Crossing random walks and stretched polymers at weak disorder

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Reference

Crossing random walks and stretched polymers at weak disorder

IOFFE, Dmitry, VELENIK, Yvan

Abstract

We consider a model of a polymer in Z^{d+1}, constrained to join 0 and a hyperplane at distance N. The polymer is subject to a quenched non-negative random environment.

Alternatively, the model describes crossing random walks in a random potential (see Chapter 5 of [Sznitman] for the original Brownian motion formulation). It was recently shown, by Flury and by Zygouras, that, in such a setting, the quenched and annealed free energies coincide in the limit N to infinity, when d is at least 3 and the temperature is sufficiently high. We first strengthen this result by proving that, under somewhat weaker assumptions on the distribution of disorder which, in particular, enable a small probability of traps, the ratio of quenched and annealed partition functions actually converges. We then conclude that, in this case, the polymer obeys a diffusive scaling, with the same diffusivity constant as the annealed model.

IOFFE, Dmitry, VELENIK, Yvan. Crossing random walks and stretched polymers at weak disorder. Annals of Probability, 2012, vol. 40, no. 2, p. 714-742

arxiv : 1002.4289v1 DOI : 10.1214/10-aop625

Available at:

http://archive-ouverte.unige.ch/unige:6406

Disclaimer: layout of this document may differ from the published version.

1 / 1

DMITRY IOFFE AND YVAN VELENIK

Abstract. We consider a model of a polymer in Z^d+1, constrained to join 0 and a hyperplane at distanceN. The polymer is subject to a quenched non-negative random environment. Alternatively, the model describes crossing random walks in a random potential, see [10] or Chapter 5 of [9] for the original Brownian motion formulation. It was recently shown [5,11] that, in such a setting, the quenched and annealed free energies coincide in the limitN → ∞, whend>3 and the temperature is sufficiently high. We first strengthen this result by proving that, under somewhat weaker assumptions on the distribution of disorder which, in particular, enable a small probability of traps, the ratio of quenched and annealed partition functions actually converges. We then conclude that, in this case, the polymer obeys a diffusive scaling, with the same diffusivity constant as the annealed model.

Contents

1. Notation and Results 2

1.1. Some open problems 3

1.2. A remark on notational conventions 4

2. Convergence of partition functions 5

2.1. Irreducible decomposition of pathsγ ∈ D_N 5

2.2. Renewal analysis of annealed partition function TN 6

2.3. Complex tilts and annealed diffusivity 7

2.4. Multi-dimensional renewal relation for quenched partition functions 8

2.5. Recursion underL²-weak disorder 9

2.6. Relation with Sinai’s representation 10

2.7. Extension to the fullD_N-ensemble 11

3. Proofs 12

3.1. The key computation 12

3.2. Proof of Theorem 2.3(1) 12

3.3. Proof of Theorem 2.3(2) 13

3.4. Proof of Theorem 2.4 14

3.5. Proof of Theorem B 14

4. L²(Ω) estimates at weak disorder 16

4.1. Preliminaries 16

4.2. Weakly interacting random walks 17

4.3. Upper bounds in terms of synchronized random walks 19

4.4. Proof of Proposition 3.1 21

4.5. Positivity of d^ω on the event{0∈Cl∞(V)} 23

References 24

Supported in part by the Israeli Science Foundation.

Supported in part by the Swiss National Science Foundation.

1

1. Notation and Results

For simplicity ¹we shall consider stretched polymers which are represented by nearest- neighbour paths onZ^d+1. Due to the presence of a preferred direction, it is convenient to decomposex∈Z^d+1 into transverse and longitudinal parts: x= (x^⊥,x^k) with x^⊥∈Z^d and x^k∈Z. GivenN ∈N, we define

H⁻_N =^∆

x∈Z^d+1 : x^k< N

and its outer vertex boundary L_N =^∆ ∂H⁻_N. We shall consider the family D_N of nearest neighbour paths from the origin 0 toL_N. The namestretchedstipulates that although the second end-point of γ ∈ D_N is constrained to lie on L_N, there are no other restrictions on the geometry of polymers, which can bend and self-intersect. In the Brownian version of this problem [9], an alternative designation often used in the literature is crossing Brownian motion.

