Last Passage Percolation, KPZ, and Competition Interfaces Peter - PowerPoint PPT Presentation

Last Passage Percolation, KPZ, and Competition Interfaces Peter Nejjar avec Patrik Ferrari ENS Paris DMA CIRM 8. 3. 2016

Totally asymmetric simple exclusion process (TASEP) v 1 v 1 v 1 v 2 v 2 -4 -3 -2 -1 0 1 2 Z ◮ Dynamics : particles on Z perform independent jumps to the right subject to the exclusion constraint ◮ We will also consider particle-dependent speeds

Totally asymmetric simple exclusion process (TASEP) v 1 v 1 v 1 v 2 v 2 3 2 1 0 -1 -4 -3 -2 -1 0 1 2 Z ◮ Dynamics : particles on Z perform independent jumps to the right subject to the exclusion constraint ◮ We will also consider particle-dependent speeds We number particles from right to left . . . < x 3 ( 0 ) < x 2 ( 0 ) < x 1 ( 0 ) < 0 ≤ x 0 ( 0 ) < x − 1 ( 0 ) < . . . x k ( t ) = position of particle k at time t

TASEP - a KPZ growth model Set h ( 0 , 0 ) = 0 and � − 1 if x + 1 is occupied at time 0 h ( x + 1 , 0 ) − h ( x , 0 ) = 1 otherwise t = 0 t = 0 -3-2-1 0 1 2 3 -3-2-1 0 1 2 3

TASEP - a KPZ growth model Set h ( 0 , 0 ) = 0 and � − 1 if x + 1 is occupied at time t h ( x + 1 , t ) − h ( x , t ) = 1 otherwise t > 0 t > 0 -3-2-1 0 1 2 3 -3-2-1 0 1 2 3

TASEP - a KPZ growth model Set h ( 0 , 0 ) = 0 and � − 1 if x + 1 is occupied at time t h ( x + 1 , t ) − h ( x , t ) = 1 otherwise t > 0 t > 0 -3-2-1 0 1 2 3 -3-2-1 0 1 2 3 Hydrodynamic theory identifies TASEP as a KPZ model

Flat TASEP and the Airy 1 process TASEP with a flat geometry ( ∂ 2 ξ h ma = 0) for periodic initial data: t = 0 t ≫ 0 For flat TASEP we have [BFPS ’07] in the sense of fin. dim. distr. x t / 4 + ξ t 2 / 3 ( t ) + 2 ξ t 2 / 3 lim = A 1 ( ξ ) , − t 1 / 3 t →∞ with A 1 ( ξ ) the Airy 1 process with one-point distribution given by the F 1 (GOE) Tracy-Widom distribution from random matrix theory.

Shocks ◮ Discontinuities of the particle density are called shocks ρ ( x , 0 ) ρ ( x , t ) t = 0 t > 0 ρ − ρ + ρ − ρ + x x vt ◮ Initial condition: Ber ( ρ + ) on N and Ber ( ρ − ) on Z − . ◮ for ρ − < ρ + there is a shock with speed v = 1 − ( ρ + + ρ − ) ◮ one can identify the microscopic shock with the position Z t of a particle fluctuating around vt : Z t − vt ∼ N ( 0 , µ 2 ) lim ( see Lig ’99 ) t 1 / 2 t →∞

Question: What are the shock fluctuations for non-random initial configuration (IC) ?

Two Speed TASEP with periodic IC t = 0 v 1 = 1 v 2 = α < 1 -4 -3 -2 -1 0 0 1 2 3 4 Z This leads to a wedge limit shape: t = 0 t ≫ 0 -3 -2 -1 0 1 2 3 shock

Shock as particle position ρ ( x , t ) t ≫ 0 ◮ The last slow particle is macroscopically at position 1 − α ( 1 − ρ ) α t = α 2 2 t . 1 2 ◮ Behind it is a jam region A of increased density x α 2 t − 1 + α A t ρ = 1 − α/ 2. 2 ◮ The particle η t , with η = 2 − α is at the macro 4 shock position. Inside the constant density regions, η ′ � = η , the fluctuations of x η ′ t are governed by the F 1 GOE Tracy-Widom distribution and live in the t 1 / 3 scale.

