Rencontre de Probabilit´es 2007 - Rouen Directed Polymers and Uniform Integrability

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Rencontre de Probabilit´ es 2007 - Rouen

Directed Polymers and Uniform Integrability

AlainCamanes

[email protected] Universit´e de Nantes

May 29, 2007

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Outline

1 The model

The polymer and its environment The energy

2 The partition function Definition

Properties

Uniform Integrability

3 2^nd order moments General method Conditional method

4 α order moments

The uniform integrability condition Examples

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Outline

1 The model

Properties

4 α order moments

The uniform integrability condition

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Le polym` ere et son milieu

n 0

Z^d

ω₁

the polymer

time

Example :g ∼Bern(1/2): g= 1 for a water particle,g=−1 for an oil particle.

g(i, x)

n time 0

Z^d

ω₁

g(1, ω1)

the environment

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Le polym` ere et son milieu

Example :g ∼Bern(1/2): g= 1 for a water particle,g=−1 for an oil g(i, x)

n time 0

Z^d

ω₁

g(1, ω₁)

the environment

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The model’s parameters

The Hamiltonian of a polymer ω of lengthn H_n(ω) =

n

X

i=1

g(i, ω_i).

The more the hydrophilic polymer visits water sites, the higherHn(ω) is.

The Boltzmann’s weight with temperatureT = 1/β e^βHⁿ^(ω)

Z_n(β)

Probability for a polymerωto have energyHn(ω) when the temperature isT

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The model’s parameters

The Hamiltonian of a polymer ω of lengthn H_n(ω) =

n

X

i=1

g(i, ω_i).

The more the hydrophilic polymer visits water sites, the higherHn(ω) is.

The Boltzmann’s weight with temperatureT = 1/β e^βHⁿ^(ω)

Z_n(β)

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Hypotheses and notations

Notations

P uniform measure on polymers paths, Q law of the environmentg,

G_n=σ g(i,x), i ≤n, x ∈Z^d .

Hypotheses

The random variables (g_n,x) are i.i.d., They have exponential moments

λ(β):= lnQ h

e^βg i

<+∞.

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Hypotheses and notations

Notations

P uniform measure on polymers paths, Q law of the environmentg,

G_n=σ g(i,x), i ≤n, x ∈Z^d .

Hypotheses

The random variables (g_n,x) are i.i.d., They have exponential moments

λ(β):= lnQ h

e^βg i

<+∞.

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Outline

1 The model

Properties

4 α order moments

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The partition function

Definition

We callpartition functionof the system the quantity Wn(β) =P

h

e^βHⁿ^{(ω)−nλ(β)} i

Questions : How doeslimnWn behave with respect to the temperatureβ? the dimension d?

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The partition function

Definition

We callpartition functionof the system the quantity Wn(β) =P

h

e^βHⁿ^{(ω)−nλ(β)} i

Questions : How doeslimnWn behave with respect to the temperatureβ? the dimension d?

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Properties of the partition function

Convergence [Bolthausen 89]: There exists a random variable W∞(β)such that

Wn(β) −→

n→∞W∞(β) p.s.

0−1 law [Bolthausen 89] :

P(W∞>0)∈{0,1}. Decreasingness [Comets, Yoshida 03] :

β 7→Qhp

W∞(β)i

isdecreasing. Critical temperature : ∃β_c ∈[0,+∞] such that

β_c = sup{β; W∞>0} ⇔ weak disorder

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Properties of the partition function

Wn(β) −→

n→∞W∞(β) p.s. 0−1 law [Bolthausen 89] :

P(W∞>0)∈{0,1}.

Decreasingness [Comets, Yoshida 03] : β 7→Qhp

W∞(β)i

isdecreasing. Critical temperature : ∃β_c ∈[0,+∞] such that

β_c = sup{β; W∞>0} ⇔ weak disorder

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Properties of the partition function

Wn(β) −→

P(W∞>0)∈{0,1}.

W∞(β)i

isdecreasing.

Critical temperature : ∃β_c ∈[0,+∞] such that β_c = sup{β; W∞>0} ⇔ weak disorder

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Properties of the partition function

Wn(β) −→

P(W∞>0)∈{0,1}.

W∞(β)i

isdecreasing.

Critical temperature : ∃β_c ∈[0,+∞] such that β_c = sup{β; W∞>0} ⇔ weak disorder

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Motivation, Results, Prospects,. . .

Why studying the partition function?

W_n=e^−nλ(β) Z

e^nβy ν_n(dy), whereνn=P

δ_H_n_/n −→ Large Deviations??

In dimension 1, 2 [Carmona, Hu 02; Comets, Yoshida 03]

βc = 0

In dimension d ≥3

cf. the rest of the talk. . .

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Motivation, Results, Prospects,. . .

W_n=e^−nλ(β) Z

βc = 0

In dimension d ≥3

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Motivation, Results, Prospects,. . .

W_n=e^−nλ(β) Z

βc = 0

In dimension d ≥3

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Uniform Integrability

Theorem (Carmona, Hu 02)

The following properties are equivalent W∞>0 a.s.

(W_n)n∈N is uniformly integrable.

Plan :

Evaluate 2^nd order moments

Evaluateα order moments,α ∈(1,2]

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Uniform Integrability

Theorem (Carmona, Hu 02)

The following properties are equivalent W∞>0 a.s.

(W_n)n∈N is uniformly integrable.

