Polynomial Chaos and Scaling Limits of Disordered Systems 2. - - PowerPoint PPT Presentation

White noise Continuum partition functions The continuum DPRE Pinning models

Polynomial Chaos and Scaling Limits of Disordered Systems

• 2. Continuum model and free energy estimates

Francesco Caravenna

Universit` a degli Studi di Milano-Bicocca YEP XIII, Eurandom ∼ March 7-11, 2016

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 1 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Overview

In the previous lecture we saw how to build continuum partition functions Zω

δ d

− − − →

δ→0

ZW (scaling limits of discrete partition functions)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 2 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Overview

In the previous lecture we saw how to build continuum partition functions Zω

δ d

− − − →

δ→0

ZW (scaling limits of discrete partition functions) In this lecture we present two interesting applications of ZW

◮ Scaling limit of the full probability measure Pω δ d

− − − →

δ→0

PW constructing a continuum version of the disordered system

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 2 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Overview

In the previous lecture we saw how to build continuum partition functions Zω

δ d

− − − →

δ→0

ZW (scaling limits of discrete partition functions) In this lecture we present two interesting applications of ZW

◮ Scaling limit of the full probability measure Pω δ d

− − − →

δ→0

PW constructing a continuum version of the disordered system We will focus on the DPRE [Alberts, Khanin, Quastel 2014b] drawing inspiration from the Pinning [C., Sun, Zygouras 2016]

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 2 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Overview

In the previous lecture we saw how to build continuum partition functions Zω

δ d

− − − →

δ→0

ZW (scaling limits of discrete partition functions) In this lecture we present two interesting applications of ZW

◮ Scaling limit of the full probability measure Pω δ d

− − − →

δ→0

PW constructing a continuum version of the disordered system We will focus on the DPRE [Alberts, Khanin, Quastel 2014b] drawing inspiration from the Pinning [C., Sun, Zygouras 2016]

◮ Sharp asymptotics on the discrete model, in terms of free energy and

critical curve

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 2 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Overview

In the previous lecture we saw how to build continuum partition functions Zω

δ d

− − − →

δ→0

ZW (scaling limits of discrete partition functions) In this lecture we present two interesting applications of ZW

◮ Scaling limit of the full probability measure Pω δ d

− − − →

δ→0

PW constructing a continuum version of the disordered system We will focus on the DPRE [Alberts, Khanin, Quastel 2014b] drawing inspiration from the Pinning [C., Sun, Zygouras 2016]

◮ Sharp asymptotics on the discrete model, in terms of free energy and

critical curve For this we will focus on Pinning models (rather than DPRE)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 2 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Outline

• 1. White noise and Wiener chaos
• 2. Continuum partition functions
• 3. The continuum DPRE
• 4. Pinning models

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 3 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Outline

• 1. White noise and Wiener chaos
• 2. Continuum partition functions
• 3. The continuum DPRE
• 4. Pinning models

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 4 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

White noise (1 dim.)

We are familiar with (1-dim.) Brownian motion B = (B(t))t≥0

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 5 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

White noise (1 dim.)

We are familiar with (1-dim.) Brownian motion B = (B(t))t≥0 We are interested in its derivative “W (t) := d

dt B(t)” called white noise

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 5 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

White noise (1 dim.)

We are familiar with (1-dim.) Brownian motion B = (B(t))t≥0 We are interested in its derivative “W (t) := d

dt B(t)” called white noise

Think of W as a stochastic process W = (W (·)) indexed by Intervals I = [a, b] − → W (I) = B(b) − B(a) ∼ N(0, b − a)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 5 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

White noise (1 dim.)

We are familiar with (1-dim.) Brownian motion B = (B(t))t≥0 We are interested in its derivative “W (t) := d

dt B(t)” called white noise

Think of W as a stochastic process W = (W (·)) indexed by Intervals I = [a, b] − → W (I) = B(b) − B(a) ∼ N(0, b − a) Borel sets A ∈ B(R) − → W (A) =

• R

1A(t) dB(t) ∼ N(0, |A|)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 5 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

White noise (1 dim.)

We are familiar with (1-dim.) Brownian motion B = (B(t))t≥0 We are interested in its derivative “W (t) := d

dt B(t)” called white noise

Think of W as a stochastic process W = (W (·)) indexed by Intervals I = [a, b] − → W (I) = B(b) − B(a) ∼ N(0, b − a) Borel sets A ∈ B(R) − → W (A) =

• R

1A(t) dB(t) ∼ N(0, |A|) W is a Gaussian process with E[W (A)] = 0 Cov[W (A), W (B)] = |A ∩ B|

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 5 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

White noise

White noise on Rd

It is a Gaussian process W = (W (A))A∈B(Rd) with E[W (A)] = 0 Cov[W (A), W (B)] = |A ∩ B|

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 6 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

White noise

White noise on Rd

It is a Gaussian process W = (W (A))A∈B(Rd) with E[W (A)] = 0 Cov[W (A), W (B)] = |A ∩ B|

◮ ∀ (An)n∈N disjoint

= ⇒ W

n∈N

An

• a.s.

=

• n∈N

W (An) Almost a random signed measure on Rd. . . (but not quite!)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 6 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

White noise

White noise on Rd

It is a Gaussian process W = (W (A))A∈B(Rd) with E[W (A)] = 0 Cov[W (A), W (B)] = |A ∩ B|

◮ ∀ (An)n∈N disjoint

= ⇒ W

n∈N

An

• a.s.

=

• n∈N

W (An) Almost a random signed measure on Rd. . . (but not quite!) We can define single stochastic integrals W (f ) :=

• f (x) W (dx)

E[W (f )] = 0 E[W (f )2] = f 2

L2(Rd)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 6 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Multiple stochastic integrals

We can define W ⊗k(g) =

• (Rd)k g(x1, . . . , xk) W (dx1) · · · W (dxk)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 7 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Multiple stochastic integrals

We can define W ⊗k(g) =

• (Rd)k g(x1, . . . , xk) W (dx1) · · · W (dxk)

For d = 1 we can restrict x1 < x2 < . . . < xk iterated Ito integrals

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 7 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Multiple stochastic integrals

We can define W ⊗k(g) =

• (Rd)k g(x1, . . . , xk) W (dx1) · · · W (dxk)

For d = 1 we can restrict x1 < x2 < . . . < xk iterated Ito integrals For symmetric functions we have E[W ⊗k(g)] = 0 E[W ⊗k(g)2] = k! g2

L2((Rd)k)

Cov[W ⊗k(f ), W ⊗k′(g)] = 0 ∀k = k′

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 7 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Multiple stochastic integrals

We can define W ⊗k(g) =

• (Rd)k g(x1, . . . , xk) W (dx1) · · · W (dxk)

For d = 1 we can restrict x1 < x2 < . . . < xk iterated Ito integrals For symmetric functions we have E[W ⊗k(g)] = 0 E[W ⊗k(g)2] = k! g2

L2((Rd)k)

Cov[W ⊗k(f ), W ⊗k′(g)] = 0 ∀k = k′

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 7 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Multiple stochastic integrals

We can define W ⊗k(g) =

• (Rd)k g(x1, . . . , xk) W (dx1) · · · W (dxk)

For d = 1 we can restrict x1 < x2 < . . . < xk iterated Ito integrals For symmetric functions we have E[W ⊗k(g)] = 0 E[W ⊗k(g)2] = k! g2

L2((Rd)k)

Cov[W ⊗k(f ), W ⊗k′(g)] = 0 ∀k = k′

Wiener chaos expansion

Any r.v. X ∈ L2(ΩW ) measurable w.r.t. σ(W ) can be written as X =

∞

• k=0

1 k!W ⊗k(fk) with fk ∈ L2

sym((Rd)k)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 7 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Discrete sums and stochastic integrals

Consider a lattice Tδ ⊆ Rd whose cells have volume vδ → 0

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 8 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Discrete sums and stochastic integrals

Consider a lattice Tδ ⊆ Rd whose cells have volume vδ → 0 Take i.i.d. random variables (X z)z∈Tδ with zero mean and unit variance

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 8 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Discrete sums and stochastic integrals

Consider a lattice Tδ ⊆ Rd whose cells have volume vδ → 0 Take i.i.d. random variables (X z)z∈Tδ with zero mean and unit variance Consider the “stochastic Riemann sum” (multi-linear polynomial) Ψδ :=

• (z1,...,zk)∈(Tδ)k

zi=zi ∀i=j

f (z1, . . . , zk) X z1 X z2 · · · X zk where f ∈ L2(Rd) is (say) continuous.

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 8 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Discrete sums and stochastic integrals

Consider a lattice Tδ ⊆ Rd whose cells have volume vδ → 0 Take i.i.d. random variables (X z)z∈Tδ with zero mean and unit variance Consider the “stochastic Riemann sum” (multi-linear polynomial) Ψδ :=

• (z1,...,zk)∈(Tδ)k

zi=zi ∀i=j

f (z1, . . . , zk) X z1 X z2 · · · X zk where f ∈ L2(Rd) is (say) continuous. (√vδ)k Ψδ

d

− − − →

δ→0

• (Rd)k g(z1, . . . , zk) W (dz1) · · · W (dzk)

(Check the variance!)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 8 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Outline

• 1. White noise and Wiener chaos
• 2. Continuum partition functions
• 3. The continuum DPRE
• 4. Pinning models

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 9 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum partition function for DPRE

1d rescaled RW Sδ

t :=

√ δSt/δ lives on Tδ =

• [0, 1] ∩ δN0
• ×

√ δZ

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 10 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum partition function for DPRE

1d rescaled RW Sδ

t :=

√ δSt/δ lives on Tδ =

• [0, 1] ∩ δN0
• ×

√ δZ Zω

δ = Eref

exp

• Hω

= Eref

• exp
• N
• n=1
• βω(n,Sn) − λ(β)
• Francesco Caravenna

Scaling Limits of Disordered Systems March 7-11, 2016 10 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum partition function for DPRE

1d rescaled RW Sδ

t :=

√ δSt/δ lives on Tδ =

• [0, 1] ∩ δN0
• ×

√ δZ Zω

δ = Eref

exp

• Hω

= Eref

• exp
• N
• n=1
• βω(n,Sn) − λ(β)
• = 1 +
• (t,x)∈Tδ

Pref(Sδ

t = x) X t,x

+ 1 2

• (t,x)=(t′,x′)∈Tδ

Pref(Sδ

t = x, Sδ t′ = x′) X t,x X t′,x′ + . . .

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 10 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum partition function for DPRE

1d rescaled RW Sδ

t :=

√ δSt/δ lives on Tδ =

• [0, 1] ∩ δN0
• ×

√ δZ Zω

δ = Eref

exp

• Hω

= Eref

• exp
• N
• n=1
• βω(n,Sn) − λ(β)
• = 1 +
• (t,x)∈Tδ

Pref(Sδ

t = x) X t,x

+ 1 2

• (t,x)=(t′,x′)∈Tδ

Pref(Sδ

t = x, Sδ t′ = x′) X t,x X t′,x′ + . . .

