Polynomial Chaos and Scaling Limits of Disordered Systems Francesco - - PowerPoint PPT Presentation

Disordered Systems Partition Function CDPM Further Developments

Polynomial Chaos and Scaling Limits of Disordered Systems

Francesco Caravenna

Universit` a degli Studi di Milano-Bicocca Nantes ∼ June 6, 2014

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 1 / 28

Disordered Systems Partition Function CDPM Further Developments

Coworkers

Joint work with Nikos Zygouras (Warwick) and Rongfeng Sun (NUS)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 2 / 28

Disordered Systems Partition Function CDPM Further Developments

Summary

We consider statistical mechanics models defined on a lattice, in which disorder (quenched randomness) enters as an external random field

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 3 / 28

Disordered Systems Partition Function CDPM Further Developments

Summary

We consider statistical mechanics models defined on a lattice, in which disorder (quenched randomness) enters as an external random field The goal is to study their scaling limits, in a suitable continuum and weak disorder regime

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 3 / 28

Disordered Systems Partition Function CDPM Further Developments

Summary

We consider statistical mechanics models defined on a lattice, in which disorder (quenched randomness) enters as an external random field The goal is to study their scaling limits, in a suitable continuum and weak disorder regime Very general framework, illustrated by 3 concrete examples

• 1. Disordered pinning models (Pinning)
• 2. Directed polymer in random environment (DPRE)
• 3. Random-field Ising model (Ising)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 3 / 28

Disordered Systems Partition Function CDPM Further Developments

Summary

We consider statistical mechanics models defined on a lattice, in which disorder (quenched randomness) enters as an external random field The goal is to study their scaling limits, in a suitable continuum and weak disorder regime Very general framework, illustrated by 3 concrete examples

• 1. Disordered pinning models (Pinning)
• 2. Directed polymer in random environment (DPRE)
• 3. Random-field Ising model (Ising)

Inspired by recent work of Alberts, Quastel and Khanin on DPRE

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 3 / 28

Disordered Systems Partition Function CDPM Further Developments

Outline

• 1. Disordered Systems and their Scaling Limits
• 2. Main Results (I): Partition Function
• 3. Main Results (II): Continuum Disordered Pinning Model
• 4. Further Developments

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 4 / 28

Disordered Systems Partition Function CDPM Further Developments

General Framework

Lattice Ω ⊆ Rd “spins” σ = (σx)x∈Ω ∈ {0, 1}Ω or {−1, +1}Ω

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 5 / 28

Disordered Systems Partition Function CDPM Further Developments

General Framework

Lattice Ω ⊆ Rd “spins” σ = (σx)x∈Ω ∈ {0, 1}Ω or {−1, +1}Ω

◮ Reference law Pref Ω (σ) on “spin configurations” (non trivial!)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 5 / 28

Disordered Systems Partition Function CDPM Further Developments

General Framework

Lattice Ω ⊆ Rd “spins” σ = (σx)x∈Ω ∈ {0, 1}Ω or {−1, +1}Ω

◮ Reference law Pref Ω (σ) on “spin configurations” (non trivial!) ◮ Disorder (ωx)x∈Zd i.i.d. random variables, independent of σ

E[ωx] = 0 Var[ωx] = 1 E[etωx] < ∞ for small |t|

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 5 / 28

Disordered Systems Partition Function CDPM Further Developments

General Framework

Lattice Ω ⊆ Rd “spins” σ = (σx)x∈Ω ∈ {0, 1}Ω or {−1, +1}Ω

◮ Reference law Pref Ω (σ) on “spin configurations” (non trivial!) ◮ Disorder (ωx)x∈Zd i.i.d. random variables, independent of σ

E[ωx] = 0 Var[ωx] = 1 E[etωx] < ∞ for small |t| (λωx + h)x∈Zd disorder with strength λ > 0 and bias h ∈ R

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 5 / 28

Disordered Systems Partition Function CDPM Further Developments

General Framework

Lattice Ω ⊆ Rd “spins” σ = (σx)x∈Ω ∈ {0, 1}Ω or {−1, +1}Ω

◮ Reference law Pref Ω (σ) on “spin configurations” (non trivial!) ◮ Disorder (ωx)x∈Zd i.i.d. random variables, independent of σ

E[ωx] = 0 Var[ωx] = 1 E[etωx] < ∞ for small |t| (λωx + h)x∈Zd disorder with strength λ > 0 and bias h ∈ R

Disordered law

Random Gibbs measure on spin configurations σ, indexed by disorder ω Pω

Ω,λ,h(σ)

∝ exp

x∈Ω

(λωx + h)σx

• Pref

Ω (σ)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 5 / 28

Disordered Systems Partition Function CDPM Further Developments

General Framework

Lattice Ω ⊆ Rd “spins” σ = (σx)x∈Ω ∈ {0, 1}Ω or {−1, +1}Ω

◮ Reference law Pref Ω (σ) on “spin configurations” (non trivial!) ◮ Disorder (ωx)x∈Zd i.i.d. random variables, independent of σ

E[ωx] = 0 Var[ωx] = 1 E[etωx] < ∞ for small |t| (λωx + h)x∈Zd disorder with strength λ > 0 and bias h ∈ R

Disordered law

Random Gibbs measure on spin configurations σ, indexed by disorder ω Pω

Ω,λ,h(σ) :=

1 Z ω

Ω,λ,h

exp

x∈Ω

(λωx + h)σx

• Pref

Ω (σ)

Partition function Z ω

Ω,λ,h = Eref Ω [e

• x∈Ω(λωx+h)σx]

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 5 / 28

Disordered Systems Partition Function CDPM Further Developments

• 1. Disordered pinning model

0 = τ0 τ1 τ2 τ3 τ4 τ5 τ6 Reference law: renewal process τ = {0 = τ0 < τ1 < τ2 < . . .} ⊆ N0 Pref (τi+1 − τi) = n

• ∼

C n1+α , tail exponent α ∈ (0, 1)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 6 / 28

Disordered Systems Partition Function CDPM Further Developments

• 1. Disordered pinning model

0 = τ0 τ1 τ2 τ3 τ4 τ5 τ6 Reference law: renewal process τ = {0 = τ0 < τ1 < τ2 < . . .} ⊆ N0 Pref (τi+1 − τi) = n

• ∼

C n1+α , tail exponent α ∈ (0, 1) “spins” σn := 1{n∈τ} ∈ {0, 1} (long-range correlations)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 6 / 28

