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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,

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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

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Exactly solvable LPP: model and main results

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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u v

We study the directed last passage percolation (LPP) on Z2. ξ(v) ∼ Exp(1), i.i.d. ∀v ∈ Z2 Passage time: Xu,v := maxγ

w∈γ ξ(w)

Geodesic: Γu,v := argmaxγ

w∈γ ξ(w)

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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u v

We study the directed last passage percolation (LPP) on Z2. ξ(v) ∼ Exp(1), i.i.d. ∀v ∈ Z2 Passage time: Xu,v := maxγ

w∈γ ξ(w)

Geodesic: Γu,v := argmaxγ

w∈γ ξ(w)

Equivalent to TASEP , exactly solvable with 1 : 2 : 3 scaling.

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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Exactly solvable in the KPZ universality class.

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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Exactly solvable in the KPZ universality class. X(0,0),(n,n) ∼ 4n (Rost, 1981).

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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Exactly solvable in the KPZ universality class. X(0,0),(n,n) ∼ 4n (Rost, 1981). 2−4/3n−1/3(X(0,0),(n,n) − 4n) converges weakly to the GUE Tracy-Widom distribution (Johansson, 2000). Point to line profile: stationary Airy2 process minus a parabola (Borodin and Ferrari, 2008) 2−4/3n−1/3

X(0,0),(n−x(2n)2/3,n+x(2n)2/3) − 4n
⇒ A2(x) − x2

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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Exactly solvable in the KPZ universality class. X(0,0),(n,n) ∼ 4n (Rost, 1981). 2−4/3n−1/3(X(0,0),(n,n) − 4n) converges weakly to the GUE Tracy-Widom distribution (Johansson, 2000). Point to line profile: stationary Airy2 process minus a parabola (Borodin and Ferrari, 2008) 2−4/3n−1/3

X(0,0),(n−x(2n)2/3,n+x(2n)2/3) − 4n
⇒ A2(x) − x2

A2 is absolute continuous with respect to Brownian motion (Corwin and Hammond, 2014).

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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Exactly solvable in the KPZ universality class. X(0,0),(n,n) ∼ 4n (Rost, 1981). 2−4/3n−1/3(X(0,0),(n,n) − 4n) converges weakly to the GUE Tracy-Widom distribution (Johansson, 2000). Point to line profile: stationary Airy2 process minus a parabola (Borodin and Ferrari, 2008) 2−4/3n−1/3

X(0,0),(n−x(2n)2/3,n+x(2n)2/3) − 4n
⇒ A2(x) − x2

A2 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

y f(y) + X(−y,y),(n−x(2n)2/3,n+x(2n)2/3) − 4n

Lingfu Zhang

Princeton LPP empirical distribution Jun 12, 2020

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We study the local behavior along geodesics.

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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We study the local behavior along geodesics.

(0, 0) (n, n) v Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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We study the local behavior along geodesics.

(0, 0) (n, n) v

v

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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We study the local behavior along geodesics.

(0, 0) (n, n) v

v ξs(v) := {ξ(u)}u∈Z2:u−v∞≤s ∈ R(2s+1)2, s ∈ Z≥0

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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We study the local behavior along geodesics.

(0, 0) (n, n) v

v ξs(v) := {ξ(u)}u∈Z2:u−v∞≤s ∈ R(2s+1)2, s ∈ Z≥0 Empirical measure µn,s :=

1 2n

v∈Γ(0,0),(n,n) δξs(v)

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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Main result ξs(v) := {ξ(u)}u∈Z2:u−v∞≤s ∈ R(2s+1)2, s ∈ Z≥0 Empirical measure µn,s :=

1 2n

v∈Γ(0,0),(n,n) δξs(v)

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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Main result ξs(v) := {ξ(u)}u∈Z2:u−v∞≤s ∈ R(2s+1)2, s ∈ Z≥0 Empirical measure µn,s :=

1 2n

v∈Γ(0,0),(n,n) δξs(v)

Question: limiting behavior of µn,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

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Main result ξs(v) := {ξ(u)}u∈Z2:u−v∞≤s ∈ R(2s+1)2, s ∈ Z≥0 Empirical measure µn,s :=

1 2n

v∈Γ(0,0),(n,n) δξs(v)

Question: limiting behavior of µn,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 µs on R(2s+1)2, such that µn,s → µs weakly in probability as n → ∞.