Recall the LLT: Pref(Sn = x) ∼

1 √n g

x

√n

• with g(z) = e− z2

2

√ 2π

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 10 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum partition function for DPRE

1d rescaled RW Sδ

t :=

√ δSt/δ lives on Tδ =

• [0, 1] ∩ δN0
• ×

√ δZ Zω

δ = Eref

exp

• Hω

= Eref

• exp
• N
• n=1
• βω(n,Sn) − λ(β)
• = 1 +
• (t,x)∈Tδ

Pref(Sδ

t = x) X t,x

+ 1 2

• (t,x)=(t′,x′)∈Tδ

Pref(Sδ

t = x, Sδ t′ = x′) X t,x X t′,x′ + . . .

Recall the LLT: Pref(Sn = x) ∼

1 √n g

x

√n

• with g(z) = e− z2

2

√ 2π

Pref(Sδ

t = x) = Pref(S t

δ =

x √ δ) ∼

√ δ gt(x) gt(x) = e− x2

2t

√ 2πt

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 10 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum partition function for DPRE

1d rescaled RW Sδ

t :=

√ δSt/δ lives on Tδ =

• [0, 1] ∩ δN0
• ×

√ δZ Zω

δ = Eref

exp

• Hω

= Eref

• exp
• N
• n=1
• βω(n,Sn) − λ(β)
• = 1 +
• (t,x)∈Tδ

Pref(Sδ

t = x) X t,x

+ 1 2

• (t,x)=(t′,x′)∈Tδ

Pref(Sδ

t = x, Sδ t′ = x′) X t,x X t′,x′ + . . .

Recall the LLT: Pref(Sn = x) ∼

1 √n g

x

√n

• with g(z) = e− z2

2

√ 2π

Pref(Sδ

t = x) = Pref(S t

δ =

x √ δ) ∼

√ δ gt(x) gt(x) = e− x2

2t

√ 2πt Replacing X t,x = e(βω(t,x)−λ(β)) − 1 ≈ βY t,x with Y t,x i.i.d. N(0, 1)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 10 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum partition function for DPRE

Zω

N = 1 + β

√ δ

• (t,x)∈Tδ

gt(x) Y t,x + 1 2 (β √ δ)2

• (t,x)=(t′,x′)∈Tδ

gt(x) gt′−t(x′ − x) Y t,x Y t′,x′ + . . .

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 11 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum partition function for DPRE

Zω

N = 1 + β

√ δ

• (t,x)∈Tδ

gt(x) Y t,x + 1 2 (β √ δ)2

• (t,x)=(t′,x′)∈Tδ

gt(x) gt′−t(x′ − x) Y t,x Y t′,x′ + . . . Cells in Tδ have volume vδ = δ √ δ = δ

3 2 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 11 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum partition function for DPRE

Zω

N = 1 + β

√ δ

• (t,x)∈Tδ

gt(x) Y t,x + 1 2 (β √ δ)2

• (t,x)=(t′,x′)∈Tδ

gt(x) gt′−t(x′ − x) Y t,x Y t′,x′ + . . . Cells in Tδ have volume vδ = δ √ δ = δ

3 2

“Stochastic Riemann sums” converge to stochastic integrals if β √ δ ≈ √vδ

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 11 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum partition function for DPRE

Zω

N = 1 + β

√ δ

• (t,x)∈Tδ

gt(x) Y t,x + 1 2 (β √ δ)2

• (t,x)=(t′,x′)∈Tδ

gt(x) gt′−t(x′ − x) Y t,x Y t′,x′ + . . . Cells in Tδ have volume vδ = δ √ δ = δ

3 2

“Stochastic Riemann sums” converge to stochastic integrals if β √ δ ≈ √vδ β ∼ ˆ β δ

1 4 =

ˆ β N

1 4 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 11 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum partition function for DPRE

Zω

N = 1 + β

√ δ

• (t,x)∈Tδ

gt(x) Y t,x + 1 2 (β √ δ)2

• (t,x)=(t′,x′)∈Tδ

gt(x) gt′−t(x′ − x) Y t,x Y t′,x′ + . . . Cells in Tδ have volume vδ = δ √ δ = δ

3 2

“Stochastic Riemann sums” converge to stochastic integrals if β √ δ ≈ √vδ β ∼ ˆ β δ

1 4 =

ˆ β N

1 4

Zω

N d

− − − →

δ→0

ZW = 1 + ˆ β

• [0,1]×R

gt(x) W (dtdx) + ˆ β2 2

• ([0,1]×R)2 gt(x) gt′−t(x′ − x) W (dtdx) W (dt′dx′)

+ . . .

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 11 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Constrained partition functions

We have constructed ZW = “free” partition function on [0, 1] × R RW paths starting at (0, 0) with no constraint on right endpoint ZW = ZW (0, 0), (1, ⋆)

• E
• ZW

= 1

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 12 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Constrained partition functions

We have constructed ZW = “free” partition function on [0, 1] × R RW paths starting at (0, 0) with no constraint on right endpoint ZW = ZW (0, 0), (1, ⋆)

• E
• ZW

= 1 Consider now constrained partition functions: for (s, y), (t, x) ∈ [0, 1] × R Discrete: Zω

δ

• (s, y), (t, x)
• = Eref

exp

• Hω

1{Sδ

t =x}

• Sδ

s = y

• Francesco Caravenna

Scaling Limits of Disordered Systems March 7-11, 2016 12 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Constrained partition functions

We have constructed ZW = “free” partition function on [0, 1] × R RW paths starting at (0, 0) with no constraint on right endpoint ZW = ZW (0, 0), (1, ⋆)

• E
• ZW

= 1 Consider now constrained partition functions: for (s, y), (t, x) ∈ [0, 1] × R Discrete: Zω

δ

• (s, y), (t, x)
• = Eref

exp

• Hω

1{Sδ

t =x}

• Sδ

s = y

• Divided by

√ δ, they converge to a continuum limit: ZW (s, y), (t, x)

• E
• ZW

(s, y), (t, x)

• = gt−s(x − y)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 12 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Constrained partition functions

We have constructed ZW = “free” partition function on [0, 1] × R RW paths starting at (0, 0) with no constraint on right endpoint ZW = ZW (0, 0), (1, ⋆)

• E
• ZW

= 1 Consider now constrained partition functions: for (s, y), (t, x) ∈ [0, 1] × R Discrete: Zω

δ

• (s, y), (t, x)
• = Eref

exp

• Hω

1{Sδ

t =x}

• Sδ

s = y

• Divided by

√ δ, they converge to a continuum limit: ZW (s, y), (t, x)

• E
• ZW

(s, y), (t, x)

• = gt−s(x − y)

This is a function of white noise in the stripe W ([s, t] × R)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 12 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Constrained partition functions

We have constructed ZW = “free” partition function on [0, 1] × R RW paths starting at (0, 0) with no constraint on right endpoint ZW = ZW (0, 0), (1, ⋆)

• E
• ZW

= 1 Consider now constrained partition functions: for (s, y), (t, x) ∈ [0, 1] × R Discrete: Zω

δ

• (s, y), (t, x)
• = Eref

exp

• Hω

1{Sδ

t =x}

• Sδ

s = y

• Divided by

√ δ, they converge to a continuum limit: ZW (s, y), (t, x)

• E
• ZW

(s, y), (t, x)

• = gt−s(x − y)

This is a function of white noise in the stripe W ([s, t] × R) Four-parameter random process ZW (s, y), (t, x)

• regularity?

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 12 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Key properties

Key properties

For a.e. realization of W the following properties hold:

◮ Continuity: ZW ((s, y), (t, x)) is jointly continuous in (s, y, t, x)

(on the domain s < t)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 13 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Key properties

Key properties

For a.e. realization of W the following properties hold:

◮ Continuity: ZW ((s, y), (t, x)) is jointly continuous in (s, y, t, x)

(on the domain s < t)

◮ Positivity: ZW ((s, y), (t, x)) > 0 for all (s, y, t, x) satisfying s < t

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 13 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Key properties

Key properties

For a.e. realization of W the following properties hold:

◮ Continuity: ZW ((s, y), (t, x)) is jointly continuous in (s, y, t, x)

(on the domain s < t)

◮ Positivity: ZW ((s, y), (t, x)) > 0 for all (s, y, t, x) satisfying s < t ◮ Semigroup (Chapman-Kolmogorov): for all s < r < t and x, y ∈ R

ZW ((s, y), (t, x)) =

• R

ZW ((s, y), (r, z)) ZW ((r, z), (t, x)) dz (Inherited from discrete partition functions: drawing!)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 13 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Key properties

Key properties

For a.e. realization of W the following properties hold:

◮ Continuity: ZW ((s, y), (t, x)) is jointly continuous in (s, y, t, x)

(on the domain s < t)

◮ Positivity: ZW ((s, y), (t, x)) > 0 for all (s, y, t, x) satisfying s < t ◮ Semigroup (Chapman-Kolmogorov): for all s < r < t and x, y ∈ R

ZW ((s, y), (t, x)) =

• R

ZW ((s, y), (r, z)) ZW ((r, z), (t, x)) dz (Inherited from discrete partition functions: drawing!) How to prove these properties?

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 13 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

The 1d Stochastic Heat Equation

The four-parameter field ZW ((s, y), (t, x)) solves the 1d SHE    ∂tZW = 1

2∆xZW + ˆ

β W ZW limt↓s ZW ((s, y), (t, x)) = δ(y − x) Checked directly from Wiener chaos expansion (mild solution) It is known that solutions to the SHE satisfy the properties above

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 14 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

The 1d Stochastic Heat Equation

The four-parameter field ZW ((s, y), (t, x)) solves the 1d SHE    ∂tZW = 1

2∆xZW + ˆ

β W ZW limt↓s ZW ((s, y), (t, x)) = δ(y − x) Checked directly from Wiener chaos expansion (mild solution) It is known that solutions to the SHE satisfy the properties above

Alternative approach (to check, OK for pinning [C., Sun, Zygouras 2016])

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 14 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

The 1d Stochastic Heat Equation

The four-parameter field ZW ((s, y), (t, x)) solves the 1d SHE    ∂tZW = 1

2∆xZW + ˆ

β W ZW limt↓s ZW ((s, y), (t, x)) = δ(y − x) Checked directly from Wiener chaos expansion (mild solution) It is known that solutions to the SHE satisfy the properties above

Alternative approach (to check, OK for pinning [C., Sun, Zygouras 2016])

◮ Prove continuity by Kolmogorov criterion, showing that

ZW ((s, y), (t, x)) gt−s(x − y) is continuous also for t = s

◮ Use continuity to prove semigroup for all times ◮ Use continuity to deduce positivity for close times, then bootstrap to

arbitrary times using semigroup

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 14 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Outline

• 1. White noise and Wiener chaos
• 2. Continuum partition functions
• 3. The continuum DPRE
• 4. Pinning models

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 15 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

A naive approach

Consider DPRE in d = 1 (random walk + disorder) Pω(S) ∝ e

N

n=1 β ω(n,Sn) Pref(S)

Can we define its continuum analogue (BM + disorder)?

