[PPT] - Directed Polymers in Random Environment with Heavy Tails A. PowerPoint Presentation

SLIDE 1

Introduction Results Idea of Proof Summary

Directed Polymers in Random Environment with Heavy Tails

A. Auffinger
O. Louidor

Courant (New York University)

Maryland Probability Seminar, 11/16/2009

SLIDE 2

Introduction Results Idea of Proof Summary

Outline

Introduction Results Idea of Proof Summary

SLIDE 3

Introduction Results Idea of Proof Summary

Outline

Introduction Results Idea of Proof Summary

SLIDE 4

Introduction Results Idea of Proof Summary

The Model

Model for the interaction between a polymer chain and

microscopic impurities in the media where it resides.

Environment σ - a random signed measure σ supported on

Z+ × Zd.

Example: σ({x, y)}) ; (x, y) ∈ Z+ × Zd are i.i.d.
Polymer s - a nearest neighbor path (bridge):

s : [0, n] ∩ Z → Zd , s(0) = s(n) = 0, s(x + 1) − s(x) ∈ {−1, +1}.

Interaction - an environment-dependent measure on polymer

paths:

SLIDE 5

Introduction Results Idea of Proof Summary

The Model

Model for the interaction between a polymer chain and

microscopic impurities in the media where it resides.

Environment σ - a random signed measure σ supported on

Z+ × Zd.

Example: σ({x, y)}) ; (x, y) ∈ Z+ × Zd are i.i.d.
Polymer s - a nearest neighbor path (bridge):

s : [0, n] ∩ Z → Zd , s(0) = s(n) = 0, s(x + 1) − s(x) ∈ {−1, +1}.

Interaction - an environment-dependent measure on polymer

paths:

SLIDE 6

Introduction Results Idea of Proof Summary

The Model

Model for the interaction between a polymer chain and

microscopic impurities in the media where it resides.

Environment σ - a random signed measure σ supported on

Z+ × Zd.

Example: σ({x, y)}) ; (x, y) ∈ Z+ × Zd are i.i.d.
Polymer s - a nearest neighbor path (bridge):

s : [0, n] ∩ Z → Zd , s(0) = s(n) = 0, s(x + 1) − s(x) ∈ {−1, +1}.

Interaction - an environment-dependent measure on polymer

paths:

SLIDE 7

Introduction Results Idea of Proof Summary

The Model

Model for the interaction between a polymer chain and

microscopic impurities in the media where it resides.

Environment σ - a random signed measure σ supported on

Z+ × Zd.

Example: σ({x, y)}) ; (x, y) ∈ Z+ × Zd are i.i.d.
Polymer s - a nearest neighbor path (bridge):

s : [0, n] ∩ Z → Zd , s(0) = s(n) = 0, s(x + 1) − s(x) ∈ {−1, +1}.

Interaction - an environment-dependent measure on polymer

paths:

SLIDE 8

Introduction Results Idea of Proof Summary

The Model - cont’d

Explicitly:

µσ

n,β(s) =

1 Qσ

n,β

exp (−βHσ(s)) ; Hσ(s) = −σ(s)

β 0 - inverse temperature, strength of interaction.
Hσ - random Hamiltonian.
σ(s) = σ(graph(s)).
Known as Directed Polymers in Random Environment (DPRE) in

d + 1 dimensions.

Introduced by Huse-Henley (85).

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Introduction Results Idea of Proof Summary

The Model - cont’d

Explicitly:

µσ

n,β(s) =

1 Qσ

n,β

exp (−βHσ(s)) ; Hσ(s) = −σ(s)

β 0 - inverse temperature, strength of interaction.
Hσ - random Hamiltonian.
σ(s) = σ(graph(s)).
Known as Directed Polymers in Random Environment (DPRE) in

d + 1 dimensions.

Introduced by Huse-Henley (85).

SLIDE 10

Introduction Results Idea of Proof Summary

The Model - cont’d

Explicitly:

µσ

n,β(s) =

1 Qσ

n,β

exp (−βHσ(s)) ; Hσ(s) = −σ(s)

β 0 - inverse temperature, strength of interaction.
Hσ - random Hamiltonian.
σ(s) = σ(graph(s)).
Known as Directed Polymers in Random Environment (DPRE) in

d + 1 dimensions.

Introduced by Huse-Henley (85).

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Introduction Results Idea of Proof Summary

The Questions

For β > 0 does s(·) behaves macroscopically differently than a

Simple Random Walk bridge (β = 0)?

Order of fluctuations.
e.g. µσ

n,β(s(n/2)2).