Disordered Systems Partition Function CDPM Further Developments

• 1. Disordered pinning model

0 = τ0 τ1 τ2 τ3 τ4 τ5 τ6 Reference law: renewal process τ = {0 = τ0 < τ1 < τ2 < . . .} ⊆ N0 Pref (τi+1 − τi) = n

• ∼

C n1+α , tail exponent α ∈ (0, 1) “spins” σn := 1{n∈τ} ∈ {0, 1} (long-range correlations) Lattice Ω := {1, . . . , N} Disordered law: disordered pinning model Pω

Ω,λ,h(τ) =

1 Z ω

Ω,λ,h

e

N

n=1(λωn+h)1{n∈τ} Pref(τ) Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 6 / 28

Disordered Systems Partition Function CDPM Further Developments

• 2. Directed polymer in random environment

Reference law: symmetric random walk X = (Xn)n≥0 on Z, in the domain of attraction

• f a stable L´

evy process with index α ∈ (0, 2] Varref(X1) < ∞ if α = 2 Pref |X1| > x

• ∼ C

xα if α ∈ (0, 2)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 7 / 28

Disordered Systems Partition Function CDPM Further Developments

• 2. Directed polymer in random environment

Reference law: symmetric random walk X = (Xn)n≥0 on Z, in the domain of attraction

• f a stable L´

evy process with index α ∈ (0, 2] Varref(X1) < ∞ if α = 2 Pref |X1| > x

• ∼ C

xα if α ∈ (0, 2) “spins” σn,x := 1{Xn=x} ∈ {0, 1} (long-range correlations)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 7 / 28

Disordered Systems Partition Function CDPM Further Developments

• 2. Directed polymer in random environment

Reference law: symmetric random walk X = (Xn)n≥0 on Z, in the domain of attraction

• f a stable L´

evy process with index α ∈ (0, 2] Varref(X1) < ∞ if α = 2 Pref |X1| > x

• ∼ C

xα if α ∈ (0, 2) “spins” σn,x := 1{Xn=x} ∈ {0, 1} (long-range correlations) Lattice Ω := {1, . . . , N} × Z Disordered law: directed polymer in random environment Pω

Ω,λ(X) =

1 Z ω

Ω,λ

e

N

n=1 λωn,x1{Xn=x} Pref(X) Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 7 / 28

Disordered Systems Partition Function CDPM Further Developments

• 3. Random field Ising model

Reference law: critical 2d Ising model with “+” boundary conditions

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + − − − − − − − − − − − − − − − − − + + + + − − − − − − − − − − − − − − − − − − − − − − − − − − − + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + σ0 −

Lattice Ω := {−N, . . . , N} × {−N, . . . , N} Pref

Ω (σ) ∝ exp

• βc
• x∼y∈Ω

σxσy

• where σx = ±1, βc = 1

2 log(1 +

√ 2)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 8 / 28

Disordered Systems Partition Function CDPM Further Developments

• 3. Random field Ising model

Reference law: critical 2d Ising model with “+” boundary conditions

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + − − − − − − − − − − − − − − − − − + + + + − − − − − − − − − − − − − − − − − − − − − − − − − − − + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + σ0 −

Lattice Ω := {−N, . . . , N} × {−N, . . . , N} Pref

Ω (σ) ∝ exp

• βc
• x∼y∈Ω

σxσy

• where σx = ±1, βc = 1

2 log(1 +

√ 2) Disordered law: random field Ising model Pω

Ω,λ,h(σ) =

1 Z ω

Ω,λ,h

e

• x∈Ω(λωx+h)σx Pref

Ω (σ)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 8 / 28

Disordered Systems Partition Function CDPM Further Developments

Continuum limit?

Common feature: reference law Pref

Ω

admits a non-trivial continuum limit

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 9 / 28

Disordered Systems Partition Function CDPM Further Developments

Continuum limit?

Common feature: reference law Pref

Ω

admits a non-trivial continuum limit Fix Ω ⊂ Rd bounded open with smooth boundary, and consider the lattice Ωδ := Ω ∩ (δZ)d i.e. rescale space by a factor δ > 0 (in the examples δ = 1

N )

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 9 / 28

Disordered Systems Partition Function CDPM Further Developments

Continuum limit?

Common feature: reference law Pref

Ω

admits a non-trivial continuum limit Fix Ω ⊂ Rd bounded open with smooth boundary, and consider the lattice Ωδ := Ω ∩ (δZ)d i.e. rescale space by a factor δ > 0 (in the examples δ = 1

N )

Under Pref

Ωδ , for a suitable γ > 0, the rescaled spins (δ−γσx)x∈Ωδ

converge in law to a (distribution-valued) continuum field (σx)x∈Ω

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 9 / 28

Disordered Systems Partition Function CDPM Further Developments

Continuum limit?

Common feature: reference law Pref

Ω

admits a non-trivial continuum limit Fix Ω ⊂ Rd bounded open with smooth boundary, and consider the lattice Ωδ := Ω ∩ (δZ)d i.e. rescale space by a factor δ > 0 (in the examples δ = 1

N )

Under Pref

Ωδ , for a suitable γ > 0, the rescaled spins (δ−γσx)x∈Ωδ

converge in law to a (distribution-valued) continuum field (σx)x∈Ω

◮ Ising model: recently proved by [Camia, Garban, Newman ’12] ◮ Pinning model: renewal processes τ regenerative set τ ◮ DPRE: random walk X

L´ evy process X

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 9 / 28

Disordered Systems Partition Function CDPM Further Developments

Continuum limit?

Common feature: reference law Pref

Ω

admits a non-trivial continuum limit Fix Ω ⊂ Rd bounded open with smooth boundary, and consider the lattice Ωδ := Ω ∩ (δZ)d i.e. rescale space by a factor δ > 0 (in the examples δ = 1

N )

Under Pref

Ωδ , for a suitable γ > 0, the rescaled spins (δ−γσx)x∈Ωδ

converge in law to a (distribution-valued) continuum field (σx)x∈Ω

◮ Ising model: recently proved by [Camia, Garban, Newman ’12] ◮ Pinning model: renewal processes τ regenerative set τ ◮ DPRE: random walk X

L´ evy process X Does the disordered model Pω

Ωδ,λ,h admit a non-trivial continuum limit?

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 9 / 28

Disordered Systems Partition Function CDPM Further Developments

A direct approach?