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 16 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

A naive approach

Consider DPRE in d = 1 (random walk + disorder) Pω(S) ∝ e

N

n=1 β ω(n,Sn) Pref(S)

Can we define its continuum analogue (BM + disorder)? Naively PW (dB) ∝ e

1

0 ˆ

β W (t,Bt) dt Pref(dB)

Pref = law of BM W (t, x) = white noise on R2 (space-time)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 16 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

A naive approach

Consider DPRE in d = 1 (random walk + disorder) Pω(S) ∝ e

N

n=1 β ω(n,Sn) Pref(S)

Can we define its continuum analogue (BM + disorder)? Naively PW (dB) ∝ e

1

0 ˆ

β W (t,Bt) dt Pref(dB)

Pref = law of BM W (t, x) = white noise on R2 (space-time)

◮ 1 0 W (t, Bt) dt ill-defined. Regularization?

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 16 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

A naive approach

Consider DPRE in d = 1 (random walk + disorder) Pω(S) ∝ e

N

n=1 β ω(n,Sn) Pref(S)

Can we define its continuum analogue (BM + disorder)? Naively PW (dB) ∝ e

1

0 ˆ

β W (t,Bt) dt Pref(dB)

Pref = law of BM W (t, x) = white noise on R2 (space-time)

◮ 1 0 W (t, Bt) dt ill-defined. Regularization?

NO! The problem is more subtle (and interesting!)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 16 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Partition functions and f.d.d.

Start from discrete: distribution of DPRE at two times 0 < t < t′ < 1 Pω

δ (Sδ t = x, Sδ t′ = x′) = Zω δ

• (0, 0), (t, x)
• Zω

δ

• (t, x), (t′, x′)
• Zω

δ

• (t′, x′), (1, ⋆)
• Zω

δ

• (0, 0), (1, ⋆)
• (drawing!) Analogous formula for any finite number of times

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 17 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Partition functions and f.d.d.

Start from discrete: distribution of DPRE at two times 0 < t < t′ < 1 Pω

δ (Sδ t = x, Sδ t′ = x′) = Zω δ

• (0, 0), (t, x)
• Zω

δ

• (t, x), (t′, x′)
• Zω

δ

• (t′, x′), (1, ⋆)
• Zω

δ

• (0, 0), (1, ⋆)
• (drawing!) Analogous formula for any finite number of times

Idea: Replace Zω

δ ZW to define the law of continuum DPRE

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 17 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Partition functions and f.d.d.

Start from discrete: distribution of DPRE at two times 0 < t < t′ < 1 Pω

δ (Sδ t = x, Sδ t′ = x′) = Zω δ

• (0, 0), (t, x)
• Zω

δ

• (t, x), (t′, x′)
• Zω

δ

• (t′, x′), (1, ⋆)
• Zω

δ

• (0, 0), (1, ⋆)
• (drawing!) Analogous formula for any finite number of times

Idea: Replace Zω

δ ZW to define the law of continuum DPRE

Recall: to define a process (Xt)t∈[0,1] it is enough (Kolmogorov) to assign finite-dimensional distributions (f.d.d.) µt1,...,tk(A1, . . . , Ak)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 17 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Partition functions and f.d.d.

Start from discrete: distribution of DPRE at two times 0 < t < t′ < 1 Pω

δ (Sδ t = x, Sδ t′ = x′) = Zω δ

• (0, 0), (t, x)
• Zω

δ

• (t, x), (t′, x′)
• Zω

δ

• (t′, x′), (1, ⋆)
• Zω

δ

• (0, 0), (1, ⋆)
• (drawing!) Analogous formula for any finite number of times

Idea: Replace Zω

δ ZW to define the law of continuum DPRE

Recall: to define a process (Xt)t∈[0,1] it is enough (Kolmogorov) to assign finite-dimensional distributions (f.d.d.) µt1,...,tk(A1, . . . , Ak) “ = P(Xt1 ∈ A1, . . . , Xtk ∈ Ak) ”

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 17 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Partition functions and f.d.d.

Start from discrete: distribution of DPRE at two times 0 < t < t′ < 1 Pω

δ (Sδ t = x, Sδ t′ = x′) = Zω δ

• (0, 0), (t, x)
• Zω

δ

• (t, x), (t′, x′)
• Zω

δ

• (t′, x′), (1, ⋆)
• Zω

δ

• (0, 0), (1, ⋆)
• (drawing!) Analogous formula for any finite number of times

Idea: Replace Zω

δ ZW to define the law of continuum DPRE

Recall: to define a process (Xt)t∈[0,1] it is enough (Kolmogorov) to assign finite-dimensional distributions (f.d.d.) µt1,...,tk(A1, . . . , Ak) “ = P(Xt1 ∈ A1, . . . , Xtk ∈ Ak) ” that are consistent µt1,...,tj,...,tk(A1, . . . , R, . . . , Ak)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 17 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Partition functions and f.d.d.

Start from discrete: distribution of DPRE at two times 0 < t < t′ < 1 Pω

δ (Sδ t = x, Sδ t′ = x′) = Zω δ

• (0, 0), (t, x)
• Zω

δ

• (t, x), (t′, x′)
• Zω

δ

• (t′, x′), (1, ⋆)
• Zω

δ

• (0, 0), (1, ⋆)
• (drawing!) Analogous formula for any finite number of times

Idea: Replace Zω

δ ZW to define the law of continuum DPRE

Recall: to define a process (Xt)t∈[0,1] it is enough (Kolmogorov) to assign finite-dimensional distributions (f.d.d.) µt1,...,tk(A1, . . . , Ak) “ = P(Xt1 ∈ A1, . . . , Xtk ∈ Ak) ” that are consistent µt1,...,tj,...,tk(A1, . . . , R, . . . , Ak) = µt1,...,tj−1,tj+1,...,tk(A1, . . . , Aj−1, Aj+1, . . . , Ak)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 17 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

The continuum 1d DPRE

◮ Fix

ˆ β ∈ (0, ∞) (on which ZW depend)

[recall that β ∼ ˆ βδ

1 4 ] Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 18 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

The continuum 1d DPRE

◮ Fix

ˆ β ∈ (0, ∞) (on which ZW depend)

[recall that β ∼ ˆ βδ

1 4 ]

◮ Fix space-time white noise W on [0, 1] × R and a realization of

continuum partition functions ZW satisfying the key properties (continuity, strict positivity, semigroup)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 18 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

The continuum 1d DPRE

◮ Fix

ˆ β ∈ (0, ∞) (on which ZW depend)

[recall that β ∼ ˆ βδ

1 4 ]

◮ Fix space-time white noise W on [0, 1] × R and a realization of

continuum partition functions ZW satisfying the key properties (continuity, strict positivity, semigroup) The Continuum DPRE is the process (Xt)t∈[0,1] with f.d.d. PW (Xt ∈ dx, Xt′ ∈ dx′) dx dx′ := ZW (0, 0), (t, x)

• ZW

(t, x), (t′, x′)

• ZW

(t′, x′), (1, ⋆)

• ZW

(0, 0), (1, ⋆)

• Francesco Caravenna

Scaling Limits of Disordered Systems March 7-11, 2016 18 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

The continuum 1d DPRE

◮ Fix

ˆ β ∈ (0, ∞) (on which ZW depend)

[recall that β ∼ ˆ βδ

1 4 ]

◮ Fix space-time white noise W on [0, 1] × R and a realization of

continuum partition functions ZW satisfying the key properties (continuity, strict positivity, semigroup) The Continuum DPRE is the process (Xt)t∈[0,1] with f.d.d. PW (Xt ∈ dx, Xt′ ∈ dx′) dx dx′ := ZW (0, 0), (t, x)

• ZW

(t, x), (t′, x′)

• ZW

(t′, x′), (1, ⋆)

• ZW

(0, 0), (1, ⋆)

• ◮ Well-defined by strict positivity of ZW

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 18 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

The continuum 1d DPRE

◮ Fix

ˆ β ∈ (0, ∞) (on which ZW depend)

[recall that β ∼ ˆ βδ

1 4 ]

◮ Fix space-time white noise W on [0, 1] × R and a realization of

continuum partition functions ZW satisfying the key properties (continuity, strict positivity, semigroup) The Continuum DPRE is the process (Xt)t∈[0,1] with f.d.d. PW (Xt ∈ dx, Xt′ ∈ dx′) dx dx′ := ZW (0, 0), (t, x)

• ZW

(t, x), (t′, x′)

• ZW

(t′, x′), (1, ⋆)

• ZW

(0, 0), (1, ⋆)

• ◮ Well-defined by strict positivity of ZW

◮ Consistent by semigroup property

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 18 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Relation with Wiener measure

The law of the continuum DPRE is a random probability PW (X ∈ · ) (quenched law) for the process X = (Xt)t∈[0,1]

[ Probab. kernel S′(R) → R[0,1] ]

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 19 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Relation with Wiener measure

The law of the continuum DPRE is a random probability PW (X ∈ · ) (quenched law) for the process X = (Xt)t∈[0,1]

[ Probab. kernel S′(R) → R[0,1] ]

Define a new law ˜ P (mutually absolutely continuous) for disorder W by d˜ P dP(W ) = ZW (0, 0), (1, ⋆)

• Francesco Caravenna

Scaling Limits of Disordered Systems March 7-11, 2016 19 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Relation with Wiener measure

The law of the continuum DPRE is a random probability PW (X ∈ · ) (quenched law) for the process X = (Xt)t∈[0,1]

[ Probab. kernel S′(R) → R[0,1] ]

Define a new law ˜ P (mutually absolutely continuous) for disorder W by d˜ P dP(W ) = ZW (0, 0), (1, ⋆)

• Key Lemma

Pann(X ∈ ·) :=

• S′(R)

PW (X ∈ · ) ˜ P(dW )

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 19 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Relation with Wiener measure

The law of the continuum DPRE is a random probability PW (X ∈ · ) (quenched law) for the process X = (Xt)t∈[0,1]

[ Probab. kernel S′(R) → R[0,1] ]

Define a new law ˜ P (mutually absolutely continuous) for disorder W by d˜ P dP(W ) = ZW (0, 0), (1, ⋆)

• Key Lemma

Pann(X ∈ ·) :=

• S′(R)

PW (X ∈ · ) ˜ P(dW ) = P(BM ∈ · )

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 19 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Relation with Wiener measure

