Random intefaces, geodesics and the directed landscape B alint Vir - - PowerPoint PPT Presentation

Random intefaces, geodesics and the directed landscape

B´ alint Vir´ ag, University of Toronto

with Duncan Dauvergne, Janosch Ortmann and Mihai Nica Palac Bedlewo May 21, 2019

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Longest increasing subsequences

Ln(·): a LIS of a uniform permutation of 1..n

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Longest increasing subsequences

Ln(·): a LIS of a uniform permutation of 1..n

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Longest increasing subsequences

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Longest increasing subsequences

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Longest increasing subsequences

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Longest increasing subsequences

Can we describe the limiting fluctuations? No known formulas! Still: yes. This talk is about how.

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Longest increasing subsequences

Can we describe the limiting fluctuations? No known formulas! Still: yes. This talk is about how.

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Longest increasing subsequences

Theorem (Shape of LIS)

Ln(⌊2t√n⌋) = nt + n1/3(γ(t) + on(t)) In some realization, with E maxt |on(t)| → 0. γ is a random continuous function [0, 1] → R. The directed geodesic.

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The directed geodesic γ

Is perhaps Brownian bridge? No, γ is Holder-2/3−. |γ(t) − γ(t + ǫ)| ≤ C ǫ2/3 log1/3 1/ǫ ∀t, ǫ P(maxt γ(t) > m) ≤ cam3, a < 1 γ is the scaling limit of geometric, Poisson, Brownian last passage percolation paths γ is the scaling limit of TASEP second class particles γ is conjectured to be the limit of all KPZ polymers γ is a geodesic in a “random metric space”

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The directed geodesic γ

Is perhaps Brownian bridge? No, γ is Holder-2/3−. |γ(t) − γ(t + ǫ)| ≤ C ǫ2/3 log1/3 1/ǫ ∀t, ǫ P(maxt γ(t) > m) ≤ cam3, a < 1 γ is the scaling limit of geometric, Poisson, Brownian last passage percolation paths γ is the scaling limit of TASEP second class particles γ is conjectured to be the limit of all KPZ polymers γ is a geodesic in a “random metric space”

B´ alint Vir´ ag Directed landscape 5/21/2019 9 / 45

The directed geodesic γ

Is perhaps Brownian bridge? No, γ is Holder-2/3−. |γ(t) − γ(t + ǫ)| ≤ C ǫ2/3 log1/3 1/ǫ ∀t, ǫ P(maxt γ(t) > m) ≤ cam3, a < 1 γ is the scaling limit of geometric, Poisson, Brownian last passage percolation paths γ is the scaling limit of TASEP second class particles γ is conjectured to be the limit of all KPZ polymers γ is a geodesic in a “random metric space”

B´ alint Vir´ ag Directed landscape 5/21/2019 9 / 45

The directed geodesic γ

Is perhaps Brownian bridge? No, γ is Holder-2/3−. |γ(t) − γ(t + ǫ)| ≤ C ǫ2/3 log1/3 1/ǫ ∀t, ǫ P(maxt γ(t) > m) ≤ cam3, a < 1 γ is the scaling limit of geometric, Poisson, Brownian last passage percolation paths γ is the scaling limit of TASEP second class particles γ is conjectured to be the limit of all KPZ polymers γ is a geodesic in a “random metric space”

B´ alint Vir´ ag Directed landscape 5/21/2019 9 / 45

The directed geodesic γ

Is perhaps Brownian bridge? No, γ is Holder-2/3−. |γ(t) − γ(t + ǫ)| ≤ C ǫ2/3 log1/3 1/ǫ ∀t, ǫ P(maxt γ(t) > m) ≤ cam3, a < 1 γ is the scaling limit of geometric, Poisson, Brownian last passage percolation paths γ is the scaling limit of TASEP second class particles γ is conjectured to be the limit of all KPZ polymers γ is a geodesic in a “random metric space”

B´ alint Vir´ ag Directed landscape 5/21/2019 9 / 45

The directed geodesic γ

Is perhaps Brownian bridge? No, γ is Holder-2/3−. |γ(t) − γ(t + ǫ)| ≤ C ǫ2/3 log1/3 1/ǫ ∀t, ǫ P(maxt γ(t) > m) ≤ cam3, a < 1 γ is the scaling limit of geometric, Poisson, Brownian last passage percolation paths γ is the scaling limit of TASEP second class particles γ is conjectured to be the limit of all KPZ polymers γ is a geodesic in a “random metric space”

