Directed Polymers

in

Random Environment:

Path Localization and Strong Disorder^{∗}

Nobuo YOSHIDA (Kyoto Univ.)

based on a joint work with Francis COMETS (Paris 7)

and

Tokuzo SHIGA (Tokyo Inst. Tech.)

∗Notes for 60 minutes talk at Shonan Village, July 2002.

♣ “Directed polymer” : a variant of RWRE (random walk in random environment)

where

• RW :{ω_{n}}_{n≥0} a simple RW in ℤ^{d} defined on (Ω, F, P).

• RE : ξ = {ξ(x, n) : x ∈ ℤ^{d}, n ∈ ℕ} real i.i.d. in space (x) and time (n) defined on (Ξ, G, Q).

• Assumption on ξ:

exp (λ(β)) := Q exp (βξ(x, n)) < ∞, ∀β ∈ ℝ.

Two typical choice of ξ;

• Bernoulli environment;

Q{ξ ; ξ(x, n) = 1} = p
Q{ξ ; ξ(x, n) = 0} = 1 − p

so that

λ(β) = ln (pe^{β} + (1 − p)) .

Remark: directed percolation ↔ “β = +∞”.

• Gaussian environment;

Q\{ξ ; ξ(x, n) ∈ dt\} = \frac{1}{√2π} exp (−t^{2}/2)dt,

so that

λ(β) = β^{2}/2.

♣ Definition of the model; Define a probability measure

μ_{n}(dω|ξ) = \frac{1}{Z_{n}(ξ)} P [e_{n}(ω, ξ) : dω],

on (Ω, F) where

e_{n}(ω, ξ) = exp (_{1≤j≤n}^{∑}(βξ(ω_{j}, j) − λ(β)))
Z_{n}(ξ) = P [e_{n}(ω, ξ)].

Note that

• μ_{n}(dω|ξ) is a r.v. under Q(dξ),

• Z_{n}(ξ) is a mean 1 martingale on (Ξ, G, Q). ♣ Physical interpretation;

• {(ω_{j}, j)}_{j=1}^{n} : a polymer chain in ℤ^{d+1} ,

• {ξ(x, n)}: inpurity in the solvent.

♣ Feature of the model; small values of βξ(ω_{j}, j) = random killing to the path

♣ Question; How does the path

((ω_{j})_{j=1}^{n} , μ_{n}(dω|ξ))

behave as n ↗ ∞, Q(dξ)-a.s?

♣ Answer to the question; •

d ≥ 3 & |β| small
⇒ similar behavior as usual RW

Imbrie and Spencer (1988), Bolthausen (1989), Song and Zhou (1996), ...

•

d ≤ 2 or |β| large
⇒ different behavior from usual RW

Carmona-Hu (2001), Comets-Shiga-Y.(2002)

How to capture the difference from the usual RW? ♣ in terms of weak/strong disorder:

• Z_{∞}(ξ) exists Q(dξ)-a.s.

• We have Q\{ξ ; Z_{∞}(ξ) > 0\} = \{ \begin{matrix}1 & “weak disorder” \\ or \\ 0 & “strong disorder” \end{matrix}

Remark: β = 0 ⇒ Z_{∞}(ξ) ≡ Z_{n}(ξ) ≡ 1.

Another way to capture the difference from the usual RW. ♣ in terms of localization:

Define

μ_{n−1}^{⊗2}(dωdω̃|ξ) = μ_{n−1}(dω|ξ) ⊗ μ_{n−1}(dω̃|ξ)

and

I_{n}(ξ) = μ_{n−1}^{⊗2}(ω_{n} = ω̃_{n}|ξ).

We formulate lim_{n↗∞} I_{n}(ξ) \{ \begin{matrix}= 0 & “delocalization” \\ > 0 & “localization” \end{matrix}

Remark: If β = 0, then I_{n}(ξ) = P^{⊗2}(ω_{n} = ω̃_{n}) = O(n^{−d/2}).

♣ Results for d ≥ 3 & small |β|

Theorem 1 Suppose: d ≥ 3 & |β| small. Then,

(a) weak disorder; Z_{∞}(ξ) > 0, Q(dξ)-a.s.

(b) CLT; Q(dξ)-a.s.,

lim_{n↗∞} μ_{n} ( \frac{ω_{n}}{√n} ∈ dx| ξ)

= (2π)^{−d/2} exp (−|x|^{2}/2) dx.

(c) delocalization;

I_{n}(ξ) = O(n^{−c}) in Q-probab.