The law of the continuum DPRE is a random probability PW (X ∈ · ) (quenched law) for the process X = (Xt)t∈[0,1]

[ Probab. kernel S′(R) → R[0,1] ]

Define a new law ˜ P (mutually absolutely continuous) for disorder W by d˜ P dP(W ) = ZW (0, 0), (1, ⋆)

• Key Lemma

Pann(X ∈ ·) :=

• S′(R)

PW (X ∈ · ) ˜ P(dW ) = P(BM ∈ · )

• Proof. The factor ZW in ˜

P cancels the denominator in the f.d.d. for PW Since E

• ZW

(s, y), (t, x)

• = gt−s(x − y) one gets f.d.d. of BM

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 19 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Absolute continuity properties

Theorem

∀A ⊆ R[0,1] : P(BM ∈ A) = 1 ⇒ PW (X ∈ A) = 1 for P-a.e. W

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 20 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Absolute continuity properties

Theorem

∀A ⊆ R[0,1] : P(BM ∈ A) = 1 ⇒ PW (X ∈ A) = 1 for P-a.e. W Any given a.s. property of BM is an a.s. property of continuum DPRE, for a.e. realization of the disorder W

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 20 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Absolute continuity properties

Theorem

∀A ⊆ R[0,1] : P(BM ∈ A) = 1 ⇒ PW (X ∈ A) = 1 for P-a.e. W Any given a.s. property of BM is an a.s. property of continuum DPRE, for a.e. realization of the disorder W

Corollary

PW (X has H¨

• lder paths with exp. 1

2−) = 1

for P-a.e. W We can thus realize PW as a law on C([0, 1], R), for P-a.e. W (More precisely: PW admits a modification with H¨

• lder paths)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 20 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Absolute continuity properties

Theorem

∀A ⊆ R[0,1] : P(BM ∈ A) = 1 ⇒ PW (X ∈ A) = 1 for P-a.e. W Any given a.s. property of BM is an a.s. property of continuum DPRE, for a.e. realization of the disorder W

Corollary

PW (X has H¨

• lder paths with exp. 1

2−) = 1

for P-a.e. W We can thus realize PW as a law on C([0, 1], R), for P-a.e. W (More precisely: PW admits a modification with H¨

• lder paths)

One is tempted to conclude that PW is absolutely continuous w.r.t. Wiener measure, for P-a.e. W . . .

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 20 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Absolute continuity properties

Theorem

∀A ⊆ R[0,1] : P(BM ∈ A) = 1 ⇒ PW (X ∈ A) = 1 for P-a.e. W Any given a.s. property of BM is an a.s. property of continuum DPRE, for a.e. realization of the disorder W

Corollary

PW (X has H¨

• lder paths with exp. 1

2−) = 1

for P-a.e. W We can thus realize PW as a law on C([0, 1], R), for P-a.e. W (More precisely: PW admits a modification with H¨

• lder paths)

One is tempted to conclude that PW is absolutely continuous w.r.t. Wiener measure, for P-a.e. W . . . NO ! “ ∀A ” and “ for P-a.e. W ” cannot be exchanged!

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 20 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Singularity properties

Theorem

The law PW is singular w.r.t. Wiener measure, for P-a.e. W .

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 21 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Singularity properties

Theorem

The law PW is singular w.r.t. Wiener measure, for P-a.e. W . for P-a.e. W ∃ A = AW ⊆ C([0, 1], R) : PW (X ∈ A) = 1 vs. P(BM ∈ A) = 0

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 21 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Singularity properties

Theorem

The law PW is singular w.r.t. Wiener measure, for P-a.e. W . for P-a.e. W ∃ A = AW ⊆ C([0, 1], R) : PW (X ∈ A) = 1 vs. P(BM ∈ A) = 0 Unlike discrete DPRE, there is no continuum Hamiltonian PW (X ∈ · ) ∝ eHW ( · )P(BM ∈ · ) Absolute continuity is lost in the scaling limit

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 21 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Singularity properties

Theorem

The law PW is singular w.r.t. Wiener measure, for P-a.e. W . for P-a.e. W ∃ A = AW ⊆ C([0, 1], R) : PW (X ∈ A) = 1 vs. P(BM ∈ A) = 0 Unlike discrete DPRE, there is no continuum Hamiltonian PW (X ∈ · ) ∝ eHW ( · )P(BM ∈ · ) Absolute continuity is lost in the scaling limit In a sense, the laws PW are just barely not absolutely continuous w.r.t. Wiener measure (“stochastically absolutely continuous”)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 21 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of singularity

Let (Xt)t∈[0,1] be the canonical process on C([0, 1], R)

[ Xt(f ) = f (t) ]

Let Fn := σ(Xtn

i : tn

i = i 2n , 0 ≤ i ≤ 2n) be the dyadic filtration

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 22 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of singularity

Let (Xt)t∈[0,1] be the canonical process on C([0, 1], R)

[ Xt(f ) = f (t) ]

Let Fn := σ(Xtn

i : tn

i = i 2n , 0 ≤ i ≤ 2n) be the dyadic filtration

Fix a typical realization of W . Setting Pref = Wiener measure RW

n (X) := dPW |Fn

dPref|Fn (X)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 22 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of singularity

Let (Xt)t∈[0,1] be the canonical process on C([0, 1], R)

[ Xt(f ) = f (t) ]

Let Fn := σ(Xtn

i : tn

i = i 2n , 0 ≤ i ≤ 2n) be the dyadic filtration

Fix a typical realization of W . Setting Pref = Wiener measure RW

n (X) := dPW |Fn

dPref|Fn (X) The process (RW

n )n∈N is a martingale w.r.t. Pref (exercise!)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 22 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of singularity

Let (Xt)t∈[0,1] be the canonical process on C([0, 1], R)

[ Xt(f ) = f (t) ]

Let Fn := σ(Xtn

i : tn

i = i 2n , 0 ≤ i ≤ 2n) be the dyadic filtration

Fix a typical realization of W . Setting Pref = Wiener measure RW

n (X) := dPW |Fn

dPref|Fn (X) The process (RW

n )n∈N is a martingale w.r.t. Pref (exercise!)

Since RW

n

≥ 0, the martingale converges: RW

n a.s.

− − − →

n→∞ RW ∞

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 22 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of singularity

Let (Xt)t∈[0,1] be the canonical process on C([0, 1], R)

[ Xt(f ) = f (t) ]

Let Fn := σ(Xtn

i : tn

i = i 2n , 0 ≤ i ≤ 2n) be the dyadic filtration

Fix a typical realization of W . Setting Pref = Wiener measure RW

n (X) := dPW |Fn

dPref|Fn (X) The process (RW

n )n∈N is a martingale w.r.t. Pref (exercise!)

Since RW

n

≥ 0, the martingale converges: RW

n a.s.

− − − →

n→∞ RW ∞ ◮ PW ≪ Pref if and only if Eref[RW ∞ ] = 1

(the martingale is UI)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 22 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of singularity

Let (Xt)t∈[0,1] be the canonical process on C([0, 1], R)

[ Xt(f ) = f (t) ]

Let Fn := σ(Xtn

i : tn

i = i 2n , 0 ≤ i ≤ 2n) be the dyadic filtration

Fix a typical realization of W . Setting Pref = Wiener measure RW

n (X) := dPW |Fn

dPref|Fn (X) The process (RW

n )n∈N is a martingale w.r.t. Pref (exercise!)

Since RW

n

≥ 0, the martingale converges: RW

n a.s.

− − − →

n→∞ RW ∞ ◮ PW ≪ Pref if and only if Eref[RW ∞ ] = 1

(the martingale is UI)

◮ PW is singular w.r.t. Pref if and only if RW ∞ = 0

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 22 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of singularity

It suffices to show that RW

n (X) −

− − →

n→∞ 0 in P ⊗ Pref-probability

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 23 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of singularity

It suffices to show that RW

n (X) −

− − →

n→∞ 0 in P ⊗ Pref-probability

Fractional moment

For Pref-a.e. X E

• RW

n (X)

γ − − − →

n→∞ 0

for some γ ∈ (0, 1)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 23 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of singularity

It suffices to show that RW

n (X) −

− − →

n→∞ 0 in P ⊗ Pref-probability

Fractional moment

For Pref-a.e. X E

• RW

n (X)

γ − − − →

n→∞ 0

for some γ ∈ (0, 1) RW

n (X) =

1 ZW (0, 0), (1, ⋆)

• 2n−1
• i=0

ZW (tn

i , Xtn

i ), (tn

i+1, Xtn

i+1)

• g 1

2n (Xtn i+1 − Xtn i ) Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 23 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of singularity

It suffices to show that RW

n (X) −

− − →

n→∞ 0 in P ⊗ Pref-probability

Fractional moment

For Pref-a.e. X E

• RW

n (X)

γ − − − →

n→∞ 0

for some γ ∈ (0, 1) RW

n (X) =

1 ZW (0, 0), (1, ⋆)

• 2n−1
• i=0

ZW (tn

i , Xtn

i ), (tn

i+1, Xtn

i+1)

• g 1

2n (Xtn i+1 − Xtn i )

◮ Switch from E to equivalent law ˜

E to cancel the denominator

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 23 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of singularity

It suffices to show that RW

n (X) −

− − →

n→∞ 0 in P ⊗ Pref-probability

Fractional moment

For Pref-a.e. X ˜ E

• RW

n (X)

γ − − − →

n→∞ 0

for some γ ∈ (0, 1) RW

n (X) =

1 ZW (0, 0), (1, ⋆)

• 2n−1
• i=0

ZW (tn

i , Xtn

i ), (tn

i+1, Xtn

i+1)

• g 1

2n (Xtn i+1 − Xtn i )

◮ Switch from E to equivalent law ˜

E to cancel the denominator

◮ For fixed X, the ZW

(tn

i , Xtn

i ), (tn

i+1, Xtn

i+1)

• ’s are independent

We need to exploit translation and scale invariance of their laws

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 23 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of singularity

Lemma 1

(Translation and scale invariance)

If we set ∆n

i :=

Xtn

i+1 − Xtn i

tn

i+1 − tn i

we have ZW

ˆ β

• (tn

i , Xtn

i ), (tn

i+1, Xtn

i+1)

• g 1

2n (Xtn i+1 − Xtn i )

d

= ZW

ˆ β 2n/4

• (0, 0), (1, ∆n

i )

• g1(∆n

i )

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 24 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of singularity

Lemma 1

(Translation and scale invariance)

If we set ∆n

i :=

Xtn

i+1 − Xtn i

tn

i+1 − tn i

we have ZW

ˆ β

• (tn

i , Xtn

i ), (tn

i+1, Xtn

i+1)

• g 1

2n (Xtn i+1 − Xtn i )

d

= ZW

ˆ β 2n/4

• (0, 0), (1, ∆n

i )

• g1(∆n

i )

Lemma 2

(Expansion)

For z ∈ R and ε ∈ [0, 1] (say) ZW

ε

• (0, 0), (1, z)
• g1(z)

= 1 + ε X z + ε2 Y ε,z E[X z] = 0 E[X ε,z] = 0 E

• X 2

z

• ≤ C

E

• Y 2

ε,z

• ≤ C
• unif. in ε, z

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 24 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of singularity

By Taylor expansion, for fixed γ ∈ (0, 1) E

• ZW

ε

• (0, 0), (1, z)
• g1(z)

γ = E

• 1 + εX z + ε2Y ε,z

γ

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 25 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of singularity

By Taylor expansion, for fixed γ ∈ (0, 1) E

• ZW

ε

• (0, 0), (1, z)
• g1(z)

γ = E

• 1 + εX z + ε2Y ε,z

γ = 1 + γ

• εE[Xz] + ε2E[Yε,z]
• + γ(γ − 1)

2

• ε2E[(X x)2] + . . .
• + . . .