The weight W_λ,β^ω (γ) of a polymer γ = (γ(0), . . . , γ(n))∈ D_N is given by W_λ,β^ω (γ)= exp^∆ n

−λn−β

n

X

l=1

V^ω(γ(l))o

. (1.1)

Here λ > λ0 ∆

= log(2d),β >0 and the random environment{V^ω(x)}_x∈Zd+1 is assumed to be i.i.d.,V^ω(x)∼^d V, and such that

Assumption (A) 0∈supp(V)⊆[0,∞] and p=^∆P(V =∞) is sufficiently small.

That the potentialV be bounded below is essential, since it guarantees attractivity of the annealed potential: For any pair of paths γ₁ and γ₂,

E W_λ,β^ω (γ₁)W_λ,β^ω (γ₂)

>E W_λ,β^ω (γ₁)

E W_λ,β^ω (γ₂)

. (1.2)

(This can be most easily deduced from the fact that decreasing functions onRare always positively correlated.) The condition on the smallness of p is also essential, since it guar- antees that we never meet situations when {x:V^ω(x)<∞} does not percolate. On the other hand, the condition inf supp(V) = 0 is just a normalization.

The corresponding quenched and annealed partition functions are defined as D^ω_N =D^ω_N(λ, β)=^∆ X

γ∈D_N

W_λ,β^ω (γ) and D_N =^∆ED^ω_N.

It has recently been proved by Flury [5] (under the additional assumption that EV^d+1<

∞), and then reproved by Zygouras [11] (under the additional assumption that supp(V) be bounded) that, in four and higher dimensions (i.e., for d>3 in our notation) and for any λ > λ₀, the annealed and quenched free energies coincide when β is small enough.

Namely, for all β sufficiently small, there existsξ =ξ(λ, β)>0 such that

− lim

N→∞

1

N logD^ω_N =ξ=− lim

N→∞

1

N logD_N. (1.3)

This is an important result: In sharp contrast with models of directed polymers, the model of stretched polymers does not have an immediate underlying martingale structure, and this makes it necessary to find different (and arguably more intrinsic) approaches to its

1We emphasize that our techniques can deal in the same way with any finite-range step distribution.

Similarly, the particular geometric setting used, with the arrival hyperplane orthogonal to some lattice direction, can easily be generalized.

analysis. The conditionEV^d+1<∞under which (1.3) was derived is inherited from [10], where it was shown to be sufficient to guarantee the existence of the quenched free energy, i.e., the left-most limit in (1.3).

In the sequel, we shall prove the following sharp version of (1.3): Let Cl∞(V) be the (unique) infinite connected cluster of sitesxwithV(x)<∞. Under Assumption (A), such a cluster P-a.s. exists and is unique.

Theorem A. Let d>3. Then, for every λ > λ₀, there exists β₀ =β₀(λ, d) and p∞>0, such that, if Assumption (A) holds withp6p∞, then, for every β∈[0, β0), the limit

d^ω = lim^∆

N→∞

D^ω_N

D_N (1.4)

exists P-a.s. and inL²(Ω). In particular, the quenched free energy−lim_N→∞ 1

NlogD^ω_N is well defined and (1.3) holds. Furthermore, d^ω >0 P-a.s. on the event {0∈Cl∞(V)}.

Our work was inspired by [5,11], however our proof of Theorem Adoes not rely on the results therein. In particular, in addition to strengthening their conclusion, Theorem A lifts some of the restrictions imposed on the potentialV in these works. In fact, under our assumptions, which do not impose any moment conditions on the distribution of V and even enable a small probability of traps, the existence of the quenched free energy needs a justification: as we have already mentioned, the corresponding existence results in [10], which is a reference work for both [5] and [11], have been established under the additional assumption EV^d+1 <∞.