Pinning to certain regions in the environment.
Entropy vs. Energy.
Quenched vs. Annealed Results.

SLIDE 12

Introduction Results Idea of Proof Summary

The Questions

For β > 0 does s(·) behaves macroscopically differently than a

Simple Random Walk bridge (β = 0)?

Order of fluctuations.
e.g. µσ

n,β(s(n/2)2).

Pinning to certain regions in the environment.
Entropy vs. Energy.
Quenched vs. Annealed Results.

SLIDE 13

Introduction Results Idea of Proof Summary

The Questions

For β > 0 does s(·) behaves macroscopically differently than a

Simple Random Walk bridge (β = 0)?

Order of fluctuations.
e.g. µσ

n,β(s(n/2)2).

Pinning to certain regions in the environment.
Entropy vs. Energy.
Quenched vs. Annealed Results.

SLIDE 14

Introduction Results Idea of Proof Summary

The Questions

For β > 0 does s(·) behaves macroscopically differently than a

Simple Random Walk bridge (β = 0)?

Order of fluctuations.
e.g. µσ

n,β(s(n/2)2).

Pinning to certain regions in the environment.
Entropy vs. Energy.
Quenched vs. Annealed Results.

SLIDE 15

Introduction Results Idea of Proof Summary

The Questions

For β > 0 does s(·) behaves macroscopically differently than a

Simple Random Walk bridge (β = 0)?

Order of fluctuations.
e.g. µσ

n,β(s(n/2)2).

Pinning to certain regions in the environment.
Entropy vs. Energy.
Quenched vs. Annealed Results.

SLIDE 16

Introduction Results Idea of Proof Summary

Conjecture

Theorem If d 2 and β > 0, or d > 2 and β ≫ 0 the polymer is super-diffusive with fluctuations of order n2/3. Otherwise, the polymer is diffusive - fluctuations of order n1/2. This is universal w.r.t a large class of distributions, including the i.i.d case with marginals that decay sufficiently fast. High disorder phase vs. low disorder phase.

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Introduction Results Idea of Proof Summary

Conjecture

Theorem If d 2 and β > 0, or d > 2 and β ≫ 0 the polymer is super-diffusive with fluctuations of order n2/3. Otherwise, the polymer is diffusive - fluctuations of order n1/2. This is universal w.r.t a large class of distributions, including the i.i.d case with marginals that decay sufficiently fast. High disorder phase vs. low disorder phase.

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Introduction Results Idea of Proof Summary

Previous Results - Low Disorder Phase

µσ

n,β(s(n/2)2) ∼ Cn1/2 and even s(n/2)/

n/2 ⇒ N(0, I).
P-a.s.
d 3, β ≪ ∞, many distributions.
Imbrie-Spencer (88). Bolthausen (89). Song-Zhou (96).
limn→∞ maxy∈Z d µσ

n,β(s(n/2) = y) = 0.

P-a.s.
d 3, β ≪ ∞, many distributions.
Carmona-Hu (02). Comets-Shiga-Yoshida (03).

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Introduction Results Idea of Proof Summary

Previous Results - Low Disorder Phase

µσ

n,β(s(n/2)2) ∼ Cn1/2 and even s(n/2)/

n/2 ⇒ N(0, I).
P-a.s.
d 3, β ≪ ∞, many distributions.
Imbrie-Spencer (88). Bolthausen (89). Song-Zhou (96).
limn→∞ maxy∈Z d µσ

n,β(s(n/2) = y) = 0.

P-a.s.
d 3, β ≪ ∞, many distributions.
Carmona-Hu (02). Comets-Shiga-Yoshida (03).

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Introduction Results Idea of Proof Summary

Previous Results - High Disorder Phase

inf{ζ > 0 : µσ

n,β(s∞ nζ) → 1 in P-prob.} ∈ [3/(4 + d), 3/4].

d 1. All β. Many distributions.
Not completely rigorous.
Piza (97)
lim supn→∞ maxy∈Z d µσ

n,β(s(n/2) = y) > C.

P-a.s.
C is non-random.
d 2 and β > 0 or d > 2 and β ≫ 0, many distributions.
Carmona-Hu (02). Comets-Shiga-Yoshida (03).

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Introduction Results Idea of Proof Summary

Previous Results - High Disorder Phase

inf{ζ > 0 : µσ

n,β(s∞ nζ) → 1 in P-prob.} ∈ [3/(4 + d), 3/4].

d 1. All β. Many distributions.
Not completely rigorous.
Piza (97)
lim supn→∞ maxy∈Z d µσ

n,β(s(n/2) = y) > C.