(⋆) First order terms vanish (⋆) γ(γ − 1) < 0

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 25 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of singularity

By Taylor expansion, for fixed γ ∈ (0, 1) E

• ZW

ε

• (0, 0), (1, z)
• g1(z)

γ = E

• 1 + εX z + ε2Y ε,z

γ = 1 + γ

• εE[Xz] + ε2E[Yε,z]
• + γ(γ − 1)

2

• ε2E[(X x)2] + . . .
• + . . .

= 1 − c ε2 ≤ e−c ε2 (⋆) First order terms vanish (⋆) γ(γ − 1) < 0 (⋆) For some c > 0

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 25 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of singularity

By Taylor expansion, for fixed γ ∈ (0, 1) E

• ZW

ε

• (0, 0), (1, z)
• g1(z)

γ = E

• 1 + εX z + ε2Y ε,z

γ = 1 + γ

• εE[Xz] + ε2E[Yε,z]
• + γ(γ − 1)

2

• ε2E[(X x)2] + . . .
• + . . .

= 1 − c ε2 ≤ e−c ε2 (⋆) First order terms vanish (⋆) γ(γ − 1) < 0 (⋆) For some c > 0 Estimate is uniform over z ∈ R

• We can set z = ∆n

i

and ε =

1 2n/4

˜ E

• RW

n (X)

γ =

2n−1

• i=0

E

• ZW

ε

• (0, 0), (1, ∆n

i )

• g1(∆n

i )

γ ≤ e−c ε2 2n = e−c 2n/2 which vanishes as n → ∞

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 25 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of Lemma 1

Introducing the dependence on ˆ β ZW

ˆ β

• (s, y), (t, x)
• d

= ZW

ˆ β

• (0, 0), (t − s, x − y)
• transl. invariance

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 26 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of Lemma 1

Introducing the dependence on ˆ β ZW

ˆ β

• (s, y), (t, x)
• d

= ZW

ˆ β

• (0, 0), (t − s, x − y)
• ZW

ˆ β

• (0, 0), (t, x)
• d

= 1 √t ZW

ˆ βt

1 4

• (0, 0),
• 1, x

√t

• transl. invariance + diffusive rescaling (prefactor, new ˆ

β)

(drawing!)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 26 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of Lemma 1

Introducing the dependence on ˆ β ZW

ˆ β

• (s, y), (t, x)
• d

= ZW

ˆ β

• (0, 0), (t − s, x − y)
• ZW

ˆ β

• (0, 0), (t, x)
• d

= 1 √t ZW

ˆ βt

1 4

• (0, 0),
• 1, x

√t

• transl. invariance + diffusive rescaling (prefactor, new ˆ

β)

(drawing!)

ZW (0, 0), (t, x)

• = gt(x) + ˆ

β

• [0,t]×R

gs(z) gt−s(x − z) W (dsdz) + . . .

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 26 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of Lemma 1

Introducing the dependence on ˆ β ZW

ˆ β

• (s, y), (t, x)
• d

= ZW

ˆ β

• (0, 0), (t − s, x − y)
• ZW

ˆ β

• (0, 0), (t, x)
• d

= 1 √t ZW

ˆ βt

1 4

• (0, 0),
• 1, x

√t

• transl. invariance + diffusive rescaling (prefactor, new ˆ

β)

(drawing!)

ZW (0, 0), (t, x)

• = gt(x) + ˆ

β

• [0,t]×R

gs(z) gt−s(x − z) W (dsdz) + . . . = 1 √t g1( x

√t ) + 1

√t ˆ β √t

[0,t]×R

g s

t ( z

√t ) g1− s

t ( x−z

√t ) W (dsdz) + . . .

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 26 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Proof of Lemma 1

Introducing the dependence on ˆ β ZW

ˆ β

• (s, y), (t, x)
• d

= ZW

ˆ β

• (0, 0), (t − s, x − y)
• ZW

ˆ β

• (0, 0), (t, x)
• d

= 1 √t ZW

ˆ βt

1 4

• (0, 0),
• 1, x

√t

• transl. invariance + diffusive rescaling (prefactor, new ˆ

β)

(drawing!)

ZW (0, 0), (t, x)

• = gt(x) + ˆ

β

• [0,t]×R

gs(z) gt−s(x − z) W (dsdz) + . . . = 1 √t g1( x

√t ) + 1

√t ˆ β t

3 4

√t

[0,t]×R

g s

t ( z

√t ) g1− s

t ( x−z

√t ) W (dsdz)

t

3 4

+ . . . = OK !

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 26 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Convergence of discrete DPRE

◮ Pω δ = law of discrete DPRE

(recall that Sδ

t :=

√ δSt/δ)

“Rescaled RW Sδ moving in an i.i.d. environment ω”

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 27 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Convergence of discrete DPRE

◮ Pω δ = law of discrete DPRE

(recall that Sδ

t :=

√ δSt/δ)

“Rescaled RW Sδ moving in an i.i.d. environment ω”

◮ PW = law of continuum DPRE

“BM moving in a white noise environment W ”

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 27 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Convergence of discrete DPRE

◮ Pω δ = law of discrete DPRE

(recall that Sδ

t :=

√ δSt/δ)

“Rescaled RW Sδ moving in an i.i.d. environment ω”

◮ PW = law of continuum DPRE

“BM moving in a white noise environment W ” Both Pω

δ

and PW are random probability laws on E := C([0, 1], R) i.e. RVs (defined on different probab. spaces) taking values in M1(E)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 27 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Convergence of discrete DPRE

◮ Pω δ = law of discrete DPRE

(recall that Sδ

t :=

√ δSt/δ)

“Rescaled RW Sδ moving in an i.i.d. environment ω”

◮ PW = law of continuum DPRE

“BM moving in a white noise environment W ” Both Pω

δ

and PW are random probability laws on E := C([0, 1], R) i.e. RVs (defined on different probab. spaces) taking values in M1(E) Does Pω

δ

converge in distribution toward PW as δ → 0 ? ∀ ψ ∈ Cb(M1(E) → R) : E

• ψ(Pω

δ )

• −

− − →

δ→0

E

• ψ(PW )
• Francesco Caravenna

Scaling Limits of Disordered Systems March 7-11, 2016 27 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Convergence of discrete DPRE

◮ Pω δ = law of discrete DPRE

(recall that Sδ

t :=

√ δSt/δ)

“Rescaled RW Sδ moving in an i.i.d. environment ω”

◮ PW = law of continuum DPRE

“BM moving in a white noise environment W ” Both Pω

δ

and PW are random probability laws on E := C([0, 1], R) i.e. RVs (defined on different probab. spaces) taking values in M1(E) Does Pω

δ

converge in distribution toward PW as δ → 0 ? ∀ ψ ∈ Cb(M1(E) → R) : E

• ψ(Pω

δ )

• −

− − →

δ→0

E

• ψ(PW )
• The answer is positive. . . almost surely ;-)

Statement for Pinning model proved in [C., Sun, Zygouras 2016] Details need to be checked for DPRE

(stronger assumptions on RW ?)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 27 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Universality

The convergence of Pω

δ

toward PW is an instance of universality

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 28 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Universality

The convergence of Pω

δ

toward PW is an instance of universality There are many discrete DPRE:

◮ any RW S

(zero mean, finite variance + technical assumptions)

◮ any (i.i.d.) disorder ω

(finite exponential moments)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 28 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Universality

The convergence of Pω

δ

toward PW is an instance of universality There are many discrete DPRE:

◮ any RW S

(zero mean, finite variance + technical assumptions)

◮ any (i.i.d.) disorder ω

(finite exponential moments)

In the continuum (δ → 0) and weak disorder (β → 0) regime, all these microscopic models Pω

δ

give rise to a unique macroscopic model PW

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 28 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Universality

The convergence of Pω

δ

toward PW is an instance of universality There are many discrete DPRE:

◮ any RW S

(zero mean, finite variance + technical assumptions)

◮ any (i.i.d.) disorder ω

(finite exponential moments)

In the continuum (δ → 0) and weak disorder (β → 0) regime, all these microscopic models Pω

δ

give rise to a unique macroscopic model PW Tomorrow we will see how the continuum model PW can tell quantitative information on discrete models Pω

δ

(free energy estimates)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 28 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Convergence

How to prove convergence in distribution Pω

δ d

− − − →

δ→0

PW ?

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 29 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Convergence

How to prove convergence in distribution Pω

δ d

− − − →

δ→0

PW ? Prove a.s. convergence through a suitable coupling of (ω, W )

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 29 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Convergence

How to prove convergence in distribution Pω

δ d

− − − →

δ→0

PW ? Prove a.s. convergence through a suitable coupling of (ω, W ) Assume we have convergence in distribution of discrete partition functions to continuum ones, in the space of continuum functions of (s, y), (t, x) Zω

δ

• (s, y), (t, x)
• d

− − − →

δ→0

ZW (s, y), (t, x)

• Francesco Caravenna

Scaling Limits of Disordered Systems March 7-11, 2016 29 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Convergence

How to prove convergence in distribution Pω

δ d

− − − →

δ→0

PW ? Prove a.s. convergence through a suitable coupling of (ω, W ) Assume we have convergence in distribution of discrete partition functions to continuum ones, in the space of continuum functions of (s, y), (t, x) Zω

δ

• (s, y), (t, x)
• d

− − − →

δ→0

ZW (s, y), (t, x)

• By Skorokhod representation theorem, there is a coupling of (ω, W )

under which this convergence holds a.s.

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 29 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Convergence

How to prove convergence in distribution Pω

δ d

− − − →

δ→0

PW ? Prove a.s. convergence through a suitable coupling of (ω, W ) Assume we have convergence in distribution of discrete partition functions to continuum ones, in the space of continuum functions of (s, y), (t, x) Zω

δ

• (s, y), (t, x)
• d

− − − →

δ→0

ZW (s, y), (t, x)

• By Skorokhod representation theorem, there is a coupling of (ω, W )

under which this convergence holds a.s. Fix such a coupling: for a.e. (ω, W ) the f.d.d. of Pω

δ converge weakly to

those of PW . It only remains to prove tightness of Pω

δ (·).