Goal: Determine the large time fluctuations of the (rescaled) particle position x n ( t ) around the shock: � x n ( t ) − vt � lim ≤ s =? t →∞ P t 1 / 3 where vt is the macroscopic position of x n ( t ) . For arbitrary fixed IC, the law of x n ( t ) is given as a Fred- holm determinant of a kernel K t [BFPS ’07], � x n ( t ) − vt � ≤ s t →∞ det ( 1 − χ s K t χ s ) ℓ 2 ( Z ) , t →∞ P lim = lim (1) t 1 / 3

Goal: Determine the large time fluctuations of the (rescaled) particle position x n ( t ) around the shock: � x n ( t ) − vt � lim ≤ s =? t →∞ P t 1 / 3 where vt is the macroscopic position of x n ( t ) . For arbitrary fixed IC, the law of x n ( t ) is given as a Fred- holm determinant of a kernel K t [BFPS ’07], � x n ( t ) − vt � ≤ s t →∞ det ( 1 − χ s K t χ s ) ℓ 2 ( Z ) , t →∞ P lim = lim (1) t 1 / 3 Problem: K t is diverging for our example (but its Fred- holm determinant will still converge), so one cannot ana- lyze (1) directly

Product structure for Two-Speed TASEP Theorem (At the F 1 – F 1 shock, Ferrari, N. ’14) Let x n ( 0 ) = − 2 n for n ∈ Z . For α < 1 let η = 2 − α and 4 v = − 1 − α 2 . Then it holds � 2 ξ � � x η t + ξ t 1 / 3 ( t ) − vt s − � � s − 2 ξ � 2 − α ≤ s = F 1 , t →∞ P lim F 1 t 1 / 3 σ 1 σ 2 2 and σ 2 = α 1 / 3 ( 2 − 2 α + α 2 ) 1 / 3 where σ 1 = 1 . 2 ( 2 − α ) 2 / 3

Product structure for Two-Speed TASEP Theorem (At the F 1 – F 1 shock, Ferrari, N. ’14) Let x n ( 0 ) = − 2 n for n ∈ Z . For α < 1 let η = 2 − α and 4 v = − 1 − α 2 . Then it holds � 2 ξ � � x η t + ξ t 1 / 3 ( t ) − vt s − � � s − 2 ξ � 2 − α ≤ s = F 1 , t →∞ P lim F 1 t 1 / 3 σ 1 σ 2 2 and σ 2 = α 1 / 3 ( 2 − 2 α + α 2 ) 1 / 3 where σ 1 = 1 . 2 ( 2 − α ) 2 / 3 One recovers GOE by changing s → s + 2 ξ and ξ → + ∞ , resp. by s → s + 2 ξ/ ( 2 − α ) and ξ → −∞

TASEP as Last Passage Percolation (LPP) ◮ Let ω i , j , ( i , j ) ∈ Z 2 , be independent weights, L ⊆ Z 2 π : L → ( m , n ) an up-right path ◮ L L→ ( m , n ) = max π � ω i , j ∈ π ω i , j = � ω i , j ∈ π max ω i , j Z L = { ( u , − u ) : u ∈ Z } = L + ∪ L − ( m , n ) π max ω i , j ∼ exp ( 1 ) (white), exp ( α ) (green). L + Z α L −

TASEP as Last Passage Percolation (LPP) ◮ Let ω i , j , ( i , j ) ∈ Z 2 , be independent weights, L ⊆ Z 2 π : L → ( m , n ) an up-right path ◮ L L→ ( m , n ) = max π � ω i , j ∈ π ω i , j = � ω i , j ∈ π max ω i , j � � Link: L L→ ( m , n ) ≤ t = P ( x n ( t ) ≥ m − n ) , P ω i , j ∼ exp ( v j ) 1 ( i , j ) ∈L c , L = { ( k + x k ( 0 ) , k ) : k ∈ Z } Z L = { ( u , − u ) : u ∈ Z } = L + ∪ L − ( m , n ) π max ω i , j ∼ exp ( 1 ) (white), exp ( α ) (green). L + Z α L −