Recall the definition of the (discrete) disordered law: Pω

Ωδ,λ,h(dσ) ∝ exp x∈Ωδ

(λωx + h)σx

• Pref

Ωδ (dσ)

Can we guess the continuum disordered law?

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 10 / 28

Disordered Systems Partition Function CDPM Further Developments

A direct approach?

Recall the definition of the (discrete) disordered law: Pω

Ωδ,λ,h(dσ) ∝ exp x∈Ωδ

(λωx + h)σx

• Pref

Ωδ (dσ)

Can we guess the continuum disordered law?

◮ replace discrete spins (σx)x∈Ωδ by continuum spins (σx)x∈Ω

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 10 / 28

Disordered Systems Partition Function CDPM Further Developments

A direct approach?

Recall the definition of the (discrete) disordered law: Pω

Ωδ,λ,h(dσ) ∝ exp x∈Ωδ

(λωx + h)σx

• Pref

Ωδ (dσ)

Can we guess the continuum disordered law?

◮ replace discrete spins (σx)x∈Ωδ by continuum spins (σx)x∈Ω ◮ replace discrete disorder (ωx)x∈Ωδ by White noise (dWx)x∈Ω

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 10 / 28

Disordered Systems Partition Function CDPM Further Developments

A direct approach?

Recall the definition of the (discrete) disordered law: Pω

Ωδ,λ,h(dσ) ∝ exp x∈Ωδ

(λωx + h)σx

• Pref

Ωδ (dσ)

Can we guess the continuum disordered law?

◮ replace discrete spins (σx)x∈Ωδ by continuum spins (σx)x∈Ω ◮ replace discrete disorder (ωx)x∈Ωδ by White noise (dWx)x∈Ω

This leads to a candidate continuum model Pω

Ω,λ,h(dσ) ∝ exp Ω

(λdWx + h)σx

• Pref

Ω (dσ)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 10 / 28

Disordered Systems Partition Function CDPM Further Developments

A direct approach?

Recall the definition of the (discrete) disordered law: Pω

Ωδ,λ,h(dσ) ∝ exp x∈Ωδ

(λωx + h)σx

• Pref

Ωδ (dσ)

Can we guess the continuum disordered law?

◮ replace discrete spins (σx)x∈Ωδ by continuum spins (σx)x∈Ω ◮ replace discrete disorder (ωx)x∈Ωδ by White noise (dWx)x∈Ω

This leads to a candidate continuum model Pω

Ω,λ,h(dσ) ∝ exp Ω

(λdWx + h)σx

• Pref

Ω (dσ)

This expression makes no sense, because σx is distribution-valued

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 10 / 28

Disordered Systems Partition Function CDPM Further Developments

A direct approach?

Recall the definition of the (discrete) disordered law: Pω

Ωδ,λ,h(dσ) ∝ exp x∈Ωδ

(λωx + h)σx

• Pref

Ωδ (dσ)

Can we guess the continuum disordered law?

◮ replace discrete spins (σx)x∈Ωδ by continuum spins (σx)x∈Ω ◮ replace discrete disorder (ωx)x∈Ωδ by White noise (dWx)x∈Ω

This leads to a candidate continuum model Pω

Ω,λ,h(dσ) ∝ exp Ω

(λdWx + h)σx

• Pref

Ω (dσ)

This expression makes no sense, because σx is distribution-valued Difficulty is substantial: Pω

Ω,λ,h can be singular w.r.t. Pref Ω

!

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 10 / 28

Disordered Systems Partition Function CDPM Further Developments

Outline

• 1. Disordered Systems and their Scaling Limits
• 2. Main Results (I): Partition Function
• 3. Main Results (II): Continuum Disordered Pinning Model
• 4. Further Developments

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 11 / 28

Disordered Systems Partition Function CDPM Further Developments

The partition function

The disordered system Pω

Ωδ,λ,h is a difficult object (a random probability)

Let us be less ambitious and focus on the partition function Z ω

Ωδ,λ,h = Eref

• exp

x∈Ωδ

(λωx + h)σx

• which is “just” a random number (i.e. a random variable)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 12 / 28

Disordered Systems Partition Function CDPM Further Developments

The partition function

The disordered system Pω

Ωδ,λ,h is a difficult object (a random probability)

Let us be less ambitious and focus on the partition function Z ω

Ωδ,λ,h = Eref

• exp

x∈Ωδ

(λωx + h)σx

• which is “just” a random number (i.e. a random variable)

Does the partition function Z ω

Ωδ,λ,h has a non-trivial limit in law as δ ↓ 0,

letting λ, h → 0 at suitable rates? (Continuum and weak disorder regime)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 12 / 28

Disordered Systems Partition Function CDPM Further Developments

The partition function

The disordered system Pω

Ωδ,λ,h is a difficult object (a random probability)

Let us be less ambitious and focus on the partition function Z ω

Ωδ,λ,h = Eref

• exp

x∈Ωδ

(λωx + h)σx

• which is “just” a random number (i.e. a random variable)

Does the partition function Z ω

Ωδ,λ,h has a non-trivial limit in law as δ ↓ 0,

letting λ, h → 0 at suitable rates? (Continuum and weak disorder regime) The answer is positive, with an explicit limit. But why should we care?

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 12 / 28

Disordered Systems Partition Function CDPM Further Developments

The partition function

The disordered system Pω

Ωδ,λ,h is a difficult object (a random probability)

Let us be less ambitious and focus on the partition function Z ω

Ωδ,λ,h = Eref

• exp

x∈Ωδ

(λωx + h)σx

• which is “just” a random number (i.e. a random variable)

Does the partition function Z ω

Ωδ,λ,h has a non-trivial limit in law as δ ↓ 0,

letting λ, h → 0 at suitable rates? (Continuum and weak disorder regime) The answer is positive, with an explicit limit. But why should we care?

◮ Z ω Ωδ,λ,h encodes large-scale properties (free energy, phase transitions)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 12 / 28

Disordered Systems Partition Function CDPM Further Developments

The partition function

The disordered system Pω

Ωδ,λ,h is a difficult object (a random probability)

Let us be less ambitious and focus on the partition function Z ω

Ωδ,λ,h = Eref

• exp

x∈Ωδ

(λωx + h)σx

• which is “just” a random number (i.e. a random variable)

Does the partition function Z ω

Ωδ,λ,h has a non-trivial limit in law as δ ↓ 0,

letting λ, h → 0 at suitable rates? (Continuum and weak disorder regime) The answer is positive, with an explicit limit. But why should we care?