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 29 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Outline

• 1. White noise and Wiener chaos
• 2. Continuum partition functions
• 3. The continuum DPRE
• 4. Pinning models

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 30 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Ingredients: renewal process & disorder

0 = τ0 τ1 τ2 τ3 τ4 τ5 τ6 Discrete renewal process τ = {0 = τ0 < τ1 < τ2 < . . .} ⊆ N0 Gaps (τi+1 − τi)i≥0 are i.i.d. with polynomial-tail distribution: Pref(τ1 = n) ∼ cK n1+α , cK > 0, α ∈ (0, 1)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 31 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Ingredients: renewal process & disorder

0 = τ0 τ1 τ2 τ3 τ4 τ5 τ6 Discrete renewal process τ = {0 = τ0 < τ1 < τ2 < . . .} ⊆ N0 Gaps (τi+1 − τi)i≥0 are i.i.d. with polynomial-tail distribution: Pref(τ1 = n) ∼ cK n1+α , cK > 0, α ∈ (0, 1) τ = {n ∈ N0 : Sn = 0} zero level set of a Markov chain S = (Sn)n≥0

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 31 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Ingredients: renewal process & disorder

0 = τ0 τ1 τ2 τ3 τ4 τ5 τ6 Discrete renewal process τ = {0 = τ0 < τ1 < τ2 < . . .} ⊆ N0 Gaps (τi+1 − τi)i≥0 are i.i.d. with polynomial-tail distribution: Pref(τ1 = n) ∼ cK n1+α , cK > 0, α ∈ (0, 1) τ = {n ∈ N0 : Sn = 0} zero level set of a Markov chain S = (Sn)n≥0 Disorder ω = (ωi)i∈N: i.i.d. real random variables with law P λ(β) := log E[eβω1] < ∞ E[ω1] = 0 Var[ω1] = 1

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 31 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Bessel random walks

For α ∈ (0, 1) the α-Bessel random walk is defined as follows:

prob.

1 2

x prob.

1 2

• 1 + cα

x

• Sn

prob.

1 2

prob.

1 2

• 1 − cα

x

• cα := 1

2 − α

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 32 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Bessel random walks

For α ∈ (0, 1) the α-Bessel random walk is defined as follows:

prob.

1 2

x prob.

1 2

• 1 + cα

x

• Sn

prob.

1 2

prob.

1 2

• 1 − cα

x

• cα := 1

2 − α ◮ (α = 1 2) no drift (cα = 0)

• simple random walk

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 32 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Bessel random walks

For α ∈ (0, 1) the α-Bessel random walk is defined as follows:

prob.

1 2

x prob.

1 2

• 1 + cα

x

• Sn

prob.

1 2

prob.

1 2

• 1 − cα

x

• cα := 1

2 − α ◮ (α = 1 2) no drift (cα = 0)

• simple random walk

◮ (α < 1 2) drift away from the origin (cα > 0)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 32 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Bessel random walks

For α ∈ (0, 1) the α-Bessel random walk is defined as follows:

prob.

1 2

x prob.

1 2

• 1 + cα

x

• Sn

prob.

1 2

prob.

1 2

• 1 − cα

x

• cα := 1

2 − α ◮ (α = 1 2) no drift (cα = 0)

• simple random walk

◮ (α < 1 2) drift away from the origin (cα > 0) ◮ (α > 1 2) drift toward the origin (cα < 0)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 32 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Disordered pinning model

N

rewards ωn > 0 penalties ωn < 0

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 33 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Disordered pinning model

N Free renewal

rewards ωn > 0 penalties ωn < 0

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 33 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Disordered pinning model

N Pinning model

rewards ωn > 0 penalties ωn < 0

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 33 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Disordered pinning model

N Pinning model

rewards ωn > 0 penalties ωn < 0

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 33 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Disordered pinning model

N Pinning model

rewards ωn > 0 penalties ωn < 0

N ∈ N (system size) β ≥ 0, h ∈ R (disorder strength, bias)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 33 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Disordered pinning model

N Pinning model

rewards ωn > 0 penalties ωn < 0

N ∈ N (system size) β ≥ 0, h ∈ R (disorder strength, bias)

The pinning model

Gibbs change of measure Pω

N = Pω N,β,h of the renewal distribution Pref

dPω

N

dPref (τ) := 1 Z ω

N

exp

• N
• n=1

(βωn + h − λ(β)

• 1{n∈τ}
• Francesco Caravenna

Scaling Limits of Disordered Systems March 7-11, 2016 33 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Disordered pinning model

N Pinning model

rewards ωn > 0 penalties ωn < 0

N ∈ N (system size) β ≥ 0, h ∈ R (disorder strength, bias)

The pinning model

Gibbs change of measure Pω

N = Pω N,β,h of the renewal distribution Pref

dPω

N

dPref (τ) := 1 Z ω

N

exp

• N
• n=1

(βωn + h − λ(β)

• 1{Sn=0}
• Francesco Caravenna

Scaling Limits of Disordered Systems March 7-11, 2016 33 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

The phase transition

How are the typical paths τ of the pinning model Pω

N?

Contact number CN :=

• τ ∩ (0, N]
• = N

n=1 1{n∈τ} = N n=1 1{Sn=0}

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 34 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

The phase transition

How are the typical paths τ of the pinning model Pω

N?

Contact number CN :=

• τ ∩ (0, N]
• = N

n=1 1{n∈τ} = N n=1 1{Sn=0}

Theorem (phase transition)

∃ continuous, non decreasing, deterministic critical curve hc(β):

◮ Localized regime: for h > hc(β) one has CN ≈ N ◮ Deocalized regime: for h < hc(β) one has CN = O(log N)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 34 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

The phase transition

How are the typical paths τ of the pinning model Pω

N?

Contact number CN :=

• τ ∩ (0, N]
• = N

n=1 1{n∈τ} = N n=1 1{Sn=0}

Theorem (phase transition)

∃ continuous, non decreasing, deterministic critical curve hc(β):

◮ Localized regime: for h > hc(β) one has CN ≈ N

∃µ = µβ,h > 0 : Pω

N

• CN

N − µ

• > ε
• −

− − − →

N→∞

ω–a.s.

◮ Deocalized regime: for h < hc(β) one has CN = O(log N)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 34 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

The phase transition

How are the typical paths τ of the pinning model Pω

N?

Contact number CN :=

• τ ∩ (0, N]
• = N

n=1 1{n∈τ} = N n=1 1{Sn=0}

Theorem (phase transition)

∃ continuous, non decreasing, deterministic critical curve hc(β):

◮ Localized regime: for h > hc(β) one has CN ≈ N

∃µ = µβ,h > 0 : Pω

N

• CN

N − µ

• > ε
• −

− − − →

N→∞

ω–a.s.

◮ Deocalized regime: for h < hc(β) one has CN = O(log N)

∃A = Aβ,h > 0 : Pω

N

CN log N > A

• −

− − − →

N→∞

ω–a.s.

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 34 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Estimates on the critical curve

For β = 0 (homogeneous pinning, no disorder) one has hc(0) = 0

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 35 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Estimates on the critical curve

For β = 0 (homogeneous pinning, no disorder) one has hc(0) = 0 What is the behavior of hc(β) for β > 0 small ?

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 35 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Estimates on the critical curve

For β = 0 (homogeneous pinning, no disorder) one has hc(0) = 0 What is the behavior of hc(β) for β > 0 small ?

Theorem ( P(τ1 = n) ∼

cK n1+α )

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 35 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Estimates on the critical curve

For β = 0 (homogeneous pinning, no disorder) one has hc(0) = 0 What is the behavior of hc(β) for β > 0 small ?

Theorem ( P(τ1 = n) ∼

cK n1+α )

◮ (α < 1 2) disorder is irrelevant: hc(β) = 0 for β > 0 small [Alexander] [Toninelli] [Lacoin] [Cheliotis, den Hollander]

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 35 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Estimates on the critical curve

For β = 0 (homogeneous pinning, no disorder) one has hc(0) = 0 What is the behavior of hc(β) for β > 0 small ?

Theorem ( P(τ1 = n) ∼

cK n1+α )

◮ (α < 1 2) disorder is irrelevant: hc(β) = 0 for β > 0 small [Alexander] [Toninelli] [Lacoin] [Cheliotis, den Hollander] ◮ (α ≥ 1 2) disorder is relevant: hc(β) > 0 for all β > 0

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 35 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Estimates on the critical curve

For β = 0 (homogeneous pinning, no disorder) one has hc(0) = 0 What is the behavior of hc(β) for β > 0 small ?

Theorem ( P(τ1 = n) ∼

cK n1+α )

◮ (α < 1 2) disorder is irrelevant: hc(β) = 0 for β > 0 small [Alexander] [Toninelli] [Lacoin] [Cheliotis, den Hollander] ◮ (α ≥ 1 2) disorder is relevant: hc(β) > 0 for all β > 0

◮ (α > 1)

hc(β) ∼ C β2 with explicit C =

α 1+α 1 2E(τ1)

[Berger, C., Poisat, Sun, Zygouras]

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 35 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Estimates on the critical curve

For β = 0 (homogeneous pinning, no disorder) one has hc(0) = 0 What is the behavior of hc(β) for β > 0 small ?

Theorem ( P(τ1 = n) ∼

cK n1+α )

◮ (α < 1 2) disorder is irrelevant: hc(β) = 0 for β > 0 small [Alexander] [Toninelli] [Lacoin] [Cheliotis, den Hollander] ◮ (α ≥ 1 2) disorder is relevant: hc(β) > 0 for all β > 0

◮ (α > 1)

hc(β) ∼ C β2 with explicit C =

α 1+α 1 2E(τ1)

[Berger, C., Poisat, Sun, Zygouras]

◮ ( 1

2 < α < 1)

C1 β

2α 2α−1 ≤ hc(β) ≤ C2 β 2α 2α−1

[Derrida, Giacomin, Lacoin, Toninelli] [Alexander, Zygouras]

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 35 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Estimates on the critical curve

For β = 0 (homogeneous pinning, no disorder) one has hc(0) = 0 What is the behavior of hc(β) for β > 0 small ?