Plan :

Evaluate 2^nd order moments

Evaluateα order moments,α ∈(1,2]

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Outline

1 The model

Properties

4 α order moments

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2

^nd

order moments

ω¹,ω² independant random walks,

q_d =P^⊗2 ∀n∈N, ω_n¹6=ω²_n .

Theorem (Bolthausen 89)

As soon asλ(2β)−2λ(β)<−ln(1−qd), W∞(β)>0.

Notation :β2∈[0,+∞] the critical value (β2≤βc)

Bernoullienvironment : λ(β) = lnch(β), β2= +∞. Gaussianenvironment : λ(β) =β²/2,

β2=p

−2 ln(1−qd).

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2

^nd

order moments

ω¹,ω² independant random walks,

q_d =P^⊗2 ∀n∈N, ω_n¹6=ω²_n .

Theorem (Bolthausen 89)

As soon asλ(2β)−2λ(β)<−ln(1−qd), W∞(β)>0.

Notation :β2∈[0,+∞] the critical value (β2≤βc)

Bernoullienvironment : λ(β) = lnch(β), β2= +∞.

Gaussianenvironment : λ(β) =β²/2, β2=p

−2 ln(1−qd).

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Conditional method : first steps

Definition

LetWn, we define

The sized-biasedvariableWcn : ∀ f ∈ B_b, Qh

f(cW_n)i

=Q[W_nf(W_n)]. The fW_n variable,

Wfn=P h

e^β(H(ω¹^)+H(ω²⁾⁾ i

e^−2nλ(β).

The following properties are equivalent

(Wn) U.I. ⇔ L(cWn) tight

⇔ L(fWn) tight

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Conditional method : first steps

Definition

LetWn, we define

The sized-biasedvariableWcn : ∀ f ∈ B_b, Qh

f(cW_n)i

=Q[W_nf(W_n)]. The fW_n variable,

Wfn=P h

e^β(H(ω¹^)+H(ω²⁾⁾ i

e^−2nλ(β).

The following properties are equivalent

(Wn) U.I. ⇔ L(cWn) tight

⇔ L(fWn) tight

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Conditional methods : problems

Identify

β∗= supn β, Qh

Wfn

i

<+∞a.s.o

Idea (Birkner 02)

Use aconditionalSanov type theorem to show λ(2β∗)−2λ(β∗) = 1 +X

n≥1

e^−H(pⁿ⁾,

where H(pn) =−P

x∈Z^dP(ωn=x) lnP(ωn=x).

Then, using Jensen’s inequality,β∗> β2.

Probleme : The large deviation principle used is not yet proven.

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Conditional methods : problems

Identify

β∗= supn β, Qh

Wfn

i

<+∞a.s.o

Idea (Birkner 02)

n≥1

e^−H(pⁿ⁾,

where H(pn) =−P

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Conditional methods : problems

Identify

β∗= supn β, Qh

Wfn

i

<+∞a.s.o

Idea (Birkner 02)

n≥1

e^−H(pⁿ⁾,

where H(pn) =−P

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Outline

1 The model

Properties

4 α order moments

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Notations and results

∃α∈(1,2] sup_nQ[W_n^α]<+∞ ⇒ (W_n)n∈N is U.I.

ω1,ω2independant random walks starting from 0 : p(t,x)=P^⊗2“

0<j<t, ω¹_j 6=ω²_j, ω_t¹=ω_t²=x” .

Theorem (Derrida, Evans 92) Ifλ(αβ)−αλ(β)<−lnP

t,xp(t,x)^α/2, then W∞>0.

Notation :βα the critical value.

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Notations and results

0<j<t, ω¹_j 6=ω²_j, ω_t¹=ω_t²=x” .

Notation :βα the critical value.

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Notations and results

0<j<t, ω¹_j 6=ω²_j, ω_t¹=ω_t²=x” .

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How better is this method?

We define

( hQ(2) = Q h e^β2g

Q[e^β2g]ln ^e^β²^g

Q[e^β2g]

i , hν(2) = −P

t,x p(t,x) 1−q_d ln^p(t,x)_1−q

d.

Theorem (C., Carmona 07) There existsα such that βα > β2 if

h_ν(2)<h_Q(2).

Remark : The optimalβ_α_∗ satisfies

α∗βα∗λ⁰(α∗βα∗)−λ(α∗βα∗) =hν(α∗).

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How better is this method?

We define

( hQ(2) = Q h e^β2g

Q[e^β2g]

i , hν(2) = −P

d.

h_ν(2)<h_Q(2).

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How better is this method?

We define

( hQ(2) = Q h e^β2g

Q[e^β2g]

i , hν(2) = −P

d.

h_ν(2)<h_Q(2).

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Some examples

Approximated values ofhν(2)

d hν(2) 3 4,6 4 3,8 5 3,6

Values ofh_Q(2) for some environments

d Gaussian Binomial Poissonian

3 2,16 4,14 6,42

4 3,29 6,23 10,30

5 4,00 7,42 12,73

Inred the environments whereβ∗ > β₂.

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Some examples

Approximated values ofhν(2)

d hν(2) 3 4,6 4 3,8 5 3,6 Values ofh_Q(2) for some environments

d Gaussian Binomial Poissonian

3 2,16 4,14 6,42

4 3,29 6,23 10,30

5 4,00 7,42 12,73

Inred the environments whereβ∗ > β₂.

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Conclusion

In general,β26=βc.

How to obtain a better bound on β_c?

Find a link between the strong, the weak disorder phases and the large deviation rate function.

What is the link with the free energy?

p_n(β) = 1

nlnW_n(β)