◮ Z ω Ωδ,λ,h encodes large-scale properties (free energy, phase transitions) ◮ Dream: scaling limit of Z ω Ωδ,λ,h scaling limit of Pω Ωδ,λ,h ???

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 12 / 28

Disordered Systems Partition Function CDPM Further Developments

The partition function

The disordered system Pω

Ωδ,λ,h is a difficult object (a random probability)

Let us be less ambitious and focus on the partition function Z ω

Ωδ,λ,h = Eref

• exp

x∈Ωδ

(λωx + h)σx

• which is “just” a random number (i.e. a random variable)

Does the partition function Z ω

Ωδ,λ,h has a non-trivial limit in law as δ ↓ 0,

letting λ, h → 0 at suitable rates? (Continuum and weak disorder regime) The answer is positive, with an explicit limit. But why should we care?

◮ Z ω Ωδ,λ,h encodes large-scale properties (free energy, phase transitions) ◮ Dream: scaling limit of Z ω Ωδ,λ,h scaling limit of Pω Ωδ,λ,h ???

YES, for Pinning and DPRE (and hopefully for Ising too)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 12 / 28

Disordered Systems Partition Function CDPM Further Developments

Assumptions

k-point function Eref

Ωδ [σx1 · · · σxk] defined on (Ωδ)k extended on Ωk

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 13 / 28

Disordered Systems Partition Function CDPM Further Developments

Assumptions

k-point function Eref

Ωδ [σx1 · · · σxk] defined on (Ωδ)k extended on Ωk

Key assumption on the reference law

The k-point functions of Pref

Ωδ converge in L2 under polynomial rescaling

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 13 / 28

Disordered Systems Partition Function CDPM Further Developments

Assumptions

k-point function Eref

Ωδ [σx1 · · · σxk] defined on (Ωδ)k extended on Ωk

Key assumption on the reference law

The k-point functions of Pref

Ωδ converge in L2 under polynomial rescaling

∃γ > 0 : Eref

Ωδ [σx1 · · · σxk]

(δγ)k

in L2(Ωk)

− − − − − − →

δ↓0

ψ(k)

Ω (x1, . . . , xk)

(⋆) ∀k ∈ N.

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 13 / 28

Disordered Systems Partition Function CDPM Further Developments

Assumptions

k-point function Eref

Ωδ [σx1 · · · σxk] defined on (Ωδ)k extended on Ωk

Key assumption on the reference law

The k-point functions of Pref

Ωδ converge in L2 under polynomial rescaling

∃γ > 0 : Eref

Ωδ [σx1 · · · σxk]

(δγ)k

in L2(Ωk)

− − − − − − →

δ↓0

ψ(k)

Ω (x1, . . . , xk)

(⋆) ∀k ∈ N. Furthermore

k∈N ψ(k) Ω 2 L2(Ωk) < ∞

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 13 / 28

Disordered Systems Partition Function CDPM Further Developments

Assumptions

k-point function Eref

Ωδ [σx1 · · · σxk] defined on (Ωδ)k extended on Ωk

Key assumption on the reference law

The k-point functions of Pref

Ωδ converge in L2 under polynomial rescaling

∃γ > 0 : Eref

Ωδ [σx1 · · · σxk]

(δγ)k

in L2(Ωk)

− − − − − − →

δ↓0

ψ(k)

Ω (x1, . . . , xk)

(⋆) ∀k ∈ N. Furthermore

k∈N ψ(k) Ω 2 L2(Ωk) < ∞

Pointwise convergence in (⋆) leads to ψ(k)

Ω (x1, . . . , xk) ≈ |xi − xj|−γ

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 13 / 28

Disordered Systems Partition Function CDPM Further Developments

Assumptions

k-point function Eref

Ωδ [σx1 · · · σxk] defined on (Ωδ)k extended on Ωk

Key assumption on the reference law

The k-point functions of Pref

Ωδ converge in L2 under polynomial rescaling

∃γ > 0 : Eref

Ωδ [σx1 · · · σxk]

(δγ)k

in L2(Ωk)

− − − − − − →

δ↓0

ψ(k)

Ω (x1, . . . , xk)

(⋆) ∀k ∈ N. Furthermore

k∈N ψ(k) Ω 2 L2(Ωk) < ∞

Pointwise convergence in (⋆) leads to ψ(k)

Ω (x1, . . . , xk) ≈ |xi − xj|−γ

L2 convergence then requires that γ < d 2

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 13 / 28

Disordered Systems Partition Function CDPM Further Developments

Main result (I): partition function

Theorem [C., Sun, Zygouras ’13]

Assume that Pref

Ωδ satisfies (⋆) with exponent γ (and dimension d)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 14 / 28

Disordered Systems Partition Function CDPM Further Developments

Main result (I): partition function

Theorem [C., Sun, Zygouras ’13]

Assume that Pref

Ωδ satisfies (⋆) with exponent γ (and dimension d) ◮ Case σx ∈ {0, 1}. Fix ˆ

λ > 0, ˆ h ∈ R and scale λ, h → 0 as λ := ˆ λ δd/2−γ h := ˆ h δd−γ − 1

2λ2

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 14 / 28

Disordered Systems Partition Function CDPM Further Developments

Main result (I): partition function

Theorem [C., Sun, Zygouras ’13]

Assume that Pref

Ωδ satisfies (⋆) with exponent γ (and dimension d) ◮ Case σx ∈ {0, 1}. Fix ˆ

λ > 0, ˆ h ∈ R and scale λ, h → 0 as λ := ˆ λ δd/2−γ h := ˆ h δd−γ − 1

2λ2

Then Z ω

Ωδ,λ,h δ↓0

= ⇒ ZW

Ω;ˆ λ,ˆ h with W (dx) := white noise on Rd and

ZW

Ω;ˆ λ,ˆ h := ∞

• k=0

1 k!

• · · ·
• Ωk ψ(k)

Ω (x1, . . . , xk) k

• i=1

ˆ λ W (dxi) + ˆ h dxi

• Wiener chaos expansion (converging in L2−)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 14 / 28

Disordered Systems Partition Function CDPM Further Developments

Main result (I): partition function

Theorem [C., Sun, Zygouras ’13]

Assume that Pref

Ωδ satisfies (⋆) with exponent γ (and dimension d) ◮ Case σx ∈ {0, 1}. Fix ˆ

λ > 0, ˆ h ∈ R and scale λ, h → 0 as λ := ˆ λ δd/2−γ h := ˆ h δd−γ − 1

2λ2

Then Z ω

Ωδ,λ,h δ↓0

= ⇒ ZW

Ω;ˆ λ,ˆ h with W (dx) := white noise on Rd and

ZW

Ω;ˆ λ,ˆ h := ∞

• k=0

1 k!