P-a.s.
C is non-random.
d 2 and β > 0 or d > 2 and β ≫ 0, many distributions.
Carmona-Hu (02). Comets-Shiga-Yoshida (03).

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Introduction Results Idea of Proof Summary

Last/First Passage Percolation

If β = ∞ then µσ

n,β is the δ measure on the maximal path

sn = arg max σ(s).

(uniform measure on maximal paths).

Model coincides with Last (first) Passage Percolation.
Supper-diffusive behavior and fluctuations of order n2/3 for

s under the same universality class are conjectured as well.

Only partially proved.
Baik-Deift, Johansson, Licea-Newman-Piza, Newman-Pize.

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Introduction Results Idea of Proof Summary

Last/First Passage Percolation

If β = ∞ then µσ

n,β is the δ measure on the maximal path

sn = arg max σ(s).

(uniform measure on maximal paths).

Model coincides with Last (first) Passage Percolation.
Supper-diffusive behavior and fluctuations of order n2/3 for

s under the same universality class are conjectured as well.

Only partially proved.
Baik-Deift, Johansson, Licea-Newman-Piza, Newman-Pize.

SLIDE 24

Introduction Results Idea of Proof Summary

Last/First Passage Percolation

If β = ∞ then µσ

n,β is the δ measure on the maximal path

sn = arg max σ(s).

(uniform measure on maximal paths).

Model coincides with Last (first) Passage Percolation.
Supper-diffusive behavior and fluctuations of order n2/3 for

s under the same universality class are conjectured as well.

Only partially proved.
Baik-Deift, Johansson, Licea-Newman-Piza, Newman-Pize.

SLIDE 25

Introduction Results Idea of Proof Summary

Last/First Passage Percolation

If β = ∞ then µσ

n,β is the δ measure on the maximal path

sn = arg max σ(s).

(uniform measure on maximal paths).

Model coincides with Last (first) Passage Percolation.
Supper-diffusive behavior and fluctuations of order n2/3 for

s under the same universality class are conjectured as well.

Only partially proved.
Baik-Deift, Johansson, Licea-Newman-Piza, Newman-Pize.

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Introduction Results Idea of Proof Summary

Outside the Universality Class

Hambley-Martin (07) investigated LPP in the case of i.i.d.

environment with heavy tails.

Specifically, suppose

Pn(σ({x, y}) > t) = t−αL(t) ; t > 0 (1)

α ∈ [0, 2).
L(t) is slowly varying (L(tu)/L(t) → 1 as t → ∞ for all u > 0.

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Introduction Results Idea of Proof Summary

Outside the Universality Class

Hambley-Martin (07) investigated LPP in the case of i.i.d.

environment with heavy tails.

Specifically, suppose

Pn(σ({x, y}) > t) = t−αL(t) ; t > 0 (1)

α ∈ [0, 2).
L(t) is slowly varying (L(tu)/L(t) → 1 as t → ∞ for all u > 0.

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Introduction Results Idea of Proof Summary

Previous Results - LPP with Heavy Tails

Theorem (Hambley-Martin (07))

1 n ·

sn 1

n

sn(n ·) ⇒ γα as n → ∞ where γα is a non-degenerate process on [0, 1].

Underlying topology is L∞.
The law of

γα, denoted M α has a constructive description.

In particular, this gives fluctuations of order n.

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Introduction Results Idea of Proof Summary

Previous Results - LPP with Heavy Tails

Theorem (Hambley-Martin (07))

1 n ·

sn 1

n

sn(n ·) ⇒ γα as n → ∞ where γα is a non-degenerate process on [0, 1].

Underlying topology is L∞.
The law of

γα, denoted M α has a constructive description.

In particular, this gives fluctuations of order n.

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Introduction Results Idea of Proof Summary

Outline

Introduction Results Idea of Proof Summary

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Introduction Results Idea of Proof Summary

Our Results - Setup

DPRE in the case of i.i.d. environment with heavy tails.
Assume same heavy tail condition (1) with α ∈ (0, 2).
For simplicity we shall assume d = 2, environment distribution

has no atoms.

How does µσ

n,β look like asymptotically?

SLIDE 32

Introduction Results Idea of Proof Summary

Our Results - Setup

DPRE in the case of i.i.d. environment with heavy tails.
Assume same heavy tail condition (1) with α ∈ (0, 2).
For simplicity we shall assume d = 2, environment distribution

has no atoms.

How does µσ

n,β look like asymptotically?