• Imbrie and Spencer (1988): variance for Bernoulli case,

• Bolthausen (1989): parts (a),(b) above for Bernoulli case,

• Song and Zhou (1996): variance for general case,

• Comets-Shiga-Y.(2002): part(c) above.

♣ Key observation in Theorem 1

[Bolthausen (1989),Song and Zhou (1996)]

For d ≥ 3, set

r = P {ω_{n} = 0 for some n ≥ 1 } < 1

and asuume that |β| is so small that

λ(2β) − 2λ(β) < ln \frac{1}{r} .

One can use this to prove that

sup_{n≥1} Q[Z_{n}^{2}] < ∞.

This implies that

Q[Z_{∞}] = lim_{n↗∞} Q[Z_{n}] = 1

and therefore that

Z_{∞}(ξ) > 0, Q(dξ)-a.s.

♣ Results for d ≤ 2 or large |β|

Theorem 2 Suppose: β ≠ 0. Then,

(a) strong disorder for large |β| ; for large |β|,

Z_{n}(ξ) = O (exp (−cn)) , Q(dξ)-a.s.

(b) strong disorder in low dimensions ; Q(dξ)-a.s.,

Z_{n}(ξ) \{ \begin{matrix}= O (exp (−cn^{1/3} )) , & d = 1 \\ ⟶ 0, & d = 2. \end{matrix}

(c) localization in low dimensions; for d = 1, 2,

lim_{n↗∞} I_{n}(ξ) > 0, Q(dξ)-a.s.

• Carmona-Hu (2001): Gaussian environment, (use of BM and Ito’s formula),

• Comets-Shiga-Y.(2002):under exponential integrability

♣ Proof of Theorem 2

• part (a) & (b): Prove a recursion for fractional moment;

Q[Z_{n}^{θ}] ≤ a_{n}Q[Z_{n−1}^{θ}] + b_{n}, n ≥ 1,

where 0 < θ < 1, a_{n} < 1, b_{n}:small.

Remark: What exactly is assuned in part (a):

βλ′(β) − λ(β) > ln(2d).

• part (c): use of the recurrence of the RW (complicated).

♣ Strong disorder and localization

Theorem 3 Let β ≠ 0. Then, up to a Q(dξ)-null set,

{ξ ; Z_{∞}(ξ) = 0} = {ξ ; _{n≥1}^{∑} I_{n}(ξ) = ∞} . Moreover, if

Q{ξ ; Z_{∞}(ξ) = 0} = 1,

then Q(dξ)-a.s., −c_{1} ln Z_{n}(ξ) ≤ _{1≤j≤n}^{∑} I_{j}(ξ) ≤ −c_{2} ln Z_{n}(ξ),

for n ≫ 1 .

• Carmona-Hu (2001): Gaussian environment (again, by use of BM and Ito’s formula),

• Comets-Shiga-Y.(2002) :under exponential integrability

♣ Proof of Theorem 3

• Doob’s decomposition; − ln Z_{n}(ξ) = M_{n} + A_{n}.

• Key estimates;

⟨ M ⟩_{n} − ⟨ M ⟩_{n−1} ≤ cI_{n}(ξ),  (1)
c_{1}I_{n}(ξ) ≤ A_{n} − A_{n−1} ≤ c_{2}I_{n}(ξ).  (2)

∘ Proofs of (1) & (2) are reduced to general estimates for i.i.d. ∘ (2) ⇒ A_{n} ≍ ∑_{j=1}^{n} I_{j}(ξ). ∘ (1) & (2) ⇒ M_{n} ≪ A_{n}. In fact,

M_{n} ≪ ⟨ M ⟩_{n} ≤ c ∑_{j=1}^{n}I_{j}(ξ) ≍ A_{n}.

cf. LLN for martigales.

• Consequently,

− ln Z_{n}(ξ) ≍ A_{n} ≍ n ; ∑ ; j=1 I_{j}(ξ).

♣ Open questions We have discussed: (PF) positive free energy;

Z_{n}(ξ) = O (exp (−cn)) , Q(dξ)-a.s.

(L) localization;

lim_{n↗∞} I_{n}(ξ) ≥ c > 0, Q(dξ)-a.s.

(SD) strong disorder;

Q{ξ ; Z_{∞}(ξ) = 0} = 1.

We see from Theorem 3 that

(PF) ⇒ (L) ⇒ (SD) .

• “⇐”? or “⇍”? i.e., ∄ or ∃ of intermediate phase?

• super-diffusivity ? cf. M.Petermann