B´ alint Vir´ ag Directed landscape 5/21/2019 9 / 45

The directed geodesic γ

Is perhaps Brownian bridge? No, γ is Holder-2/3−. |γ(t) − γ(t + ǫ)| ≤ C ǫ2/3 log1/3 1/ǫ ∀t, ǫ P(maxt γ(t) > m) ≤ cam3, a < 1 γ is the scaling limit of geometric, Poisson, Brownian last passage percolation paths γ is the scaling limit of TASEP second class particles γ is conjectured to be the limit of all KPZ polymers γ is a geodesic in a “random metric space”

B´ alint Vir´ ag Directed landscape 5/21/2019 9 / 45

The directed geodesic γ

Is perhaps Brownian bridge? No, γ is Holder-2/3−. |γ(t) − γ(t + ǫ)| ≤ C ǫ2/3 log1/3 1/ǫ ∀t, ǫ P(maxt γ(t) > m) ≤ cam3, a < 1 γ is the scaling limit of geometric, Poisson, Brownian last passage percolation paths γ is the scaling limit of TASEP second class particles γ is conjectured to be the limit of all KPZ polymers γ is a geodesic in a “random metric space”

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What is KPZ universality?

Kardar-Parisi-Zhang (1980) A set of models that “behave similarly”. 1,2,3 appear in exponents, Tracy-Widom distribution. Longest increasing subsequences Directed polymers, stochastic heat equation Last passage problems Growing interfaces

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What is KPZ universality?

Kardar-Parisi-Zhang (1980) A set of models that “behave similarly”. 1,2,3 appear in exponents, Tracy-Widom distribution. Longest increasing subsequences Directed polymers, stochastic heat equation Last passage problems Growing interfaces

B´ alint Vir´ ag Directed landscape 5/21/2019 10 / 45

What is KPZ universality?

Kardar-Parisi-Zhang (1980) A set of models that “behave similarly”. 1,2,3 appear in exponents, Tracy-Widom distribution. Longest increasing subsequences Directed polymers, stochastic heat equation Last passage problems Growing interfaces

B´ alint Vir´ ag Directed landscape 5/21/2019 10 / 45

What is KPZ universality?

Kardar-Parisi-Zhang (1980) A set of models that “behave similarly”. 1,2,3 appear in exponents, Tracy-Widom distribution. Longest increasing subsequences Directed polymers, stochastic heat equation Last passage problems Growing interfaces

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Silde by I. Corwin – thanks!

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Slide by I. Corwin – thanks!

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Tetris is KPZ

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Particle systems - TASEP

particles move to the right free sites at rate 1 height function

• Wavelike. second class particles = peaks

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Particle systems - TASEP

particles move to the right free sites at rate 1 height function

• Wavelike. second class particles = peaks

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Particle systems - TASEP

particles move to the right free sites at rate 1 height function

• Wavelike. second class particles = peaks

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Poisson last passage percolation

L(x → y) = 3 L(y → z) = 2 L(x → z) = maxy L(x → y) + L(y → z)

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Poisson last passage percolation

notation R4

↑ = {(x, s; y, t) : s < t},

L : R4

↑ → N

metric composition s < t < u L(x, s; z, u) = maxy L(x, s; y, t) + L(y, t; z, u) L is a “directed metric” :( wrong sign, asymmetry :) triangle inequality, geodesics Perelman’s L-distance

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Poisson last passage percolation

notation R4

↑ = {(x, s; y, t) : s < t},

L : R4

↑ → N

metric composition s < t < u L(x, s; z, u) = maxy L(x, s; y, t) + L(y, t; z, u) L is a “directed metric” :( wrong sign, asymmetry :) triangle inequality, geodesics Perelman’s L-distance

B´ alint Vir´ ag Directed landscape 5/21/2019 16 / 45

Poisson last passage percolation

notation R4

↑ = {(x, s; y, t) : s < t},

L : R4

↑ → N

metric composition s < t < u L(x, s; z, u) = maxy L(x, s; y, t) + L(y, t; z, u) L is a “directed metric” :( wrong sign, asymmetry :) triangle inequality, geodesics Perelman’s L-distance

B´ alint Vir´ ag Directed landscape 5/21/2019 16 / 45

Poisson last passage percolation

notation R4

↑ = {(x, s; y, t) : s < t},

L : R4

↑ → N

metric composition s < t < u L(x, s; z, u) = maxy L(x, s; y, t) + L(y, t; z, u) L is a “directed metric” :( wrong sign, asymmetry :) triangle inequality, geodesics Perelman’s L-distance