SLIDE 33

Introduction Results Idea of Proof Summary

Our Results - Pinning

Theorem (Auffinger, L (09)) For all β > 0: µσ

n,β

s −

s∞ > δn

→ 0

in P-probability, for all δ > 0.

Pinning to LPP path with margin of order o(n).
Not so interesting.
Scale β with n !.
“Right” scaling is βn = βn−2/α+1L0(n).
L0(n) is another (related) slow varying function.

SLIDE 34

Introduction Results Idea of Proof Summary

Our Results - Pinning

Theorem (Auffinger, L (09)) For all β > 0: µσ

n,β

s −

s∞ > δn

→ 0

in P-probability, for all δ > 0.

Pinning to LPP path with margin of order o(n).
Not so interesting.
Scale β with n !.
“Right” scaling is βn = βn−2/α+1L0(n).
L0(n) is another (related) slow varying function.

SLIDE 35

Introduction Results Idea of Proof Summary

Our Results - Pinning

Theorem (Auffinger, L (09)) For all β > 0: µσ

n,β

s −

s∞ > δn

→ 0

in P-probability, for all δ > 0.

Pinning to LPP path with margin of order o(n).
Not so interesting.
Scale β with n !.
“Right” scaling is βn = βn−2/α+1L0(n).
L0(n) is another (related) slow varying function.

SLIDE 36

Introduction Results Idea of Proof Summary

Our Results - Pinning

Theorem (Auffinger, L (09)) For all β > 0: µσ

n,β

s −

s∞ > δn

→ 0

in P-probability, for all δ > 0.

Pinning to LPP path with margin of order o(n).
Not so interesting.
Scale β with n !.
“Right” scaling is βn = βn−2/α+1L0(n).
L0(n) is another (related) slow varying function.

SLIDE 37

Introduction Results Idea of Proof Summary

Our Results - Pinning

Theorem (Auffinger, L (09)) For all β > 0: µσ

n,β

s −

s∞ > δn

→ 0

in P-probability, for all δ > 0.

Pinning to LPP path with margin of order o(n).
Not so interesting.
Scale β with n !.
“Right” scaling is βn = βn−2/α+1L0(n).
L0(n) is another (related) slow varying function.

SLIDE 38

Introduction Results Idea of Proof Summary

Energy-Entropy Balance

Let L0

n be the set of functions γ : [0, n] → Rd which are 1-Lipchitz

and satisfy s(0) = s(n) = 0 with · ∞ topology.

The Entropy of γ ∈ L0

n is:

E(γ) = n e(γ′(x))dx where e : [−1, 1] → R is defined as e(x) = 1

2[(1 + x) log(1 + x) + (1 − x) log(1 − x)] .

The Energy-Entropy Balance of a path is:

W σ

β (γ) = βσ(γ) − E(γ)

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Introduction Results Idea of Proof Summary

Energy-Entropy Balance

Let L0

n be the set of functions γ : [0, n] → Rd which are 1-Lipchitz

and satisfy s(0) = s(n) = 0 with · ∞ topology.

The Entropy of γ ∈ L0

n is:

E(γ) = n e(γ′(x))dx where e : [−1, 1] → R is defined as e(x) = 1

2[(1 + x) log(1 + x) + (1 − x) log(1 − x)] .

The Energy-Entropy Balance of a path is:

W σ

β (γ) = βσ(γ) − E(γ)

SLIDE 40

Introduction Results Idea of Proof Summary

Energy-Entropy Balance

Let L0

n be the set of functions γ : [0, n] → Rd which are 1-Lipchitz

and satisfy s(0) = s(n) = 0 with · ∞ topology.

The Entropy of γ ∈ L0

n is:

E(γ) = n e(γ′(x))dx where e : [−1, 1] → R is defined as e(x) = 1

2[(1 + x) log(1 + x) + (1 − x) log(1 − x)] .

The Energy-Entropy Balance of a path is:

W σ

β (γ) = βσ(γ) − E(γ)

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Introduction Results Idea of Proof Summary

Pinning to Optimal Path

Theorem (Auffinger, L (09)) Suppose βn = βn−2/α+1L0(n). µσ

n,βn

s − γ∗

n,βn∞ > δn

→ 0

in P-probability, for all δ > 0, where γ∗

n,β = arg max γ∈L0

n

W σ

β (γ)

SLIDE 42

Introduction Results Idea of Proof Summary

Convergence of Scaled Optimal Paths

Is γ∗

n,βn really different from

s?

Let M n,βn be the distribution of 1

n · γ∗ n,βn

A measures on L0 L0

1.

Theorem (Auffinger, L (09)) If βn = βn−2/α+1L0(n) then M n,βn ⇒ M α,β as n → ∞.

