� I V [ ρ ] = θ ln | x − y | ρ ( dx ) ρ ( dy ) − V ( x ) ρ ( dx ) . x � = y −∞ in appropriate class of measures taking into account filling fractions

Law of Large Numbers N 1 Γ( ℓ j − ℓ i + 1)Γ( ℓ j − ℓ i + θ ) � � w ( ℓ i ; N ) , Z Γ( ℓ j − ℓ i )Γ( ℓ j − ℓ i + 1 − θ ) 1 ≤ i < j ≤ N i =1 Theorem. Suppose that all data regularly depends on N → ∞ , then the LLN holds: There exists µ ( x ) dx with 0 ≤ µ ( x ) ≤ θ − 1 , such that for any Lipshitz f and any ε > 0 � N � 1 � ℓ i � � � � N →∞ N 1 / 2 − ε � lim − f ( x ) µ ( x ) dx � = 0 f � � N N � � � i =1 µ ( x ) dx is the unique maximizer of the functional I V � ∞ �� I V [ ρ ] = θ ln | x − y | ρ ( dx ) ρ ( dy ) − V ( x ) ρ ( dx ) . x � = y −∞ This is a very general statement. Lots of analogues.

Law of Large Numbers: example (Pictures by L. Petrov)

Law of Large Numbers: example Graph of λ i = ℓ i − i (green lozenges) along the middle vertical empty

Law of Large Numbers: example (Pictures by L. Petrov)

Law of Large Numbers: example Graph of λ i = ℓ i − i (green lozenges) along the vertical axis of hole empty The filling fractions above and below the hole are fixed .

Law of Large Numbers: example Averaged λ i = ℓ i − i (green lozenges) along the vertical axis of hole • Frozen region: void. No particles, µ ( x ) = 0 . • Frozen region: saturation. Dense packing, µ ( x ) = θ − 1 . • Band.