• · · ·
• Ωk ψ(k)

Ω (x1, . . . , xk) k

• i=1

ˆ λ W (dxi) + ˆ h dxi

• Wiener chaos expansion (converging in L2−)

◮ Case σx ∈ {−1, 1}. The same, up to minor modifications (cf. below)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 14 / 28

Disordered Systems Partition Function CDPM Further Developments

Motivating models: Pinning and DPRE

◮ Pinning. Dimension d = 1, exponent γ = 1 − α,

ψ(k)

Ω (x1, . . . , xk) =

ck x1−α

1

(x2 − x1)1−α · · · (xk − xk−1)1−α (pointwise conv.) by renewal theory

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 15 / 28

Disordered Systems Partition Function CDPM Further Developments

Motivating models: Pinning and DPRE

◮ Pinning. Dimension d = 1, exponent γ = 1 − α,

ψ(k)

Ω (x1, . . . , xk) =

ck x1−α

1

(x2 − x1)1−α · · · (xk − xk−1)1−α (pointwise conv.) by renewal theory

◮ DPRE. Effective dimension d = 1 + 1/α, exponent γ = 1/α,

ψ(k)

Ω (t1, x1, . . . , tk, xk) = gt1(x1)gt2−t1(x2 − x1) · · · gtk−tk−1(xk − xk−1)

(pointwise conv.) with gt(x) density of α-stable L´ evy process Xt

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 15 / 28

Disordered Systems Partition Function CDPM Further Developments

Motivating models: Pinning and DPRE

◮ Pinning. Dimension d = 1, exponent γ = 1 − α,

ψ(k)

Ω (x1, . . . , xk) =

ck x1−α

1

(x2 − x1)1−α · · · (xk − xk−1)1−α (pointwise conv.) by renewal theory

◮ DPRE. Effective dimension d = 1 + 1/α, exponent γ = 1/α,

ψ(k)

Ω (t1, x1, . . . , tk, xk) = gt1(x1)gt2−t1(x2 − x1) · · · gtk−tk−1(xk − xk−1)

(pointwise conv.) with gt(x) density of α-stable L´ evy process Xt For assumption (⋆), L2 convergence (γ < d

2 ) forces

α ∈ ( 1

2, 1) [Pinning]

α ∈ (1, 2] [DPRE]

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 15 / 28

Disordered Systems Partition Function CDPM Further Developments

Motivating models: Pinning and DPRE

◮ Pinning. Dimension d = 1, exponent γ = 1 − α,

ψ(k)

Ω (x1, . . . , xk) =

ck x1−α

1

(x2 − x1)1−α · · · (xk − xk−1)1−α (pointwise conv.) by renewal theory

◮ DPRE. Effective dimension d = 1 + 1/α, exponent γ = 1/α,

ψ(k)

Ω (t1, x1, . . . , tk, xk) = gt1(x1)gt2−t1(x2 − x1) · · · gtk−tk−1(xk − xk−1)

(pointwise conv.) with gt(x) density of α-stable L´ evy process Xt For assumption (⋆), L2 convergence (γ < d

2 ) forces

α ∈ ( 1

2, 1) [Pinning]

α ∈ (1, 2] [DPRE] These restrictions are not technical, but substantial (physical)!

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 15 / 28

Disordered Systems Partition Function CDPM Further Developments

Motivating models: Ising

Pointwise convergence of k-point function, with exponent γ = 1

8, toward

ψ(k)

Ω (x1, . . . , xk) conformally covariant,

was proved in [Chelkak, Hongler, Izyurov ’12].

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 16 / 28

Disordered Systems Partition Function CDPM Further Developments

Motivating models: Ising

Pointwise convergence of k-point function, with exponent γ = 1

8, toward

ψ(k)

Ω (x1, . . . , xk) conformally covariant,

was proved in [Chelkak, Hongler, Izyurov ’12]. This convergence holds in L2(Ωk), for bounded open Ω ⊆ R2 with piecewise smooth boundary (we provide a uniform domination)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 16 / 28

Disordered Systems Partition Function CDPM Further Developments

Motivating models: Ising

Pointwise convergence of k-point function, with exponent γ = 1

8, toward

ψ(k)

Ω (x1, . . . , xk) conformally covariant,

was proved in [Chelkak, Hongler, Izyurov ’12]. This convergence holds in L2(Ωk), for bounded open Ω ⊆ R2 with piecewise smooth boundary (we provide a uniform domination) Recall that we consider random field 2d Ising model at the critical point, with external field (λωx + h)x∈Ωδ

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 16 / 28

Disordered Systems Partition Function CDPM Further Developments

Motivating models: Ising

Pointwise convergence of k-point function, with exponent γ = 1

8, toward

ψ(k)

Ω (x1, . . . , xk) conformally covariant,

was proved in [Chelkak, Hongler, Izyurov ’12]. This convergence holds in L2(Ωk), for bounded open Ω ⊆ R2 with piecewise smooth boundary (we provide a uniform domination) Recall that we consider random field 2d Ising model at the critical point, with external field (λωx + h)x∈Ωδ We fix continuous functions ˆ λ : Ω → (0, ∞) and ˆ h : Ω → R and set λ = ˆ λ(x) δ7/8 h = ˆ h(x) δ15/8

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 16 / 28

Disordered Systems Partition Function CDPM Further Developments

Motivating models: Ising

Theorem [C., Sun, Zygouras ’13]

As δ ↓ 0 one has the convergence in law e− 1

2 ˆ

λ2

2 δ−1/4Z ω

Ωδ,λ,h =

⇒ ZW

Ω;ˆ λ,ˆ h

where W (dx) is white noise on Rd and ZW

Ω;ˆ λ,ˆ h := ∞

• k=0

1 k!

• · · ·
• Ωkψ(k)

Ω (x1, . . . , xk) k

• i=1

ˆ λ(xi) W (dxi) + ˆ h(xi) dxi

• Francesco Caravenna

Scaling Limits of Disordered Systems June 6, 2014 17 / 28

Disordered Systems Partition Function CDPM Further Developments

Motivating models: Ising

Theorem [C., Sun, Zygouras ’13]

As δ ↓ 0 one has the convergence in law e− 1

2 ˆ

λ2

2 δ−1/4Z ω

Ωδ,λ,h =

⇒ ZW

Ω;ˆ λ,ˆ h

where W (dx) is white noise on Rd and ZW

Ω;ˆ λ,ˆ h := ∞

• k=0

1 k!