What is M α,β?

SLIDE 43

Introduction Results Idea of Proof Summary

Convergence of Scaled Optimal Paths

Is γ∗

n,βn really different from

s?

Let M n,βn be the distribution of 1

n · γ∗ n,βn

A measures on L0 L0

1.

Theorem (Auffinger, L (09)) If βn = βn−2/α+1L0(n) then M n,βn ⇒ M α,β as n → ∞.

What is M α,β?

SLIDE 44

Introduction Results Idea of Proof Summary

Convergence of Scaled Optimal Paths

Is γ∗

n,βn really different from

s?

Let M n,βn be the distribution of 1

n · γ∗ n,βn

A measures on L0 L0

1.

Theorem (Auffinger, L (09)) If βn = βn−2/α+1L0(n) then M n,βn ⇒ M α,β as n → ∞.

What is M α,β?

SLIDE 45

Introduction Results Idea of Proof Summary

Convergence of Scaled Optimal Paths

Is γ∗

n,βn really different from

s?

Let M n,βn be the distribution of 1

n · γ∗ n,βn

A measures on L0 L0

1.

Theorem (Auffinger, L (09)) If βn = βn−2/α+1L0(n) then M n,βn ⇒ M α,β as n → ∞.

What is M α,β?

SLIDE 46

Introduction Results Idea of Proof Summary

Description of the Limit Environment

We follow Hambley-Martine (07).
All curves γ ∈ L0 live in

D = {(x, y) ∈ [0, 1]2 : |y| min(x, 1 − x)}.

Define a collection of mutually independent random variables:

((Z i, V i); i = 1, 2, . . . ):

Z i ∼ Uniform(D).
V i d

=(E1 + · · · + Ei)− 1

α where Ej are i.i.d exponentials with rate 1.

The limit environment is:

π =

i

V iδ

· − Z i

SLIDE 47

Introduction Results Idea of Proof Summary

Description of the Limit Environment

We follow Hambley-Martine (07).
All curves γ ∈ L0 live in

D = {(x, y) ∈ [0, 1]2 : |y| min(x, 1 − x)}.

Define a collection of mutually independent random variables:

((Z i, V i); i = 1, 2, . . . ):

Z i ∼ Uniform(D).
V i d

=(E1 + · · · + Ei)− 1

α where Ej are i.i.d exponentials with rate 1.

The limit environment is:

π =

i

V iδ

· − Z i

SLIDE 48

Introduction Results Idea of Proof Summary

Description of the Limit Environment

We follow Hambley-Martine (07).
All curves γ ∈ L0 live in

D = {(x, y) ∈ [0, 1]2 : |y| min(x, 1 − x)}.

Define a collection of mutually independent random variables:

((Z i, V i); i = 1, 2, . . . ):

Z i ∼ Uniform(D).
V i d

=(E1 + · · · + Ei)− 1

α where Ej are i.i.d exponentials with rate 1.

The limit environment is:

π =

i

V iδ

· − Z i

SLIDE 49

Introduction Results Idea of Proof Summary

Description of the Limit Environment

We follow Hambley-Martine (07).
All curves γ ∈ L0 live in

D = {(x, y) ∈ [0, 1]2 : |y| min(x, 1 − x)}.

Define a collection of mutually independent random variables:

((Z i, V i); i = 1, 2, . . . ):

Z i ∼ Uniform(D).
V i d

=(E1 + · · · + Ei)− 1

α where Ej are i.i.d exponentials with rate 1.

The limit environment is:

π =

i

V iδ

· − Z i

SLIDE 50

Introduction Results Idea of Proof Summary

Description of the Limit Environment

We follow Hambley-Martine (07).
All curves γ ∈ L0 live in

D = {(x, y) ∈ [0, 1]2 : |y| min(x, 1 − x)}.

Define a collection of mutually independent random variables:

((Z i, V i); i = 1, 2, . . . ):

Z i ∼ Uniform(D).
V i d

=(E1 + · · · + Ei)− 1

α where Ej are i.i.d exponentials with rate 1.

The limit environment is:

π =

i

V iδ

· − Z i

SLIDE 51

Introduction Results Idea of Proof Summary

Description of the Limit Environment

We follow Hambley-Martine (07).
All curves γ ∈ L0 live in

D = {(x, y) ∈ [0, 1]2 : |y| min(x, 1 − x)}.

Define a collection of mutually independent random variables:

((Z i, V i); i = 1, 2, . . . ):

Z i ∼ Uniform(D).
V i d

=(E1 + · · · + Ei)− 1

α where Ej are i.i.d exponentials with rate 1.