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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Ingredients of the proof

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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General idea Weights along the geodesic are asymptotically i.i.d.

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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General idea Weights along the geodesic are asymptotically i.i.d.

(0, 0) (n, n) x + y = βn x + y = αn vβ vα Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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General idea Weights along the geodesic are asymptotically i.i.d.

(0, 0) (n, n) x + y = βn x + y = αn vβ vα

Find some Ψn,s, s.t. ∀α, β, as n → ∞, the joint law of ξs(vα), ξs(vβ) is close to Ψn,s × Ψn,s.

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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General idea Weights along the geodesic are asymptotically i.i.d.

(0, 0) (n, n) x + y = βn x + y = αn vβ vα

Find some Ψn,s, s.t. ∀α, β, as n → ∞, the joint law of ξs(vα), ξs(vβ) is close to Ψn,s × Ψn,s. Ψn,s converges as n → ∞.

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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Mostly depends on a strip

(0, 0) (n, n) x + y = βn Sα Sβ x + y = αn vβ vα Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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Mostly depends on a strip

(0, 0) (n, n) x + y = βn Sα Sβ x + y = αn vβ vα

Conditioned on ξ(v) for v ∈ Sα, the law of ξs(vα) is close to Ψn,s,∀α.

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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Mostly depends on a strip

(0, 0) (n, n) x + y = βn Sα Sβ x + y = αn vβ vα

Conditioned on ξ(v) for v ∈ Sα, the law of ξs(vα) is close to Ψn,s,∀α. Sα being disjoint from Sβ = ⇒ asymptotic independence.

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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A closer look at the strip

(0, 0) (n, n) L+ L− Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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A closer look at the strip

(0, 0) (n, n) L+ L−

Take L−, L+ being δn away from x + y = αn. Consider the passage times from (0, 0) to L− and from (n, n) to L+: H−, H+. X(0,0),(n,n) = maxu∈L−,w∈L+ Xu,w + H−(u) + H+(w). Geodesic between L− and L+: ΓH−,H+.

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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A closer look at the strip

(0, 0) (n, n) L+ L−

L− L+ H−/B− H+/B+

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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A closer look at the strip

(0, 0) (n, n) L+ L−

L− L+ H−/B− H+/B+

Conditioned on ξ(v) for v ∈ Sα, 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

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A closer look at the strip

(0, 0) (n, n) L+ L−

L− L+ H−/B− H+/B+

Replace H−, H+ by B−, B+. With high prob ΓB−,B+ largely overlaps with ΓH−,H+. With high prob vα = ΓH−,H+ ∩ {x + y = αn} = ΓB−,B+ ∩ {x + y = αn}.

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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Convergence of Ψn,s

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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Convergence of Ψn,s Cover geodesics by length ∼ m geodesics, for large fixed m.

(0, 0) (n, n) Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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Convergence of Ψn,s Cover geodesics by length ∼ m geodesics, for large fixed m.

(0, 0) (n, n) Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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Convergence of Ψn,s Cover geodesics by length ∼ m geodesics, for large fixed m.

(0, 0) (n, n)

A 1 − ǫ portion of vertices in Γ(0,0),(n,n) are covered = ⇒ Ψn,s is close to Ψm,s.

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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Thank you!

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020

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AimPL. (2015). First passage percolation. [available at http://aimpl.org/firstpercolation].

Borodin, A., & Ferrari, P . (2008). Large time asymptotics of growth models on space-like paths I: PushASEP. Electron. J. Probab., 13, 1380–1418. Corwin, I., & Hammond, A. (2014). Brownian Gibbs property for Airy line ensembles.

Invent. Math., 195(2), 441–508.

Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys., 209(2), 437–476. Matetski, K., Quastel, J., & Remenik, D. (2017). The KPZ fixed point [arXiv preprint arXiv:1701.00018]. arXiv preprint arXiv:1701.00018. Rost, H. (1981). Non-equilibrium behaviour of a many particle process: Density profile and local equilibria. Zeitschrift f. Warsch. Verw. Gebiete, 58(1), 41–53. Sly, A., & Z., L. (2020). Empirical distribution along geodesics in exponential last pas- sage percolation.

Lingfu Zhang Princeton LPP empirical distribution Jun 12, 2020