• · · ·
• Ωkψ(k)

Ω (x1, . . . , xk) k

• i=1

ˆ λ(xi) W (dxi) + ˆ h(xi) dxi

• Conformal covariance: if φ : ˜

Ω → Ω is a conformal map, ZW

Ω;ˆ λ,ˆ h dist.

= ZW

˜ Ω;˜ λ,˜ h

where ˜ λ(x) := |φ′(x)|7/8ˆ λ(φ(x)) and ˜ h(x) := |φ′(x)|15/8ˆ h(φ(x))

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 17 / 28

Disordered Systems Partition Function CDPM Further Developments

Sketch of the proof

• 1. Linearization. Since σx ∈ {0, 1}, every function of σx is linear

Z ω

Ωδ,λ,h = Eref Ωδ x∈Ωδ

e(λωx+h)σx

• = Eref

Ωδ x∈Ωδ

• 1 + ǫxσx
• where ǫx := eλωx+h − 1.

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 18 / 28

Disordered Systems Partition Function CDPM Further Developments

Sketch of the proof

• 1. Linearization. Since σx ∈ {0, 1}, every function of σx is linear

Z ω

Ωδ,λ,h = Eref Ωδ x∈Ωδ

e(λωx+h)σx

• = Eref

Ωδ x∈Ωδ

• 1 + ǫxσx
• where ǫx := eλωx+h − 1. Note that

E[ǫx] ≃ h + 1

2λ2 =: h′

Var[ǫx] ≃ λ2

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 18 / 28

Disordered Systems Partition Function CDPM Further Developments

Sketch of the proof

• 1. Linearization. Since σx ∈ {0, 1}, every function of σx is linear

Z ω

Ωδ,λ,h = Eref Ωδ x∈Ωδ

e(λωx+h)σx

• = Eref

Ωδ x∈Ωδ

• 1 + ǫxσx
• where ǫx := eλωx+h − 1. Note that

E[ǫx] ≃ h + 1

2λ2 =: h′

Var[ǫx] ≃ λ2

• 2. High-temperature expansion. By a binomial expansion of the product

Z ω

Ωδ,λ,h = |Ωδ|

• k=0

1 k!

• (x1,...,xk)∈(Ωδ)k

Eref

Ωδ

• σx1 · · · σxk
• ǫx1 · · · ǫxk

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 18 / 28

Disordered Systems Partition Function CDPM Further Developments

Sketch of the proof

• 1. Linearization. Since σx ∈ {0, 1}, every function of σx is linear

Z ω

Ωδ,λ,h = Eref Ωδ x∈Ωδ

e(λωx+h)σx

• = Eref

Ωδ x∈Ωδ

• 1 + ǫxσx
• where ǫx := eλωx+h − 1. Note that

E[ǫx] ≃ h + 1

2λ2 =: h′

Var[ǫx] ≃ λ2

• 2. High-temperature expansion. By a binomial expansion of the product

Z ω

Ωδ,λ,h = |Ωδ|

• k=0

1 k!

• (x1,...,xk)∈(Ωδ)k

Eref

Ωδ

• σx1 · · · σxk
• ǫx1 · · · ǫxk

Partition function is a multilinear polynomial of the random variables ǫx, with coefficient given by the k-point functions of Pref

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 18 / 28

Disordered Systems Partition Function CDPM Further Developments

Sketch of the proof

• 3. Lindeberg principle, extending [Mossel, ODonnell, Oleszkiewicz ’10]

The law of a multilinear polynomial is insensitive toward the distribution

• f the ǫx (keeping same mean and variance) independent Gaussians

ǫx N(h′, λ)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 19 / 28

Disordered Systems Partition Function CDPM Further Developments

Sketch of the proof

• 3. Lindeberg principle, extending [Mossel, ODonnell, Oleszkiewicz ’10]

The law of a multilinear polynomial is insensitive toward the distribution

• f the ǫx (keeping same mean and variance) independent Gaussians

ǫx N(h′, λ) ∼ λ δ−d/2 W (∆x) + h′ δ−d Leb(∆x) white noise W integrated on cell ∆x := (x − δ

2, x + δ 2)d

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 19 / 28

Disordered Systems Partition Function CDPM Further Developments

Sketch of the proof

• 3. Lindeberg principle, extending [Mossel, ODonnell, Oleszkiewicz ’10]

The law of a multilinear polynomial is insensitive toward the distribution

• f the ǫx (keeping same mean and variance) independent Gaussians

ǫx N(h′, λ) ∼ λ δ−d/2 W (∆x) + h′ δ−d Leb(∆x) white noise W integrated on cell ∆x := (x − δ

2, x + δ 2)d

Since the k-point function is piecewise constant on cells ∆x, we get Z ω

Ωδ,λ,h ≃ ∞

• k=0

1 k!

• · · ·
• Ωk Eref

Ωδ

• σx1 · · · σxk
• k
• i=1
• λ δ− d

2 W (dxi) + h′ δ−d dxi

• Francesco Caravenna

Scaling Limits of Disordered Systems June 6, 2014 19 / 28

Disordered Systems Partition Function CDPM Further Developments

Sketch of the proof

• 3. Lindeberg principle, extending [Mossel, ODonnell, Oleszkiewicz ’10]

The law of a multilinear polynomial is insensitive toward the distribution

• f the ǫx (keeping same mean and variance) independent Gaussians

ǫx N(h′, λ) ∼ λ δ−d/2 W (∆x) + h′ δ−d Leb(∆x) white noise W integrated on cell ∆x := (x − δ

2, x + δ 2)d

Since the k-point function is piecewise constant on cells ∆x, we get Z ω

Ωδ,λ,h ≃ ∞

• k=0

1 k!