The limit environment is:

π =

i

V iδ

· − Z i

SLIDE 52

Introduction Results Idea of Proof Summary

Description of the Limit Optimal Path

M α,β is the distribution of

γ∗

α,β = arg max γ∈L0

W π

β = arg max γ∈L0

βπ(γ) − E(γ)

Well defined (existence, uniqueness, finiteness) a.s.
M α,0 = δ{y≡0}.
M α,∞ =

M α

LHS defined as the limit of M n,βn when βn/(n−2/α+1L0(n)) → ∞.
RHS is the scaling limit of the LPP path.
M α,β1=M α,β2 if β1=β2.
In other words {M α,β; α ∈ (0, 2), β ∈ [0, ∞]} is a two parameters

family of distributions.

SLIDE 53

Introduction Results Idea of Proof Summary

Description of the Limit Optimal Path

M α,β is the distribution of

γ∗

α,β = arg max γ∈L0

W π

β = arg max γ∈L0

βπ(γ) − E(γ)

Well defined (existence, uniqueness, finiteness) a.s.
M α,0 = δ{y≡0}.
M α,∞ =

M α

LHS defined as the limit of M n,βn when βn/(n−2/α+1L0(n)) → ∞.
RHS is the scaling limit of the LPP path.
M α,β1=M α,β2 if β1=β2.
In other words {M α,β; α ∈ (0, 2), β ∈ [0, ∞]} is a two parameters

family of distributions.

SLIDE 54

Introduction Results Idea of Proof Summary

Description of the Limit Optimal Path

M α,β is the distribution of

γ∗

α,β = arg max γ∈L0

W π

β = arg max γ∈L0

βπ(γ) − E(γ)

Well defined (existence, uniqueness, finiteness) a.s.
M α,0 = δ{y≡0}.
M α,∞ =

M α

LHS defined as the limit of M n,βn when βn/(n−2/α+1L0(n)) → ∞.
RHS is the scaling limit of the LPP path.
M α,β1=M α,β2 if β1=β2.
In other words {M α,β; α ∈ (0, 2), β ∈ [0, ∞]} is a two parameters

family of distributions.

SLIDE 55

Introduction Results Idea of Proof Summary

Description of the Limit Optimal Path

M α,β is the distribution of

γ∗

α,β = arg max γ∈L0

W π

β = arg max γ∈L0

βπ(γ) − E(γ)

Well defined (existence, uniqueness, finiteness) a.s.
M α,0 = δ{y≡0}.
M α,∞ =

M α

LHS defined as the limit of M n,βn when βn/(n−2/α+1L0(n)) → ∞.
RHS is the scaling limit of the LPP path.
M α,β1=M α,β2 if β1=β2.
In other words {M α,β; α ∈ (0, 2), β ∈ [0, ∞]} is a two parameters

family of distributions.

SLIDE 56

Introduction Results Idea of Proof Summary

Description of the Limit Optimal Path

M α,β is the distribution of

γ∗

α,β = arg max γ∈L0

W π

β = arg max γ∈L0

βπ(γ) − E(γ)

Well defined (existence, uniqueness, finiteness) a.s.
M α,0 = δ{y≡0}.
M α,∞ =

M α

LHS defined as the limit of M n,βn when βn/(n−2/α+1L0(n)) → ∞.
RHS is the scaling limit of the LPP path.
M α,β1=M α,β2 if β1=β2.
In other words {M α,β; α ∈ (0, 2), β ∈ [0, ∞]} is a two parameters

family of distributions.

SLIDE 57

Introduction Results Idea of Proof Summary

Description of the Limit Optimal Path

M α,β is the distribution of

γ∗

α,β = arg max γ∈L0

W π

β = arg max γ∈L0

βπ(γ) − E(γ)

Well defined (existence, uniqueness, finiteness) a.s.
M α,0 = δ{y≡0}.
M α,∞ =

M α

LHS defined as the limit of M n,βn when βn/(n−2/α+1L0(n)) → ∞.
RHS is the scaling limit of the LPP path.
M α,β1=M α,β2 if β1=β2.
In other words {M α,β; α ∈ (0, 2), β ∈ [0, ∞]} is a two parameters

family of distributions.

SLIDE 58

Introduction Results Idea of Proof Summary

Corollaries

Fluctuations of order n.
Annealed weak convergence:

1 n · s ⇒ γ∗ α,β

as n → ∞

s is sampled according to Pµσ

n,βn.