B´ alint Vir´ ag Directed landscape 5/21/2019 16 / 45

Poisson last passage percolation

notation R4

↑ = {(x, s; y, t) : s < t},

L : R4

↑ → N

metric composition s < t < u L(x, s; z, u) = maxy L(x, s; y, t) + L(y, t; z, u) L is a “directed metric” :( wrong sign, asymmetry :) triangle inequality, geodesics Perelman’s L-distance

B´ alint Vir´ ag Directed landscape 5/21/2019 16 / 45

Poisson last passage percolation

notation R4

↑ = {(x, s; y, t) : s < t},

L : R4

↑ → N

metric composition s < t < u L(x, s; z, u) = maxy L(x, s; y, t) + L(y, t; z, u) L is a “directed metric” :( wrong sign, asymmetry :) triangle inequality, geodesics Perelman’s L-distance

B´ alint Vir´ ag Directed landscape 5/21/2019 16 / 45

The metric composition semigroup M

Elements: a, b : R2 → R ∪ {+∞}. Composition: a ⋆ b(x, z) = sup

y a(x, y) + b(y, z).

Example: Fix t. at(x, y) := −(x, 0) − (y, t)2 Then at ⋆ as = as+t. Works for any path metric on R2

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The metric composition semigroup M

Elements: a, b : R2 → R ∪ {+∞}. Composition: a ⋆ b(x, z) = sup

y a(x, y) + b(y, z).

Example: Fix t. at(x, y) := −(x, 0) − (y, t)2 Then at ⋆ as = as+t. Works for any path metric on R2

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The metric composition semigroup M

Elements: a, b : R2 → R ∪ {+∞}. Composition: a ⋆ b(x, z) = sup

y a(x, y) + b(y, z).

Example: Fix t. at(x, y) := −(x, 0) − (y, t)2 Then at ⋆ as = as+t. Works for any path metric on R2

B´ alint Vir´ ag Directed landscape 5/21/2019 17 / 45

The metric composition semigroup M

Elements: a, b : R2 → R ∪ {+∞}. Composition: a ⋆ b(x, z) = sup

y a(x, y) + b(y, z).

Example: Fix t. at(x, y) := −(x, 0) − (y, t)2 Then at ⋆ as = as+t. Works for any path metric on R2

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Poisson last passage, algebraically

M = ({a : R2 → R ∪ {+∞}}, ⋆). Poisson LP is a stationary independent increment process on M. Random walk on M (Levy process) In semigroups, two time parameters are needed to document increments Ls,t = L(·, s; ·, t). Groups, no: Xt − Xs = (Xt − X0)−(Xs − X0)

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Poisson last passage, algebraically

M = ({a : R2 → R ∪ {+∞}}, ⋆). Poisson LP is a stationary independent increment process on M. Random walk on M (Levy process) In semigroups, two time parameters are needed to document increments Ls,t = L(·, s; ·, t). Groups, no: Xt − Xs = (Xt − X0)−(Xs − X0)

B´ alint Vir´ ag Directed landscape 5/21/2019 18 / 45

Poisson last passage, algebraically

M = ({a : R2 → R ∪ {+∞}}, ⋆). Poisson LP is a stationary independent increment process on M. Random walk on M (Levy process) In semigroups, two time parameters are needed to document increments Ls,t = L(·, s; ·, t). Groups, no: Xt − Xs = (Xt − X0)−(Xs − X0)

B´ alint Vir´ ag Directed landscape 5/21/2019 18 / 45

Poisson last passage, algebraically

M = ({a : R2 → R ∪ {+∞}}, ⋆). Poisson LP is a stationary independent increment process on M. Random walk on M (Levy process) In semigroups, two time parameters are needed to document increments Ls,t = L(·, s; ·, t). Groups, no: Xt − Xs = (Xt − X0)−(Xs − X0)

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Poisson last passage, algebraically

M = ({a : R2 → R ∪ {+∞}}, ⋆). Poisson LP is random walk on M What is Brownian motion on M? What is a Gaussian on M? “BM” is the directed landscape “Gaussian” is the Airy sheet sheet → landscape: Levy’s construction

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Poisson last passage, algebraically

M = ({a : R2 → R ∪ {+∞}}, ⋆). Poisson LP is random walk on M What is Brownian motion on M? What is a Gaussian on M? “BM” is the directed landscape “Gaussian” is the Airy sheet sheet → landscape: Levy’s construction

B´ alint Vir´ ag Directed landscape 5/21/2019 19 / 45

Poisson last passage, algebraically

M = ({a : R2 → R ∪ {+∞}}, ⋆). Poisson LP is random walk on M What is Brownian motion on M? What is a Gaussian on M? “BM” is the directed landscape “Gaussian” is the Airy sheet sheet → landscape: Levy’s construction