Last Passage Percolation in combinatorics There is a bijection between integer matrices M k � M , N = { A | A = ( a i , j ) 1 ≤ i ≤ M 1 ≤ j ≤ N , a i , j ∈ N 0 , = k } i , j and generalized permutations σ � i 1 i 2 i 3 ··· i k − 1 i k � { σ : σ = , i l ∈ [ N ] , j l ∈ [ M ] , either i l < i l + 1 j 1 j 2 j 3 ··· j k − 1 j k or i l = i l + 1 , j l ≤ j l + 1 } � i r 1 � i rm where [ M ] = { 1 , 2 , . . . M } . Call � · · · � an increasing j r 1 j rm subsequence of length m if r 1 < r 2 < · · · < r m and j 1 ≤ j 2 · · · ≤ j m , and denote ℓ ( σ ) a longest increasing subsequence.

Last Passage Percolation in combinatorics There is a bijection between integer matrices M k � M , N = { A | A = ( a i , j ) 1 ≤ i ≤ M 1 ≤ j ≤ N , a i , j ∈ N 0 , = k } i , j and generalized permutations σ � i 1 i 2 i 3 ··· i k − 1 i k � { σ : σ = , i l ∈ [ N ] , j l ∈ [ M ] , either i l < i l + 1 j 1 j 2 j 3 ··· j k − 1 j k or i l = i l + 1 , j l ≤ j l + 1 } � i r 1 � i rm where [ M ] = { 1 , 2 , . . . M } . Call � · · · � an increasing j r 1 j rm subsequence of length m if r 1 < r 2 < · · · < r m and j 1 ≤ j 2 · · · ≤ j m , and denote ℓ ( σ ) a longest increasing subsequence. If we set ω i , j = a i , j then under the above bijection L { ( 1 , 1 ) }→ ( M , N ) = ℓ ( σ ) .

Generic Theorem ( η 0 t , t ) Z Assume that there exists some µ such that L + � L L + → ( η 0 t , t ) − µ t � ≤ s t →∞ P lim = G 1 ( s ) , t 1 / 3 Z � L L − → ( η 0 t , t ) − µ t � ≤ s t →∞ P lim = G 2 ( s ) . t 1 / 3 L − Theorem (Ferrari, N. ’14) Under some assumptions we have � L L→ ( η 0 t , t ) − µ t � lim ≤ s = G 1 ( s ) G 2 ( s ) , t →∞ P t 1 / 3 where L = L + ∪ L − .

On the assumptions Z ( η 0 t , t ) L + Z L −

On the assumptions Z I. Assume that we have a point E + = ( η 0 t − κ t ν , t − t ν ) such that E + ( η 0 t , t ) for some µ 0 , and ν ∈ ( 1 / 3 , 1 ) it holds L L + → E + − µ t + µ 0 t ν L + → G 1 t 1 / 3 L E + → ( η 0 t , t ) − µ 0 t ν Z → G 0 , t ν/ 3 L −

On the assumptions Z I. Slow Decorrelation E + E + ( η 0 t , t ) L + Z L −

On the assumptions Z I. Slow Decorrelation E + E + ( η 0 t , t ) II. Assume there is a point D = ( η 0 ( t − t β ) , t − t β ) with η 0 t β ≤ D κ t ν such that π max and π max π max cross L + + − + π max − ( 0 , 0 ) D with vanishing probability. Z L −

On the assumptions Z I. Slow Decorrelation E + E + ( η 0 t , t ) II. Assume there is a point D = ( η 0 ( t − t β ) , t − t β ) with η 0 t β ≤ D κ t ν such that π max and π max π max cross L + + − + π max − ( 0 , 0 ) D with vanishing probability. Z L −

On the assumptions Z I. Slow Decorrelation II. No crossing E + E + ( η 0 t , t ) D π max L + + π max − Z L −

Some remarks: ◮ I. is related to the universal phenomenon known as slow decorrelation [CFP ’12] ◮ II. follows if we have that the ’characteristic lines’ of the two LPP problems meet at ( η 0 t , t ) , together with the transversal fluctuations which are only O ( t 2 / 3 ) [Jo ’00] ◮ III. An extension to joint laws � m � � { L L→ ( η t + u k t 1 / 3 , t ) ≤ µ t + s k t 1 / 3 } P k = 1 is available (Ferrari, N. ’16) and based on controling local fluctuations in LPP