βn = βn−2/α+1L0(n)

SLIDE 59

Introduction Results Idea of Proof Summary

Corollaries

Fluctuations of order n.
Annealed weak convergence:

1 n · s ⇒ γ∗ α,β

as n → ∞

s is sampled according to Pµσ

n,βn.

βn = βn−2/α+1L0(n)

SLIDE 60

Introduction Results Idea of Proof Summary

Phase Transition Phenomenon

By definition for fixed α, all γ∗

α,β for β ∈ [0, ∞] can be coupled

together (almost true).

Define:

βc = inf{β 0 : γ∗

α,β ≡ 0}.

The threshold value for when the polymer’s shape becomes

macroscopically effected by the environment.

Random.

Theorem (Auffinger, L (09))

if α ∈ [ 1

2, 2) then βc = 0 a.s.

if α ∈ (0, 1

3) then βc > 0 a.s.

SLIDE 61

Introduction Results Idea of Proof Summary

Phase Transition Phenomenon

By definition for fixed α, all γ∗

α,β for β ∈ [0, ∞] can be coupled

together (almost true).

Define:

βc = inf{β 0 : γ∗

α,β ≡ 0}.

The threshold value for when the polymer’s shape becomes

macroscopically effected by the environment.

Random.

Theorem (Auffinger, L (09))

if α ∈ [ 1

2, 2) then βc = 0 a.s.

if α ∈ (0, 1

3) then βc > 0 a.s.

SLIDE 62

Introduction Results Idea of Proof Summary

Phase Transition Phenomenon

By definition for fixed α, all γ∗

α,β for β ∈ [0, ∞] can be coupled

together (almost true).

Define:

βc = inf{β 0 : γ∗

α,β ≡ 0}.

The threshold value for when the polymer’s shape becomes

macroscopically effected by the environment.

Random.

Theorem (Auffinger, L (09))

if α ∈ [ 1

2, 2) then βc = 0 a.s.

if α ∈ (0, 1

3) then βc > 0 a.s.

SLIDE 63

Introduction Results Idea of Proof Summary

Phase Transition Phenomenon

By definition for fixed α, all γ∗

α,β for β ∈ [0, ∞] can be coupled

together (almost true).

Define:

βc = inf{β 0 : γ∗

α,β ≡ 0}.

The threshold value for when the polymer’s shape becomes

macroscopically effected by the environment.

Random.

Theorem (Auffinger, L (09))

if α ∈ [ 1

2, 2) then βc = 0 a.s.

if α ∈ (0, 1

3) then βc > 0 a.s.

SLIDE 64

Introduction Results Idea of Proof Summary

Outline

Introduction Results Idea of Proof Summary

SLIDE 65

Introduction Results Idea of Proof Summary

Extreme Value Theory

((Ui

n, Pi n) : i = 1, . . . , n) extreme values and positions of

{σ({(x, y)}) : (x, y) ∈ Dn}

Dn - sites reachable in n steps.
Standard Extreme Value Theory:

((V i

n, Z i n))k i=1

L0(n)

n2/α Ui n, 1 nPi n

k

i=1 ⇒ ((V i, Z i))k i=1

as n → ∞.

Equivalently πk

n ⇒ πk as n → ∞ where

πk

n = k

i=1

V i

nδ

· − Z i

n

;

πk =

k

i=1

V iδ

· − Z i

SLIDE 66

Introduction Results Idea of Proof Summary

Extreme Value Theory

((Ui

n, Pi n) : i = 1, . . . , n) extreme values and positions of

{σ({(x, y)}) : (x, y) ∈ Dn}

Dn - sites reachable in n steps.
Standard Extreme Value Theory:

((V i

n, Z i n))k i=1

L0(n)

n2/α Ui n, 1 nPi n

k

i=1 ⇒ ((V i, Z i))k i=1

as n → ∞.

Equivalently πk

n ⇒ πk as n → ∞ where

πk

n = k

i=1

V i

nδ

· − Z i

n

;

πk =

k

i=1

V iδ

· − Z i

SLIDE 67

Introduction Results Idea of Proof Summary

Extreme Value Theory

((Ui

n, Pi n) : i = 1, . . . , n) extreme values and positions of

{σ({(x, y)}) : (x, y) ∈ Dn}

Dn - sites reachable in n steps.
Standard Extreme Value Theory:

((V i

n, Z i n))k i=1

L0(n)

n2/α Ui n, 1 nPi n

k

i=1 ⇒ ((V i, Z i))k i=1

as n → ∞.