• · · ·
• Ωk Eref

Ωδ

• σx1 · · · σxk
• k
• i=1
• λ δ− d

2 W (dxi) + h′ δ−d dxi

• 4. Wiener chaos expansion. Plugging the assumption

Eref

Ωδ

• σx1 · · · σxk
• ≃ (δγ)kψ(k)

Ω (x1, . . . , xk)

yields a Wiener chaos expansion with ˆ λ = λδγ− d

2

and ˆ h = h′δγ−d

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 19 / 28

Disordered Systems Partition Function CDPM Further Developments

Outline

• 1. Disordered Systems and their Scaling Limits
• 2. Main Results (I): Partition Function
• 3. Main Results (II): Continuum Disordered Pinning Model
• 4. Further Developments

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 20 / 28

Disordered Systems Partition Function CDPM Further Developments

Back to pinning models

0 = τ0 τ1 τ2 τ3 τ4 τ5 τ6 τ = {τ0 < τ1 < τ2 < . . .} random element of E := {closed subsets of R}

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 21 / 28

Disordered Systems Partition Function CDPM Further Developments

Back to pinning models

0 = τ0 τ1 τ2 τ3 τ4 τ5 τ6 τ = {τ0 < τ1 < τ2 < . . .} random element of E := {closed subsets of R} Rescaled set (δτ, Pref)

δ↓0

= ⇒ (τ, Pref) α-stable regenerative set

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 21 / 28

Disordered Systems Partition Function CDPM Further Developments

Back to pinning models

0 = τ0 τ1 τ2 τ3 τ4 τ5 τ6 τ = {τ0 < τ1 < τ2 < . . .} random element of E := {closed subsets of R} Rescaled set (δτ, Pref)

δ↓0

= ⇒ (τ, Pref) α-stable regenerative set What happens for the disordered model Pω

Ωδ,λ,h?

(Ω = (0, 1))

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 21 / 28

Disordered Systems Partition Function CDPM Further Developments

Back to pinning models

0 = τ0 τ1 τ2 τ3 τ4 τ5 τ6 τ = {τ0 < τ1 < τ2 < . . .} random element of E := {closed subsets of R} Rescaled set (δτ, Pref)

δ↓0

= ⇒ (τ, Pref) α-stable regenerative set What happens for the disordered model Pω

Ωδ,λ,h?

(Ω = (0, 1))

Restrict α ∈ ( 1

2, 1). Fix ˆ

λ > 0, ˆ h ∈ R and set λ := ˆ λ δα− 1

2

h := ˆ h δα − 1

2λ2

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 21 / 28

Disordered Systems Partition Function CDPM Further Developments

Continuum Disordered Pinning Model

[C., Sun, Zygouras ’14]

E := {closed subsets of R} equipped with the Hausdorff distance

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 22 / 28

Disordered Systems Partition Function CDPM Further Developments

Continuum Disordered Pinning Model

[C., Sun, Zygouras ’14]

E := {closed subsets of R} equipped with the Hausdorff distance

Theorem (existence and universality of the CDPM)

As δ ↓ 0, the rescaled discrete set (δτ, Pω

Ωδ,λ,h) converges in distribution

• n E to a universal random closed set (τ, PW

Ω,ˆ λ,ˆ h), called CDPM

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 22 / 28

Disordered Systems Partition Function CDPM Further Developments

Continuum Disordered Pinning Model

[C., Sun, Zygouras ’14]

E := {closed subsets of R} equipped with the Hausdorff distance

Theorem (existence and universality of the CDPM)

As δ ↓ 0, the rescaled discrete set (δτ, Pω

Ωδ,λ,h) converges in distribution

• n E to a universal random closed set (τ, PW

Ω,ˆ λ,ˆ h), called CDPM

Theorem (a.s. properties)

The CDPM has any a.s. property of the α-stable regenerative set Pref A ⊆ E, Pref(A) = 1 = ⇒ PW

Ω,ˆ λ,ˆ h(A) = 1 ,

P(dW )-a.s.

Example: A =

• A ⊆ R : Hausdorff dim. of A = α
• Francesco Caravenna

Scaling Limits of Disordered Systems June 6, 2014 22 / 28

Disordered Systems Partition Function CDPM Further Developments

Continuum Disordered Pinning Model

[C., Sun, Zygouras ’14]

E := {closed subsets of R} equipped with the Hausdorff distance

Theorem (existence and universality of the CDPM)

As δ ↓ 0, the rescaled discrete set (δτ, Pω

Ωδ,λ,h) converges in distribution

• n E to a universal random closed set (τ, PW

Ω,ˆ λ,ˆ h), called CDPM

Theorem (a.s. properties)

The CDPM has any a.s. property of the α-stable regenerative set Pref A ⊆ E, Pref(A) = 1 = ⇒ PW

Ω,ˆ λ,ˆ h(A) = 1 ,

P(dW )-a.s.

Example: A =

• A ⊆ R : Hausdorff dim. of A = α
• Theorem (singularity)

The CDPM PW

Ω,ˆ λ,ˆ h law is singular w.r.t. Pref for P-a.e. W

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 22 / 28

Disordered Systems Partition Function CDPM Further Developments

Construction strategy

Macroscopic observables (finite-dimensional distributions) expressed using partition functions with suitable boundary conditions

t x y N

Pω

Ωδ,λ,h(. . .) =

Z cond

0,x C (y−x)1+α Zy,N

Z0,N

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 23 / 28

Disordered Systems Partition Function CDPM Further Developments

Construction strategy

Macroscopic observables (finite-dimensional distributions) expressed using partition functions with suitable boundary conditions

t x y N

Pω

Ωδ,λ,h(. . .) =

Z cond

0,x C (y−x)1+α Zy,N

Z0,N Scaling limit (at the process level) of (Z cond

x,y

, Zx,y)0≤x<y≤N Definition of CDPM via “finite-dimensional distributions” The same can be done for DPRE, cf. [Alberts, Khanin, Quastel ’12]

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 23 / 28

Disordered Systems Partition Function CDPM Further Developments

Continuum random field Ising model?

Analogous procedure for Ising? Need joint scaling limit of partition functions for “many” domains and boundary conditions

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 24 / 28

Disordered Systems Partition Function CDPM Further Developments

Continuum random field Ising model?

Analogous procedure for Ising? Need joint scaling limit of partition functions for “many” domains and boundary conditions Possible alternative approach: define continuum disordered law PW

Ω,ˆ λ,ˆ h

assigning its k-point function E W

Ω,ˆ λ,ˆ h

• σx1 · · · σxk
• ?

A generalization of our theorem about the scaling limit of partition functions yields the corresponding scaling limit of correlations: Eω

Ωδ,λ,h

• σx1 · · · σxk
• d

− − →

δ↓0 E W Ω,ˆ λ,ˆ h

• σx1 · · · σxk
• := Wiener chaos expansion

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 24 / 28

Disordered Systems Partition Function CDPM Further Developments

Outline

• 1. Disordered Systems and their Scaling Limits
• 2. Main Results (I): Partition Function
• 3. Main Results (II): Continuum Disordered Pinning Model
• 4. Further Developments

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 25 / 28

Disordered Systems Partition Function CDPM Further Developments

Disorder relevance vs. irrelevance

Why the restriction α > 1

2 for pinning?