B´ alint Vir´ ag Directed landscape 5/21/2019 19 / 45

Poisson last passage, algebraically

M = ({a : R2 → R ∪ {+∞}}, ⋆). Poisson LP is random walk on M What is Brownian motion on M? What is a Gaussian on M? “BM” is the directed landscape “Gaussian” is the Airy sheet sheet → landscape: Levy’s construction

B´ alint Vir´ ag Directed landscape 5/21/2019 19 / 45

Poisson last passage, algebraically

M = ({a : R2 → R ∪ {+∞}}, ⋆). Poisson LP is random walk on M What is Brownian motion on M? What is a Gaussian on M? “BM” is the directed landscape “Gaussian” is the Airy sheet sheet → landscape: Levy’s construction

B´ alint Vir´ ag Directed landscape 5/21/2019 19 / 45

Poisson last passage, algebraically

M = ({a : R2 → R ∪ {+∞}}, ⋆). Poisson LP is random walk on M What is Brownian motion on M? What is a Gaussian on M? “BM” is the directed landscape “Gaussian” is the Airy sheet sheet → landscape: Levy’s construction

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Last passage across functions

Definition

For a sequence of functions fk and n ≤ m, s < t define the last passage value as f [(s, m) → (t, n)] = sup

π

• k∈Z:π−1(k)o=(u,v)

fk(v) − fk(u)

• ver nonincreasing π : [s, t] → Z with π(s) = m,

π(t) = n.

k

→ for k nonintersecting paths.

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Last passage across functions

Definition

For a sequence of functions fk and n ≤ m, s < t define the last passage value as f [(s, m) → (t, n)] = sup

π

• k∈Z:π−1(k)o=(u,v)

fk(v) − fk(u)

• ver nonincreasing π : [s, t] → Z with π(s) = m,

π(t) = n.

k

→ for k nonintersecting paths.

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Brownian last passage

π : [s, t] → Z nonincreasing, π(s) = 4, π(t) = 1 f are BMs some symmetry lost, some gained

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RSK: the melon Wf of f

Fix n, fk : R+ → R, k ∈ {1, . . . , n}, Define the melon Wf by Wf1(y) + . . . + Wfk(y) = f [(0, n)

k

→ (y, 1)] One of two parts of a continuous RSK Key property: for all x < y f [(x, n) → (y, 1)] = Wf [(x, n) → (y, 1)].

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O’Connell-Yor, 2002

W applied to B1, . . . , Bn. WB has the law of n Brownian motions conditioned not to intersect. Proof, essentially: RSK is a bijection Pre-limit Airy sheet: (x, y) → WB[(x, n) → (y, 1)].

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O’Connell-Yor, 2002

W applied to B1, . . . , Bn. WB has the law of n Brownian motions conditioned not to intersect. Proof, essentially: RSK is a bijection Pre-limit Airy sheet: (x, y) → WB[(x, n) → (y, 1)].

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O’Connell-Yor, 2002

W applied to B1, . . . , Bn. WB has the law of n Brownian motions conditioned not to intersect. Proof, essentially: RSK is a bijection Pre-limit Airy sheet: (x, y) → WB[(x, n) → (y, 1)].

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The Brownian melon

The melon of Brownian paths Non-intersecting BM Eigenvalues of Hermitian BM Dyson’s BM Warren process marginal The Airy line ensemble is the limit of the top.

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The Brownian melon

The melon of Brownian paths Non-intersecting BM Eigenvalues of Hermitian BM Dyson’s BM Warren process marginal The Airy line ensemble is the limit of the top.

B´ alint Vir´ ag Directed landscape 5/21/2019 24 / 45

The Brownian melon

The melon of Brownian paths Non-intersecting BM Eigenvalues of Hermitian BM Dyson’s BM Warren process marginal The Airy line ensemble is the limit of the top.

B´ alint Vir´ ag Directed landscape 5/21/2019 24 / 45

The Brownian melon

The melon of Brownian paths Non-intersecting BM Eigenvalues of Hermitian BM Dyson’s BM Warren process marginal The Airy line ensemble is the limit of the top.

B´ alint Vir´ ag Directed landscape 5/21/2019 24 / 45

The Brownian melon

The melon of Brownian paths Non-intersecting BM Eigenvalues of Hermitian BM Dyson’s BM Warren process marginal The Airy line ensemble is the limit of the top.

