Empirical distribution along geodesics in exponential last passage - PowerPoint PPT Presentation

Empirical distribution along geodesics in exponential last passage percolation Lingfu Zhang (Joint work with Allan Sly) Princeton University Department of Mathematics Jun 12, 2020 Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

Exactly solvable LPP: model and main results Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

v u We study the directed last passage percolation (LPP) on Z 2 . ξ ( v ) ∼ Exp( 1 ) , i.i.d. ∀ v ∈ Z 2 � Passage time: X u , v := max γ w ∈ γ ξ ( w ) � Geodesic: Γ u , v := argmax γ w ∈ γ ξ ( w ) Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

v u We study the directed last passage percolation (LPP) on Z 2 . ξ ( v ) ∼ Exp( 1 ) , i.i.d. ∀ v ∈ Z 2 � Passage time: X u , v := max γ w ∈ γ ξ ( w ) � Geodesic: Γ u , v := argmax γ w ∈ γ ξ ( w ) Equivalent to TASEP , exactly solvable with 1 : 2 : 3 scaling. Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

Exactly solvable in the KPZ universality class. Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

Exactly solvable in the KPZ universality class. X ( 0 , 0 ) , ( n , n ) ∼ 4 n (Rost, 1981). Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

Exactly solvable in the KPZ universality class. X ( 0 , 0 ) , ( n , n ) ∼ 4 n (Rost, 1981). 2 − 4 / 3 n − 1 / 3 ( X ( 0 , 0 ) , ( n , n ) − 4 n ) converges weakly to the GUE Tracy-Widom distribution (Johansson, 2000). Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

Exactly solvable in the KPZ universality class. X ( 0 , 0 ) , ( n , n ) ∼ 4 n (Rost, 1981). 2 − 4 / 3 n − 1 / 3 ( X ( 0 , 0 ) , ( n , n ) − 4 n ) converges weakly to the GUE Tracy-Widom distribution (Johansson, 2000). Point to line profile: stationary Airy 2 process minus a parabola (Borodin and Ferrari, 2008) � � 2 − 4 / 3 n − 1 / 3 ⇒ A 2 ( x ) − x 2 X ( 0 , 0 ) , ( n − x ( 2 n ) 2 / 3 , n + x ( 2 n ) 2 / 3 ) − 4 n Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

Exactly solvable in the KPZ universality class. X ( 0 , 0 ) , ( n , n ) ∼ 4 n (Rost, 1981). 2 − 4 / 3 n − 1 / 3 ( X ( 0 , 0 ) , ( n , n ) − 4 n ) converges weakly to the GUE Tracy-Widom distribution (Johansson, 2000). Point to line profile: stationary Airy 2 process minus a parabola (Borodin and Ferrari, 2008) � � 2 − 4 / 3 n − 1 / 3 ⇒ A 2 ( x ) − x 2 X ( 0 , 0 ) , ( n − x ( 2 n ) 2 / 3 , n + x ( 2 n ) 2 / 3 ) − 4 n A 2 is absolute continuous with respect to Brownian motion (Corwin and Hammond, 2014). Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

Exactly solvable in the KPZ universality class. X ( 0 , 0 ) , ( n , n ) ∼ 4 n (Rost, 1981). 2 − 4 / 3 n − 1 / 3 ( X ( 0 , 0 ) , ( n , n ) − 4 n ) converges weakly to the GUE Tracy-Widom distribution (Johansson, 2000). Point to line profile: stationary Airy 2 process minus a parabola (Borodin and Ferrari, 2008) � � 2 − 4 / 3 n − 1 / 3 ⇒ A 2 ( x ) − x 2 X ( 0 , 0 ) , ( n − x ( 2 n ) 2 / 3 , n + x ( 2 n ) 2 / 3 ) − 4 n A 2 is absolute continuous with respect to Brownian motion (Corwin and Hammond, 2014). General initial data: KPZ fixed point (Matetski, Quastel, and Remenik, 2017). � � x �→ n − 1 / 3 sup y f ( y ) + X ( − y , y ) , ( n − x ( 2 n ) 2 / 3 , n + x ( 2 n ) 2 / 3 ) − 4 n Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

We study the local behavior along geodesics. Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

We study the local behavior along geodesics. ( n , n ) v ( 0 , 0 ) Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

We study the local behavior along geodesics. ( n , n ) v v ( 0 , 0 ) Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

We study the local behavior along geodesics. ( n , n ) v v ( 0 , 0 ) ξ s ( v ) := { ξ ( u ) } u ∈ Z 2 : � u − v � ∞ ≤ s ∈ R ( 2 s + 1 ) 2 , s ∈ Z ≥ 0 Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

We study the local behavior along geodesics. ( n , n ) v v ( 0 , 0 ) ξ s ( v ) := { ξ ( u ) } u ∈ Z 2 : � u − v � ∞ ≤ s ∈ R ( 2 s + 1 ) 2 , s ∈ Z ≥ 0 1 � Empirical measure µ n , s := v ∈ Γ ( 0 , 0 ) , ( n , n ) δ ξ s ( v ) 2 n Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