Theorem ( P(τ1 = n) ∼

cK n1+α )

◮ (α < 1 2) disorder is irrelevant: hc(β) = 0 for β > 0 small [Alexander] [Toninelli] [Lacoin] [Cheliotis, den Hollander] ◮ (α ≥ 1 2) disorder is relevant: hc(β) > 0 for all β > 0

◮ (α > 1)

hc(β) ∼ C β2 with explicit C =

α 1+α 1 2E(τ1)

[Berger, C., Poisat, Sun, Zygouras]

◮ ( 1

2 < α < 1)

C1 β

2α 2α−1 ≤ hc(β) ≤ C2 β 2α 2α−1

[Derrida, Giacomin, Lacoin, Toninelli] [Alexander, Zygouras]

◮ (α = 1

2) hc(β) = e − c+o(1)

β2

[Giacomin, Lacoin, Toninelli] [Berger, Lacoin]

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 35 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Estimates on the critical curve

For β = 0 (homogeneous pinning, no disorder) one has hc(0) = 0 What is the behavior of hc(β) for β > 0 small ?

Theorem ( P(τ1 = n) ∼

cK n1+α )

◮ (α < 1 2) disorder is irrelevant: hc(β) = 0 for β > 0 small [Alexander] [Toninelli] [Lacoin] [Cheliotis, den Hollander] ◮ (α ≥ 1 2) disorder is relevant: hc(β) > 0 for all β > 0

◮ (α > 1)

hc(β) ∼ C β2 with explicit C =

α 1+α 1 2E(τ1)

[Berger, C., Poisat, Sun, Zygouras]

◮ ( 1

2 < α < 1)

hc(β) ∼ ˆ C β

2α 2α−1

using continuum part. funct.!

[C., Torri, Toninelli]

◮ (α = 1

2) hc(β) = e − c+o(1)

β2

[Giacomin, Lacoin, Toninelli] [Berger, Lacoin]

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 35 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Discrete free energy and critical curve

Partition function Zω

N := E

• eHN(τ)

= E

• e

N

n=1(h+βωn−Λ(β))1{n∈τ}

• Francesco Caravenna

Scaling Limits of Disordered Systems March 7-11, 2016 36 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Discrete free energy and critical curve

Partition function Zω

N := E

• eHN(τ)

= E

• e

N

n=1(h+βωn−Λ(β))1{n∈τ}

• Consider first the regime of N → ∞ with fixed β, h

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 36 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Discrete free energy and critical curve

Partition function Zω

N := E

• eHN(τ)

= E

• e

N

n=1(h+βωn−Λ(β))1{n∈τ}

• Consider first the regime of N → ∞ with fixed β, h

◮ Free energy

F(β, h) := lim

N→∞ 1 N log Zω N ≥ 0

P(dω)-a.s.

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 36 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Discrete free energy and critical curve

Partition function Zω

N := E

• eHN(τ)

= E

• e

N

n=1(h+βωn−Λ(β))1{n∈τ}

• Consider first the regime of N → ∞ with fixed β, h

◮ Free energy

F(β, h) := lim

N→∞ 1 N log Zω N ≥ 0

P(dω)-a.s. Zω

N ≥ E

• eHN(τ) 1{τ∩(0,N]=∅}
• Francesco Caravenna

Scaling Limits of Disordered Systems March 7-11, 2016 36 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Discrete free energy and critical curve

Partition function Zω

N := E

• eHN(τ)

= E

• e

N

n=1(h+βωn−Λ(β))1{n∈τ}

• Consider first the regime of N → ∞ with fixed β, h

◮ Free energy

F(β, h) := lim

N→∞ 1 N log Zω N ≥ 0

P(dω)-a.s. Zω

N ≥ E

• eHN(τ) 1{τ∩(0,N]=∅}
• = P(τ ∩ (0, N] = ∅)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 36 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Discrete free energy and critical curve

Partition function Zω

N := E

• eHN(τ)

= E

• e

N

n=1(h+βωn−Λ(β))1{n∈τ}

• Consider first the regime of N → ∞ with fixed β, h

◮ Free energy

F(β, h) := lim

N→∞ 1 N log Zω N ≥ 0

P(dω)-a.s. Zω

N ≥ E

• eHN(τ) 1{τ∩(0,N]=∅}
• = P(τ ∩ (0, N] = ∅) ∼ (const.)

Nα

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 36 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Discrete free energy and critical curve

Partition function Zω

N := E

• eHN(τ)

= E

• e

N

n=1(h+βωn−Λ(β))1{n∈τ}

• Consider first the regime of N → ∞ with fixed β, h

◮ Free energy

F(β, h) := lim

N→∞ 1 N log Zω N ≥ 0

P(dω)-a.s. Zω

N ≥ E

• eHN(τ) 1{τ∩(0,N]=∅}
• = P(τ ∩ (0, N] = ∅) ∼ (const.)

Nα ◮ Critical curve

hc(β) = sup{h ∈ R : F(β, h) = 0} non analiticity!

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 36 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Discrete free energy and critical curve

Partition function Zω

N := E

• eHN(τ)

= E

• e

N

n=1(h+βωn−Λ(β))1{n∈τ}

• Consider first the regime of N → ∞ with fixed β, h

◮ Free energy

F(β, h) := lim

N→∞ 1 N log Zω N ≥ 0

P(dω)-a.s. Zω

N ≥ E

• eHN(τ) 1{τ∩(0,N]=∅}
• = P(τ ∩ (0, N] = ∅) ∼ (const.)

Nα ◮ Critical curve

hc(β) = sup{h ∈ R : F(β, h) = 0} non analiticity! (convexity) ∂F(β, h) ∂h = lim

N→∞ Eω N

CN N

• Francesco Caravenna

Scaling Limits of Disordered Systems March 7-11, 2016 36 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Discrete free energy and critical curve

Partition function Zω

N := E

• eHN(τ)

= E

• e

N

n=1(h+βωn−Λ(β))1{n∈τ}

• Consider first the regime of N → ∞ with fixed β, h

◮ Free energy

F(β, h) := lim

N→∞ 1 N log Zω N ≥ 0

P(dω)-a.s. Zω

N ≥ E

• eHN(τ) 1{τ∩(0,N]=∅}
• = P(τ ∩ (0, N] = ∅) ∼ (const.)

Nα ◮ Critical curve

hc(β) = sup{h ∈ R : F(β, h) = 0} non analiticity! (convexity) ∂F(β, h) ∂h = lim

N→∞ Eω N

CN N > 0 if h > hc(β) = 0 if h < hc(β)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 36 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Discrete free energy and critical curve

Partition function Zω

N := E

• eHN(τ)

= E

• e

N

n=1(h+βωn−Λ(β))1{n∈τ}

• Consider first the regime of N → ∞ with fixed β, h

◮ Free energy

F(β, h) := lim

N→∞ 1 N log Zω N ≥ 0

P(dω)-a.s. Zω

N ≥ E

• eHN(τ) 1{τ∩(0,N]=∅}
• = P(τ ∩ (0, N] = ∅) ∼ (const.)

Nα ◮ Critical curve

hc(β) = sup{h ∈ R : F(β, h) = 0} non analiticity! (convexity) ∂F(β, h) ∂h = lim

N→∞ Eω N

CN N > 0 if h > hc(β) = 0 if h < hc(β) F(β, h) and hc(β) depend on the law of τ and ω Universality as β, h → 0 ?

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 36 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Discrete free energy and critical curve

Partition function Zω

N := E

• eHN(τ)

= E

• e

N

n=1(h+βωn−Λ(β))1{n∈τ}

• Consider first the regime of N → ∞ with fixed β, h

◮ Free energy

F(β, h) := lim

N→∞ 1 N log Zω N ≥ 0

P(dω)-a.s. Zω

N ≥ E

• eHN(τ) 1{τ∩(0,N]=∅}
• = P(τ ∩ (0, N] = ∅) ∼ (const.)

Nα ◮ Critical curve

hc(β) = sup{h ∈ R : F(β, h) = 0} non analiticity! (convexity) ∂F(β, h) ∂h = lim

N→∞ Eω N

CN N > 0 if h > hc(β) = 0 if h < hc(β) F(β, h) and hc(β) depend on the law of τ and ω Universality as β, h → 0 ? YES, connected to continuum model

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 36 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum partition functions

Build continuum partition functions for Pinning Model with α ∈ ( 1

2, 1)

(disorder relevant) following “usual” approach [C, Sun, Zygouras 2015+]

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 37 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum partition functions

Build continuum partition functions for Pinning Model with α ∈ ( 1

2, 1)

(disorder relevant) following “usual” approach [C, Sun, Zygouras 2015+] We need to rescale β = βN = ˆ β Nα−1/2 h = hN = ˆ h Nα

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 37 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum partition functions

Build continuum partition functions for Pinning Model with α ∈ ( 1

2, 1)

(disorder relevant) following “usual” approach [C, Sun, Zygouras 2015+] We need to rescale β = βN = ˆ β Nα−1/2 h = hN = ˆ h Nα One has Zω

N d

− − − − →

N→∞

ZW with ZW := 1 + C

• 0<t<1

dW

ˆ β,ˆ h t

t1−α + C 2

• 0<t<t′<1

dW

ˆ β,ˆ h t

dW

ˆ β,ˆ h t′

t1−α(t′ − t)1−α + . . . where W

ˆ β,ˆ h t

:= ˆ βW t + ˆ h t and C = α sin(απ)

π cK

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 37 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum free energy

In analogy with the discrete model, define Continuum free energy F(ˆ β, ˆ h) := lim

t→∞

1 t log ZW

ˆ β,ˆ h(0, t)

a.s. (existence and self-averaging need some work)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 38 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum free energy

In analogy with the discrete model, define Continuum free energy F(ˆ β, ˆ h) := lim

t→∞

1 t log ZW

ˆ β,ˆ h(0, t)

a.s. (existence and self-averaging need some work) Again F(ˆ β, ˆ h) ≥ 0 and define Continuum critical curve Hc(ˆ β) := sup ˆ h ∈ R : F(ˆ β, ˆ h) = 0

• Francesco Caravenna

Scaling Limits of Disordered Systems March 7-11, 2016 38 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum free energy

In analogy with the discrete model, define Continuum free energy F(ˆ β, ˆ h) := lim

t→∞

1 t log ZW

ˆ β,ˆ h(0, t)

a.s. (existence and self-averaging need some work) Again F(ˆ β, ˆ h) ≥ 0 and define Continuum critical curve Hc(ˆ β) := sup ˆ h ∈ R : F(ˆ β, ˆ h) = 0

• Scaling relations

∀c > 0 : ZW

ˆ β,ˆ h(c t) d

= ZW

cα− 1

2 ˆ

β,cαˆ h(t)

(Wiener chaos exp.)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 38 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum free energy

In analogy with the discrete model, define Continuum free energy F(ˆ β, ˆ h) := lim

t→∞

1 t log ZW

ˆ β,ˆ h(0, t)

a.s. (existence and self-averaging need some work) Again F(ˆ β, ˆ h) ≥ 0 and define Continuum critical curve Hc(ˆ β) := sup ˆ h ∈ R : F(ˆ β, ˆ h) = 0

• Scaling relations

∀c > 0 : ZW

ˆ β,ˆ h(c t) d

= ZW

cα− 1

2 ˆ

β,cαˆ h(t)

(Wiener chaos exp.)