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The Airy line ensemble

A·(t) + t2/4 is stationary Ak(0) ∼ −(3πk/2)2/3

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The Airy line ensemble

A·(t) + t2/4 is stationary Ak(0) ∼ −(3πk/2)2/3

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The Airy line ensemble

Theorem

Let WB be a Brownian n-melon. Define the rescaled melon by An

i (t) = n1/6(WBi(1 + tn−1/3) − 2√n − tn1/6).

Then An converges in law to a random sequence functions A, the Airy line ensemble. Proof: formulas + tightness. (Prahofer-Spohn, Adler-Van Moerbeke, Corwin-Hammond).

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The Airy line ensemble

Theorem

Let WB be a Brownian n-melon. Define the rescaled melon by An

i (t) = n1/6(WBi(1 + tn−1/3) − 2√n − tn1/6).

Then An converges in law to a random sequence functions A, the Airy line ensemble. Proof: formulas + tightness. (Prahofer-Spohn, Adler-Van Moerbeke, Corwin-Hammond).

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Defintion of the Airy sheet S

A random continuous function so that S has the same law as S(· + z, · + z). S can be coupled with an Airy line ensemble so that S(0, ·) = A1(·) and for all (x, y, z) ∈ Q+ × Q2 a.s. S(x, z) − S(x, y) = A[(−

• k/2x, k) → (z, 1)]−

A[(−

• k/2x, k) → (y, 1)]

for all large enough k.

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Defintion of the Airy sheet S

A random continuous function so that S has the same law as S(· + z, · + z). S can be coupled with an Airy line ensemble so that S(0, ·) = A1(·) and for all (x, y, z) ∈ Q+ × Q2 a.s. S(x, z) − S(x, y) = A[(−

• k/2x, k) → (z, 1)]−

A[(−

• k/2x, k) → (y, 1)]

for all large enough k.

B´ alint Vir´ ag Directed landscape 5/21/2019 27 / 45

Defintion of the Airy sheet S

A random continuous function so that S has the same law as S(· + z, · + z). S can be coupled with an Airy line ensemble so that S(0, ·) = A1(·) and for all (x, y, z) ∈ Q+ × Q2 a.s. S(x, z) − S(x, y) = A[(−

• k/2x, k) → (z, 1)]−

A[(−

• k/2x, k) → (y, 1)]

for all large enough k.

B´ alint Vir´ ag Directed landscape 5/21/2019 27 / 45

Defintion of the Airy sheet S

A random continuous function so that S has the same law as S(· + z, · + z). S can be coupled with an Airy line ensemble so that S(0, ·) = A1(·) and for all (x, y, z) ∈ Q+ × Q2 a.s. S(x, z) − S(x, y) = A[(−

• k/2x, k) → (z, 1)]−

A[(−

• k/2x, k) → (y, 1)]

for all large enough k.

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Hello, I am the Airy sheet

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Hello, in a different dress

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Airy sheet

Theorem

For every n, there exists a coupling so that B[(2x/n1/3, n) → (1 + 2y/n1/3, 1)] = 2√n + (y − x)n1/6 + n−1/6(S + on)(x, y),

• n every compact K ⊂ R2 there exists a > 1 with

EasupK |on|3/2 → 1.

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Properties of the Airy sheet

S(x, y) + (x − y)2 has Tracy-Widom law. y → S(x, x + y) are Airy-2 processes. x, y-swap symmetry Skew symmetry: S(x, y) + (x − y)2 is shift-invariant in R2! Quadrangle inequality Local sum structure S(x, y) is the CDF of the random shock measure!

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Properties of the Airy sheet

S(x, y) + (x − y)2 has Tracy-Widom law. y → S(x, x + y) are Airy-2 processes. x, y-swap symmetry Skew symmetry: S(x, y) + (x − y)2 is shift-invariant in R2! Quadrangle inequality Local sum structure S(x, y) is the CDF of the random shock measure!

B´ alint Vir´ ag Directed landscape 5/21/2019 31 / 45

Properties of the Airy sheet

S(x, y) + (x − y)2 has Tracy-Widom law. y → S(x, x + y) are Airy-2 processes. x, y-swap symmetry Skew symmetry: S(x, y) + (x − y)2 is shift-invariant in R2! Quadrangle inequality Local sum structure S(x, y) is the CDF of the random shock measure!

B´ alint Vir´ ag Directed landscape 5/21/2019 31 / 45

Properties of the Airy sheet

S(x, y) + (x − y)2 has Tracy-Widom law. y → S(x, x + y) are Airy-2 processes. x, y-swap symmetry Skew symmetry: S(x, y) + (x − y)2 is shift-invariant in R2! Quadrangle inequality Local sum structure S(x, y) is the CDF of the random shock measure!