Main result ξ s ( v ) := { ξ ( u ) } u ∈ Z 2 : � u − v � ∞ ≤ s ∈ R ( 2 s + 1 ) 2 , s ∈ Z ≥ 0 1 Empirical measure µ n , s := � v ∈ Γ ( 0 , 0 ) , ( n , n ) δ ξ s ( v ) 2 n Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

Main result ξ s ( v ) := { ξ ( u ) } u ∈ Z 2 : � u − v � ∞ ≤ s ∈ R ( 2 s + 1 ) 2 , s ∈ Z ≥ 0 1 Empirical measure µ n , s := � v ∈ Γ ( 0 , 0 ) , ( n , n ) δ ξ s ( v ) 2 n Question: limiting behavior of µ n , s as n → ∞ ? (First asked for the first passage percolation (FPP) model (e.g. AimPL, 2015)) Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

Main result ξ s ( v ) := { ξ ( u ) } u ∈ Z 2 : � u − v � ∞ ≤ s ∈ R ( 2 s + 1 ) 2 , s ∈ Z ≥ 0 1 Empirical measure µ n , s := � v ∈ Γ ( 0 , 0 ) , ( n , n ) δ ξ s ( v ) 2 n Question: limiting behavior of µ n , s as n → ∞ ? (First asked for the first passage percolation (FPP) model (e.g. AimPL, 2015)) Theorem (Sly and Z., 2020) For each s ∈ Z ≥ 0 , there exists a (deterministic) measure µ s on R ( 2 s + 1 ) 2 , such that µ n , s → µ s weakly in probability as n → ∞ . Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

Ingredients of the proof Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

General idea Weights along the geodesic are asymptotically i.i.d. Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

General idea Weights along the geodesic are asymptotically i.i.d. x + y = β n ( n , n ) v β x + y = α n v α ( 0 , 0 ) Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

General idea Weights along the geodesic are asymptotically i.i.d. x + y = β n ( n , n ) v β x + y = α n v α ( 0 , 0 ) Find some Ψ n , s , s.t. ∀ α, β , as n → ∞ , the joint law of ξ s ( v α ) , ξ s ( v β ) is close to Ψ n , s × Ψ n , s . Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

General idea Weights along the geodesic are asymptotically i.i.d. x + y = β n ( n , n ) v β x + y = α n v α ( 0 , 0 ) Find some Ψ n , s , s.t. ∀ α, β , as n → ∞ , the joint law of ξ s ( v α ) , ξ s ( v β ) is close to Ψ n , s × Ψ n , s . Ψ n , s converges as n → ∞ . Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

Mostly depends on a strip x + y = β n ( n , n ) S β v β x + y = α n S α v α ( 0 , 0 ) Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

Mostly depends on a strip x + y = β n ( n , n ) S β v β x + y = α n S α v α ( 0 , 0 ) Conditioned on ξ ( v ) for v �∈ S α , the law of ξ s ( v α ) is close to Ψ n , s , ∀ α . Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

Mostly depends on a strip x + y = β n ( n , n ) S β v β x + y = α n S α v α ( 0 , 0 ) Conditioned on ξ ( v ) for v �∈ S α , the law of ξ s ( v α ) is close to Ψ n , s , ∀ α . S α being disjoint from S β = ⇒ asymptotic independence. Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

A closer look at the strip ( n , n ) L + L − ( 0 , 0 ) Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

A closer look at the strip ( n , n ) L + L − ( 0 , 0 ) Take L − , L + being δ n away from x + y = α n . Consider the passage times from ( 0 , 0 ) to L − and from ( n , n ) to L + : H − , H + . X ( 0 , 0 ) , ( n , n ) = max u ∈ L − , w ∈ L + X u , w + H − ( u ) + H + ( w ) . Geodesic between L − and L + : Γ H − , H + . Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

A closer look at the strip ( n , n ) L + L + L − H + / B + L − H − / B − ( 0 , 0 ) Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

A closer look at the strip ( n , n ) L + L + L − H + / B + L − H − / B − ( 0 , 0 ) Conditioned on ξ ( v ) for v �∈ S α , H − , H + are locally Brownian. Around argmax H − + H + , (with rescaling) the law of H − , H + is close to B − , B + , where B − + B + is 3D-Bessel and B − − B + is Brownian motion. Using KPZ fixed point formulae. Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

A closer look at the strip ( n , n ) L + L + L − H + / B + L − H − / B − ( 0 , 0 ) Replace H − , H + by B − , B + . With high prob Γ B − , B + largely overlaps with Γ H − , H + . With high prob v α = Γ H − , H + ∩ { x + y = α n } = Γ B − , B + ∩ { x + y = α n } . Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

Convergence of Ψ n , s Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

Convergence of Ψ n , s Cover geodesics by length ∼ m geodesics, for large fixed m . ( n , n ) ( 0 , 0 ) Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

Convergence of Ψ n , s Cover geodesics by length ∼ m geodesics, for large fixed m . ( n , n ) ( 0 , 0 ) Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020