Equivalently πk

n ⇒ πk as n → ∞ where

πk

n = k

i=1

V i

nδ

· − Z i

n

;

πk =

k

i=1

V iδ

· − Z i

SLIDE 68

Introduction Results Idea of Proof Summary

Key Lemma 1

Lemma (Hambley-Martin 07) sup

γ∈L0(πn n − πk n)(γ) → 0 as k → ∞

in P-probability uniformly in n ∞.

We can work with truncated = finite approximations to

environments.

Only possible with heavy tails and α < 2.

SLIDE 69

Introduction Results Idea of Proof Summary

Key Lemma 1

Lemma (Hambley-Martin 07) sup

γ∈L0(πn n − πk n)(γ) → 0 as k → ∞

in P-probability uniformly in n ∞.

We can work with truncated = finite approximations to

environments.

Only possible with heavy tails and α < 2.

SLIDE 70

Introduction Results Idea of Proof Summary

Key Lemma 1

Lemma (Hambley-Martin 07) sup

γ∈L0(πn n − πk n)(γ) → 0 as k → ∞

in P-probability uniformly in n ∞.

We can work with truncated = finite approximations to

environments.

Only possible with heavy tails and α < 2.

SLIDE 71

Introduction Results Idea of Proof Summary

Large Deviation Analysis

(Mogulskii) (µn,0(n ·))n 1 satisfies a LDP on L0 ( · ∞) with

good rate function E µn,0(γ) ∼ = e−nE(n−1·γ) for γ ∈ Ln

0.

Let σk

n = k i=1 Ui nδ

· − Pi

n

. Then βnσk

n(γ) = βnπk n(n−1 · γ). and

exp

−βnHσk

n(γ)

∼

= e+nβπk

n(n−1·γ)

for γ ∈ Ln

0.

Then

µσk

n

n,β(γ) ∼

= exp

n
(βπk

n − E − const)(n−1 · γ)

SLIDE 72

Introduction Results Idea of Proof Summary

Large Deviation Analysis

(Mogulskii) (µn,0(n ·))n 1 satisfies a LDP on L0 ( · ∞) with

good rate function E µn,0(γ) ∼ = e−nE(n−1·γ) for γ ∈ Ln

0.

Let σk

n = k i=1 Ui nδ

· − Pi

n

. Then βnσk

n(γ) = βnπk n(n−1 · γ). and

exp

−βnHσk

n(γ)

∼

= e+nβπk

n(n−1·γ)

for γ ∈ Ln

0.

Then

µσk

n

n,β(γ) ∼

= exp

n
(βπk

n − E − const)(n−1 · γ)

SLIDE 73

Introduction Results Idea of Proof Summary

Large Deviation Analysis

(Mogulskii) (µn,0(n ·))n 1 satisfies a LDP on L0 ( · ∞) with

good rate function E µn,0(γ) ∼ = e−nE(n−1·γ) for γ ∈ Ln

0.

Let σk

n = k i=1 Ui nδ

· − Pi

n

. Then βnσk

n(γ) = βnπk n(n−1 · γ). and

exp

−βnHσk

n(γ)

∼

= e+nβπk

n(n−1·γ)

for γ ∈ Ln

0.

Then

µσk

n

n,β(γ) ∼

= exp

n
(βπk

n − E − const)(n−1 · γ)

SLIDE 74

Introduction Results Idea of Proof Summary

Key Lemma 2

µσ

n,β is exponentially concentrated around the maximizer of

(βπk

n − E)(n−1 · γ) = n−1(βnσk n − E)(γ)

(Varadhan-Laplace Lemma).

Call γk

n,βn this maximizer. Then

Lemma For all δ > 0, ǫ > 0 there exists ν > 0 such that µσk

n

n,βn

s : s − γk

n,βn∞ > δn

e−νn

with P-probability at least 1 − ǫ uniformly in k.

SLIDE 75

Introduction Results Idea of Proof Summary

Key Lemma 2

µσ

n,β is exponentially concentrated around the maximizer of

(βπk

n − E)(n−1 · γ) = n−1(βnσk n − E)(γ)

(Varadhan-Laplace Lemma).

Call γk

n,βn this maximizer. Then

Lemma For all δ > 0, ǫ > 0 there exists ν > 0 such that µσk

n

n,βn

s : s − γk

n,βn∞ > δn

e−νn

with P-probability at least 1 − ǫ uniformly in k.

SLIDE 76

Introduction Results Idea of Proof Summary

Outline

Introduction Results Idea of Proof Summary

SLIDE 77

Introduction Results Idea of Proof Summary

Open Questions

Extend for α ∈ [2, 5).

SLIDE 78

Introduction Results Idea of Proof Summary