B´ alint Vir´ ag Directed landscape 5/21/2019 31 / 45

Properties of the Airy sheet

S(x, y) + (x − y)2 has Tracy-Widom law. y → S(x, x + y) are Airy-2 processes. x, y-swap symmetry Skew symmetry: S(x, y) + (x − y)2 is shift-invariant in R2! Quadrangle inequality Local sum structure S(x, y) is the CDF of the random shock measure!

B´ alint Vir´ ag Directed landscape 5/21/2019 31 / 45

Properties of the Airy sheet

S(x, y) + (x − y)2 has Tracy-Widom law. y → S(x, x + y) are Airy-2 processes. x, y-swap symmetry Skew symmetry: S(x, y) + (x − y)2 is shift-invariant in R2! Quadrangle inequality Local sum structure S(x, y) is the CDF of the random shock measure!

B´ alint Vir´ ag Directed landscape 5/21/2019 31 / 45

Properties of the Airy sheet

S(x, y) + (x − y)2 has Tracy-Widom law. y → S(x, x + y) are Airy-2 processes. x, y-swap symmetry Skew symmetry: S(x, y) + (x − y)2 is shift-invariant in R2! Quadrangle inequality Local sum structure S(x, y) is the CDF of the random shock measure!

B´ alint Vir´ ag Directed landscape 5/21/2019 31 / 45

The 1-2-3 scaling

Airy sheet of scale s is Ss(x, y) = s S(x/s2, y/s2). Metric composition. r 3 = s3 + t3. Sr(x, z) = max

y∈R Ss(x, y) + St(y, z).

Is the Airy sheet uniquely defined by this property?

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The 1-2-3 scaling

Airy sheet of scale s is Ss(x, y) = s S(x/s2, y/s2). Metric composition. r 3 = s3 + t3. Sr(x, z) = max

y∈R Ss(x, y) + St(y, z).

Is the Airy sheet uniquely defined by this property?

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The directed landscape

The directed landscape is a stationary independent increment process with respect to metric composition. The increments are Airy sheets. L(x, t; y, s) continuous, no technical issues. Increment L(·, t; ·, t + s3): Airy sheet of scale s Increments are independent over disjoint time-intervals.

B´ alint Vir´ ag Directed landscape 5/21/2019 34 / 45

The directed landscape

The directed landscape is a stationary independent increment process with respect to metric composition. The increments are Airy sheets. L(x, t; y, s) continuous, no technical issues. Increment L(·, t; ·, t + s3): Airy sheet of scale s Increments are independent over disjoint time-intervals.

B´ alint Vir´ ag Directed landscape 5/21/2019 34 / 45

The directed landscape

The directed landscape is a stationary independent increment process with respect to metric composition. The increments are Airy sheets. L(x, t; y, s) continuous, no technical issues. Increment L(·, t; ·, t + s3): Airy sheet of scale s Increments are independent over disjoint time-intervals.

B´ alint Vir´ ag Directed landscape 5/21/2019 34 / 45

The directed landscape

The directed landscape is a stationary independent increment process with respect to metric composition. The increments are Airy sheets. L(x, t; y, s) continuous, no technical issues. Increment L(·, t; ·, t + s3): Airy sheet of scale s Increments are independent over disjoint time-intervals.

B´ alint Vir´ ag Directed landscape 5/21/2019 34 / 45

Geodesics

Length of a path is |π|L = inf

k∈N

inf

t=t0<···<tk=s k

• i=1

L(π(ti−1), ti−1; π(ti), ti) Most paths have length −∞ Geodesic if all equalities hold without infs The geodesic between (0, 0) and (0, 1) is γ (i.e. (γ(t), t)). Almost all geodesics are unique! Point pairs with non-unique geodesics exist.

B´ alint Vir´ ag Directed landscape 5/21/2019 35 / 45

Geodesics

Length of a path is |π|L = inf

k∈N

inf

t=t0<···<tk=s k

• i=1

L(π(ti−1), ti−1; π(ti), ti) Most paths have length −∞ Geodesic if all equalities hold without infs The geodesic between (0, 0) and (0, 1) is γ (i.e. (γ(t), t)). Almost all geodesics are unique! Point pairs with non-unique geodesics exist.