F

• cα− 1

2 ˆ

β, cαˆ h

• = c F(ˆ

β, ˆ h)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 38 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Continuum free energy

In analogy with the discrete model, define Continuum free energy F(ˆ β, ˆ h) := lim

t→∞

1 t log ZW

ˆ β,ˆ h(0, t)

a.s. (existence and self-averaging need some work) Again F(ˆ β, ˆ h) ≥ 0 and define Continuum critical curve Hc(ˆ β) := sup ˆ h ∈ R : F(ˆ β, ˆ h) = 0

• Scaling relations

∀c > 0 : ZW

ˆ β,ˆ h(c t) d

= ZW

cα− 1

2 ˆ

β,cαˆ h(t)

(Wiener chaos exp.)

F

• cα− 1

2 ˆ

β, cαˆ h

• = c F(ˆ

β, ˆ h) Hc(ˆ β) = Hc(1) ˆ β

2α 2α−1 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 38 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Can we relate continuum free energy to the discrete one?

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 39 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Can we relate continuum free energy to the discrete one? By construction of continuum partition functions ZW

ˆ β,ˆ h(t) d

= lim

N→∞ Zω βN,hN(Nt)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 39 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Can we relate continuum free energy to the discrete one? By construction of continuum partition functions ZW

ˆ β,ˆ h(t) d

= lim

N→∞ Zω βN,hN(Nt)

Assuming uniform integrability of log Zω (OK) F(ˆ β, ˆ h) = lim

t→∞

1 t E

• log ZW

ˆ β,ˆ h(t)

• Francesco Caravenna

Scaling Limits of Disordered Systems March 7-11, 2016 39 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Can we relate continuum free energy to the discrete one? By construction of continuum partition functions ZW

ˆ β,ˆ h(t) d

= lim

N→∞ Zω βN,hN(Nt)

Assuming uniform integrability of log Zω (OK) F(ˆ β, ˆ h) = lim

t→∞

1 t E

• log ZW

ˆ β,ˆ h(t)

• = lim

t→∞

1 t lim

N→∞ E

• log Zω

βN,hN(Nt)

• Francesco Caravenna

Scaling Limits of Disordered Systems March 7-11, 2016 39 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Can we relate continuum free energy to the discrete one? By construction of continuum partition functions ZW

ˆ β,ˆ h(t) d

= lim

N→∞ Zω βN,hN(Nt)

Assuming uniform integrability of log Zω (OK) F(ˆ β, ˆ h) = lim

t→∞

1 t E

• log ZW

ˆ β,ˆ h(t)

• = lim

t→∞

1 t lim

N→∞ E

• log Zω

βN,hN(Nt)

• Assuming we can interchange the limits N → ∞ and t → ∞

F(ˆ β, ˆ h) = lim

N→∞

lim

t→∞

1 t E

• log Zω

βN,hN(Nt)

• Francesco Caravenna

Scaling Limits of Disordered Systems March 7-11, 2016 39 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Can we relate continuum free energy to the discrete one? By construction of continuum partition functions ZW

ˆ β,ˆ h(t) d

= lim

N→∞ Zω βN,hN(Nt)

Assuming uniform integrability of log Zω (OK) F(ˆ β, ˆ h) = lim

t→∞

1 t E

• log ZW

ˆ β,ˆ h(t)

• = lim

t→∞

1 t lim

N→∞ E

• log Zω

βN,hN(Nt)

• Assuming we can interchange the limits N → ∞ and t → ∞

F(ˆ β, ˆ h) = lim

N→∞ N lim t→∞

1 Nt E

• log Zω

βN,hN(Nt)

• Francesco Caravenna

Scaling Limits of Disordered Systems March 7-11, 2016 39 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Can we relate continuum free energy to the discrete one? By construction of continuum partition functions ZW

ˆ β,ˆ h(t) d

= lim

N→∞ Zω βN,hN(Nt)

Assuming uniform integrability of log Zω (OK) F(ˆ β, ˆ h) = lim

t→∞

1 t E

• log ZW

ˆ β,ˆ h(t)

• = lim

t→∞

1 t lim

N→∞ E

• log Zω

βN,hN(Nt)

• Assuming we can interchange the limits N → ∞ and t → ∞

F(ˆ β, ˆ h) = lim

N→∞ N lim t→∞

1 Nt E

• log Zω

βN,hN(Nt)

• =

lim

N→∞ N F(βN, hN)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 39 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Can we relate continuum free energy to the discrete one? By construction of continuum partition functions ZW

ˆ β,ˆ h(t) d

= lim

N→∞ Zω βN,hN(Nt)

Assuming uniform integrability of log Zω (OK) F(ˆ β, ˆ h) = lim

t→∞

1 t E

• log ZW

ˆ β,ˆ h(t)

• = lim

t→∞

1 t lim

N→∞ E

• log Zω

βN,hN(Nt)

• Assuming we can interchange the limits N → ∞ and t → ∞

F(ˆ β, ˆ h) = lim

N→∞ N lim t→∞

1 Nt E

• log Zω

βN,hN(Nt)

• =

lim

N→∞ N F(βN, hN)

Setting δ = 1

N for clarity, we arrive at. . .

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 39 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Conjecture

F(ˆ β, ˆ h) = lim

δ→0

F ˆ βδα− 1

2 , ˆ

hδα δ

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 40 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Conjecture

F(ˆ β, ˆ h) = lim

δ→0

F ˆ βδα− 1

2 , ˆ

hδα δ

Theorem

[C., Toninelli, Torri 2015]

For all ˆ β > 0, ˆ h ∈ R and η > 0 there is δ0 > 0 such that ∀δ < δ0 F(ˆ β, ˆ h − η) ≤ F ˆ βδα− 1

2 , ˆ

hδα δ ≤ F(ˆ β, ˆ h + η)

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 40 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Conjecture

F(ˆ β, ˆ h) = lim

δ→0

F ˆ βδα− 1

2 , ˆ

hδα δ

Theorem

[C., Toninelli, Torri 2015]

For all ˆ β > 0, ˆ h ∈ R and η > 0 there is δ0 > 0 such that ∀δ < δ0 F(ˆ β, ˆ h − η) ≤ F ˆ βδα− 1

2 , ˆ

hδα δ ≤ F(ˆ β, ˆ h + η) This implies Conj. and hc(β) ∼ Hc(β) ∼ Hc(1) β

2α 2α−1 Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 40 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Conjecture

F(ˆ β, ˆ h) = lim

δ→0

F ˆ βδα− 1

2 , ˆ

hδα δ

Theorem

[C., Toninelli, Torri 2015]

For all ˆ β > 0, ˆ h ∈ R and η > 0 there is δ0 > 0 such that ∀δ < δ0 F(ˆ β, ˆ h − η) ≤ F ˆ βδα− 1

2 , ˆ

hδα δ ≤ F(ˆ β, ˆ h + η) This implies Conj. and hc(β) ∼ Hc(β) ∼ Hc(1) β

2α 2α−1

For any discrete Pinning model with α ∈ ( 1

2, 1), the free energy F(β, h)

and the critical curve hc(β) have a universal shape in the regime β, h → 0

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 40 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Very delicate result. How to prove it?

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 41 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Very delicate result. How to prove it?

◮ Assume that there is a continuum Hamiltonian:

Zω = E

• eHω

Nt

ZW = E

• eHW

t Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 41 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Very delicate result. How to prove it?

◮ Assume that there is a continuum Hamiltonian:

Zω = E

• eHω

Nt

ZW = E

• eHW

t

◮ Couple Hω Nt and HW t

• n the same probability space in such a way

that the difference ∆N,t := Hω

Nt − HW t

is “small”

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 41 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Very delicate result. How to prove it?

◮ Assume that there is a continuum Hamiltonian:

Zω = E

• eHω

Nt

ZW = E

• eHW

t

◮ Couple Hω Nt and HW t

• n the same probability space in such a way

that the difference ∆N,t := Hω

Nt − HW t

is “small”

◮ Deduce that

E

• log Zω

≤ E

• log ZW

+ log EE

• e∆N,t

and show that the last term is “negligible”

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 41 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Very delicate result. How to prove it?

◮ Assume that there is a continuum Hamiltonian:

Zω = E

• eHω

Nt

ZW = E

• eHW

t

◮ Couple Hω Nt and HW t

• n the same probability space in such a way

that the difference ∆N,t := Hω

Nt − HW t

is “small”

◮ Deduce that

E

• log Zω

≤ E

• log ZW

+ log EE

• e∆N,t

and show that the last term is “negligible” Problem: there is no continuum Hamiltonian!

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 41 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

Interchanging the limits

Very delicate result. How to prove it?

◮ Assume that there is a continuum Hamiltonian:

Zω = E

• eHω

Nt

ZW = E

• eHW

t

◮ Couple Hω Nt and HW t

• n the same probability space in such a way

that the difference ∆N,t := Hω

Nt − HW t

is “small”

◮ Deduce that

E

• log Zω

≤ E

• log ZW

+ log EE

• e∆N,t

and show that the last term is “negligible” Problem: there is no continuum Hamiltonian! Solution: perform coarse-graining and define an “effective” Hamiltonian

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 41 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

The DPRE case

What about the DPRE? We can still define discrete F(β) and continuum F(ˆ β) free energy

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 42 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

The DPRE case

What about the DPRE? We can still define discrete F(β) and continuum F(ˆ β) free energy Since F(ˆ β) ∼ F(1) β4 we can hope that F(β) ∼ F(1) β4 as β → 0 provided the “interchanging of limits” is justified

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 42 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

The DPRE case

What about the DPRE? We can still define discrete F(β) and continuum F(ˆ β) free energy Since F(ˆ β) ∼ F(1) β4 we can hope that F(β) ∼ F(1) β4 as β → 0 provided the “interchanging of limits” is justified

• N. Torri is currently working on this problem. A finer coarse-graining is

needed, together with sharper estimates on continuum partition functions

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 42 / 43

White noise Continuum partition functions The continuum DPRE Pinning models

References

◮ T. Alberts, K. Khanin, J. Quastel

The intermediate disorder regime for directed polymers in dimension 1 + 1

• Ann. Probab. 42 (2014), 1212–1256

◮ T. Alberts, K. Khanin, J. Quastel

The Continuum Directed Random Polymer

• J. Stat. Phys. 154 (2014), 305–326

◮ F. Caravenna, R. Sun, N. Zygouras

Polynomial chaos and scaling limits of disordered systems

• J. Eur. Math. Soc. (JEMS), to appear

◮ F. Caravenna, R. Sun, N. Zygouras

The continuum disordered pinning model

• Probab. Theory Related Fields 164 (2016), 17-59.

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 43 / 43