[And α ∈ (1, 2] for DPRE]

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 26 / 28

Disordered Systems Partition Function CDPM Further Developments

Disorder relevance vs. irrelevance

Why the restriction α > 1

2 for pinning?

[And α ∈ (1, 2] for DPRE]

◮ The regime α < 1 2 is disorder-irrelevant for pinning models

If λ > 0 is small, the disordered model Pω

Ωδ,λ,h has same properties

(e.g. critical exponents) as the non-disordered model (λ = 0)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 26 / 28

Disordered Systems Partition Function CDPM Further Developments

Disorder relevance vs. irrelevance

Why the restriction α > 1

2 for pinning?

[And α ∈ (1, 2] for DPRE]

◮ The regime α < 1 2 is disorder-irrelevant for pinning models

If λ > 0 is small, the disordered model Pω

Ωδ,λ,h has same properties

(e.g. critical exponents) as the non-disordered model (λ = 0) Conj.: scaling limit of Pω

Ωδ,λ,h is non-disordered

[Proved for DPRE]

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 26 / 28

Disordered Systems Partition Function CDPM Further Developments

Disorder relevance vs. irrelevance

Why the restriction α > 1

2 for pinning?

[And α ∈ (1, 2] for DPRE]

◮ The regime α < 1 2 is disorder-irrelevant for pinning models

If λ > 0 is small, the disordered model Pω

Ωδ,λ,h has same properties

(e.g. critical exponents) as the non-disordered model (λ = 0) Conj.: scaling limit of Pω

Ωδ,λ,h is non-disordered

[Proved for DPRE]

◮ The regime α > 1 2 is disorder-relevant for pinning models

For any λ > 0, the disordered model Pω

Ωδ,λ,h has different properties

(e.g. critical exponents) than the non-disordered model (λ = 0)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 26 / 28

Disordered Systems Partition Function CDPM Further Developments

Disorder relevance vs. irrelevance

Why the restriction α > 1

2 for pinning?

[And α ∈ (1, 2] for DPRE]

◮ The regime α < 1 2 is disorder-irrelevant for pinning models

If λ > 0 is small, the disordered model Pω

Ωδ,λ,h has same properties

(e.g. critical exponents) as the non-disordered model (λ = 0) Conj.: scaling limit of Pω

Ωδ,λ,h is non-disordered

[Proved for DPRE]

◮ The regime α > 1 2 is disorder-relevant for pinning models

For any λ > 0, the disordered model Pω

Ωδ,λ,h has different properties

(e.g. critical exponents) than the non-disordered model (λ = 0) Our results fit this picture nicely: even though λ → 0 as δ ↓ 0, disordered survives in the scaling limit

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 26 / 28

Disordered Systems Partition Function CDPM Further Developments

Disorder relevance vs. irrelevance

Why the restriction α > 1

2 for pinning?

[And α ∈ (1, 2] for DPRE]

◮ The regime α < 1 2 is disorder-irrelevant for pinning models

If λ > 0 is small, the disordered model Pω

Ωδ,λ,h has same properties

(e.g. critical exponents) as the non-disordered model (λ = 0) Conj.: scaling limit of Pω

Ωδ,λ,h is non-disordered

[Proved for DPRE]

◮ The regime α > 1 2 is disorder-relevant for pinning models

For any λ > 0, the disordered model Pω

Ωδ,λ,h has different properties

(e.g. critical exponents) than the non-disordered model (λ = 0) Our results fit this picture nicely: even though λ → 0 as δ ↓ 0, disordered survives in the scaling limit Our restriction involving L2 convergence of k-point function (γ < d

2 )

matches with Harris criterion ν < 2

d for disorder relevance

(ν correlation length exponent ν =

1 d−γ )

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 26 / 28

Disordered Systems Partition Function CDPM Further Developments

Continuum free energy and critical exponents

Continuum partition function ZW

Ω,ˆ λ,ˆ h continuum free energy

F(ˆ λ, ˆ h) := lim

Ω↑Rd

1 Leb(Ω) log ZW

Ω,ˆ λ,ˆ h

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 27 / 28

Disordered Systems Partition Function CDPM Further Developments

Continuum free energy and critical exponents

Continuum partition function ZW

Ω,ˆ λ,ˆ h continuum free energy

F(ˆ λ, ˆ h) := lim

Ω↑Rd

1 Leb(Ω) log ZW

Ω,ˆ λ,ˆ h

Discrete free energy F(λ, h) := lim

Ω↑Zd

1 |Ω| log Z W

Ω,λ,h

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 27 / 28

Disordered Systems Partition Function CDPM Further Developments

Continuum free energy and critical exponents

Continuum partition function ZW

Ω,ˆ λ,ˆ h continuum free energy

F(ˆ λ, ˆ h) := lim

Ω↑Rd

1 Leb(Ω) log ZW

Ω,ˆ λ,ˆ h

Discrete free energy F(λ, h) := lim

Ω↑Zd

1 |Ω| log Z W

Ω,λ,h

Interchanging of limits (Ising)

lim

δ↓0

F(ˆ λ δ

7 8 , ˆ

h δ

15 8 )

δ2 = F(ˆ λ, ˆ h)

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 27 / 28

Disordered Systems Partition Function CDPM Further Developments

Continuum free energy and critical exponents

Continuum partition function ZW

Ω,ˆ λ,ˆ h continuum free energy

F(ˆ λ, ˆ h) := lim

Ω↑Rd

1 Leb(Ω) log ZW

Ω,ˆ λ,ˆ h

Discrete free energy F(λ, h) := lim

Ω↑Zd

1 |Ω| log Z W

Ω,λ,h

Interchanging of limits (Ising)

lim

δ↓0

F(ˆ λ δ

7 8 , ˆ

h δ

15 8 )

δ2 = F(ˆ λ, ˆ h)

Conjecture

lim

h↓0

σ0ˆ

λ h

7 15 ,h

h

1 15

= ∂F ∂h (ˆ λ, 1) refining [Camia, Garban, Newman ’12]

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 27 / 28

Disordered Systems Partition Function CDPM Further Developments

Thanks

Francesco Caravenna Scaling Limits of Disordered Systems June 6, 2014 28 / 28