B´ alint Vir´ ag Directed landscape 5/21/2019 35 / 45

Geodesics

Length of a path is |π|L = inf

k∈N

inf

t=t0<···<tk=s k

• i=1

L(π(ti−1), ti−1; π(ti), ti) Most paths have length −∞ Geodesic if all equalities hold without infs The geodesic between (0, 0) and (0, 1) is γ (i.e. (γ(t), t)). Almost all geodesics are unique! Point pairs with non-unique geodesics exist.

B´ alint Vir´ ag Directed landscape 5/21/2019 35 / 45

Geodesics

Length of a path is |π|L = inf

k∈N

inf

t=t0<···<tk=s k

• i=1

L(π(ti−1), ti−1; π(ti), ti) Most paths have length −∞ Geodesic if all equalities hold without infs The geodesic between (0, 0) and (0, 1) is γ (i.e. (γ(t), t)). Almost all geodesics are unique! Point pairs with non-unique geodesics exist.

B´ alint Vir´ ag Directed landscape 5/21/2019 35 / 45

Geodesics

Length of a path is |π|L = inf

k∈N

inf

t=t0<···<tk=s k

• i=1

L(π(ti−1), ti−1; π(ti), ti) Most paths have length −∞ Geodesic if all equalities hold without infs The geodesic between (0, 0) and (0, 1) is γ (i.e. (γ(t), t)). Almost all geodesics are unique! Point pairs with non-unique geodesics exist.

B´ alint Vir´ ag Directed landscape 5/21/2019 35 / 45

Geodesic trees

B´ alint Vir´ ag Directed landscape 5/21/2019 36 / 45

Airy sheet

Theorem

For every n, there exists a coupling so that B[(2x/n1/3, n) → (1 + 2y/n1/3, 1)] = 2√n + (y − x)n1/6 + n−1/6(S + on)(x, y),

• n every compact K ⊂ R2 there exists a > 1 with

EasupK |on|3/2 → 1.

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The directed landscape as a limit

Let (x, s)n = (s + 2x/n1/3, −⌊sn⌋), translation between locations.

Theorem

There exists a coupling of Brownian last passage percolation and the directed landcape L so that Bn[(x, s)n → (y, t)n] = 2(t − s)√n + (y − x)n1/6 + n−1/6(L + on)(x, s; y, t). P

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Last passage path as a limit

πn denote an optimizing path for B[(0, n) → (1, 1)].

Theorem

In law, as random functions in the uniform norm πn(s) − n(1 − s) n2/3

d

→ γ(s).

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The limit of TASEP

ht(y) = sup

x∈R

h0(x) + L(x, 0; y, t) Like the variational formula for Burger’s equation. KPZ fixed point. (Matetski-Quastel-Remenik)

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Previous work

Baik-Deift-Johansson (99): TW limit of the longest increasing subsequence Prahofer-Spohn (02): Airy process Corwin-Questel-Remenik (13): Conjectures. Matetski-Quastel-Remenik (17+): “KPZ fixed point”: 2 parameter limit. ht(y) = sup

x∈R

h0(x) + L(x, 0; y, t)

B´ alint Vir´ ag Directed landscape 5/21/2019 41 / 45

Previous work

Baik-Deift-Johansson (99): TW limit of the longest increasing subsequence Prahofer-Spohn (02): Airy process Corwin-Questel-Remenik (13): Conjectures. Matetski-Quastel-Remenik (17+): “KPZ fixed point”: 2 parameter limit. ht(y) = sup

x∈R

h0(x) + L(x, 0; y, t)

B´ alint Vir´ ag Directed landscape 5/21/2019 41 / 45

Previous work

Baik-Deift-Johansson (99): TW limit of the longest increasing subsequence Prahofer-Spohn (02): Airy process Corwin-Questel-Remenik (13): Conjectures. Matetski-Quastel-Remenik (17+): “KPZ fixed point”: 2 parameter limit. ht(y) = sup

x∈R

h0(x) + L(x, 0; y, t)

B´ alint Vir´ ag Directed landscape 5/21/2019 41 / 45

Previous work

Baik-Deift-Johansson (99): TW limit of the longest increasing subsequence Prahofer-Spohn (02): Airy process Corwin-Questel-Remenik (13): Conjectures. Matetski-Quastel-Remenik (17+): “KPZ fixed point”: 2 parameter limit. ht(y) = sup

x∈R

h0(x) + L(x, 0; y, t)

B´ alint Vir´ ag Directed landscape 5/21/2019 41 / 45

Previous work

Baik-Deift-Johansson (99): TW limit of the longest increasing subsequence Prahofer-Spohn (02): Airy process Corwin-Questel-Remenik (13): Conjectures. Matetski-Quastel-Remenik (17+): “KPZ fixed point”: 2 parameter limit. ht(y) = sup

x∈R

h0(x) + L(x, 0; y, t)

B´ alint Vir´ ag Directed landscape 5/21/2019 41 / 45

Previous work II

Random functional wrt. metric composition: g → gS MQR gives marginals for fixed g, used to construct a Markov process. Here: full distribution of functional. Baik-Liu: (17,JAMS 19) Two-parameter function on the cylinder

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Previous work II

Random functional wrt. metric composition: g → gS MQR gives marginals for fixed g, used to construct a Markov process. Here: full distribution of functional. Baik-Liu: (17,JAMS 19) Two-parameter function on the cylinder

B´ alint Vir´ ag Directed landscape 5/21/2019 42 / 45

Previous work II

Random functional wrt. metric composition: g → gS MQR gives marginals for fixed g, used to construct a Markov process. Here: full distribution of functional. Baik-Liu: (17,JAMS 19) Two-parameter function on the cylinder

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What is new?

Directed landscape: 4-parameter function, path limits, second class particle limits (full information). Universality: no new formulas, but existing formulas extend to all models! Example: Baik-Liu formulas have a limit, give distribution for multiple spacetime. Geometric structure. Greek approach. Main interest: phenomena, not formulas.

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What is new?

Directed landscape: 4-parameter function, path limits, second class particle limits (full information). Universality: no new formulas, but existing formulas extend to all models! Example: Baik-Liu formulas have a limit, give distribution for multiple spacetime. Geometric structure. Greek approach. Main interest: phenomena, not formulas.

B´ alint Vir´ ag Directed landscape 5/21/2019 43 / 45

What is new?

Directed landscape: 4-parameter function, path limits, second class particle limits (full information). Universality: no new formulas, but existing formulas extend to all models! Example: Baik-Liu formulas have a limit, give distribution for multiple spacetime. Geometric structure. Greek approach. Main interest: phenomena, not formulas.

B´ alint Vir´ ag Directed landscape 5/21/2019 43 / 45

What is new?

Directed landscape: 4-parameter function, path limits, second class particle limits (full information). Universality: no new formulas, but existing formulas extend to all models! Example: Baik-Liu formulas have a limit, give distribution for multiple spacetime. Geometric structure. Greek approach. Main interest: phenomena, not formulas.

B´ alint Vir´ ag Directed landscape 5/21/2019 43 / 45

Summary

The directed landscape should be the full scaling limit of KPZ models Brownian last passage converges to the directed landscape First to contain all relevant information Geometric structure of prelimiting models

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Summary

The directed landscape should be the full scaling limit of KPZ models Brownian last passage converges to the directed landscape First to contain all relevant information Geometric structure of prelimiting models

B´ alint Vir´ ag Directed landscape 5/21/2019 44 / 45

Summary

The directed landscape should be the full scaling limit of KPZ models Brownian last passage converges to the directed landscape First to contain all relevant information Geometric structure of prelimiting models

B´ alint Vir´ ag Directed landscape 5/21/2019 44 / 45

Summary

The directed landscape should be the full scaling limit of KPZ models Brownian last passage converges to the directed landscape First to contain all relevant information Geometric structure of prelimiting models

B´ alint Vir´ ag Directed landscape 5/21/2019 44 / 45

Where are we? Brownain motion

Bachelier (1900), Einstein (1905) Wiener (1923) first construction But: Cameron-Martin, Feinman-Kac, Wiener chaos, Ito, Malliavin? Expect alternative constructions (Levy)

B´ alint Vir´ ag Directed landscape 5/21/2019 45 / 45

Where are we? Brownain motion

Bachelier (1900), Einstein (1905) Wiener (1923) first construction But: Cameron-Martin, Feinman-Kac, Wiener chaos, Ito, Malliavin? Expect alternative constructions (Levy)

B´ alint Vir´ ag Directed landscape 5/21/2019 45 / 45

Where are we? Brownain motion

Bachelier (1900), Einstein (1905) Wiener (1923) first construction But: Cameron-Martin, Feinman-Kac, Wiener chaos, Ito, Malliavin? Expect alternative constructions (Levy)

B´ alint Vir´ ag Directed landscape 5/21/2019 45 / 45

Where are we? Brownain motion

Bachelier (1900), Einstein (1905) Wiener (1923) first construction But: Cameron-Martin, Feinman-Kac, Wiener chaos, Ito, Malliavin? Expect alternative constructions (Levy)

B´ alint Vir´ ag Directed landscape 5/21/2019 45 / 45