Our second result confirms the prediction that stretched polymers should be diffusive at weak disorder: On the event 0∈Cl∞(V), the random weights (1.1) induce a (random) probability distribution µ^ω_N on D_N. For a polymer γ = (γ(0), . . . , γ(n))∈ D_N, we define π^⊥(γ) as the (Z^d-valued) transverse component of its end-point, and π^k(γ) = N as its longitudinal component, so that γ(n) = (π^⊥(γ), π^k(γ)).

Theorem B. Let d>3. Then, for every λ > λ0, there exist βˆ0 = ˆβ0(λ, d) and pˆ∞ >0 such that, if Assumption (A) holds with p 6 pˆ∞, then, for every β ∈ [0,βˆ0), the distribution of π^⊥ displays diffusive scaling with a non-random non-degenerate diffusivity matrix Σ and, accordingly, a positive diffusivity constant σ² = σ²(β, λ) =^∆ Tr(Σ) > 0.

Namely, define P^∗(·)=^∆ P(·|0∈Cl∞(V)). Then, P^*-lim

N→∞ µ^ω_N

|π^⊥(γ)|² N

=σ², (1.5)

where P^*-limdenotes convergence inP^∗-probability. Furthermore, for any bounded contin- uous function f onR^d,

P^*-lim

N→∞

X

x∈Z^d

µ^ω_N(π^⊥(γ) =x)f x

√ N

= 1

pdet(2πΣ) Z

R^d

f(x)e^{−12(Σ−1x,x)}dx. (1.6) Σ and σ² above are precisely the diffusion matrix and the diffusivity constant of the cor- responding annealed polymer model, see (2.11) below.

We expect both (1.5) and (1.6) to hold not only in P^∗-probability, but also in L²(Ω) and P^∗-a.s..

1.1. Some open problems. In this subsection, we briefly discuss some points that are left untouched in the present work.

Stronger modes of convergence. As already mentioned above, we expect our diffusivity results to hold also a.s. in the environment and in L²(Ω). Such results are known in the directed case, as a consequence of the much simpler martingale structure [1]. Furthermore, we expect the P^∗-a.s. validity of a local CLT, or equivalently, of a (random) Ornstein- Zernike type formula for long-range quenched connections, see the discussion at the end of Subsection 3.5.

Invariance principle. Once equipped with a local CLT and thanks to our good control on the path geometry, it should be mostly straightforward to obtain a full invariance principle for the path.

“Real” stretched polymer. In the present work, we have focused on ensembles of paths of

“point-to-plane” type (the set D_N). It would be physically quite interesting to analyze also the case of fixed-length polymers, stretched by an external force (notice that in the directed case there is no difference between “point-to-plane” and “fixed-length” scenarios);

in particular, it would be interesting to obtain a local limit theorem for the free endpoint.

Such questions have been investigated in the annealed setting in our previous work [6]. In the quenched setting coincidence of Lyapunov exponents (under the additionalEV^d+1<∞ assumption) has been established in [5].

Nonperturbative proof. Our results are only valid at very high temperatures. It would be quite interesting (and probably challenging) to push them to the full weak-disorder regime. Results of that type have been obtained in the directed case [4].

Strong disorder. We only consider the weak disorder case here. Obtaining some informa- tion on the behaviour of typical paths in the strong disorder regime would also be quite interesting, and is the subject of some work in progress. See [3] for such results, in the full strong disorder regime, in the directed case.

1.2. A remark on notational conventions. Given two sequences{a_n(w)}and{b_n(w)}

of positive real numbers indexed by w from some set of parameters W_n, we say that an(w).bn(w), if

lim sup

n→∞

an(w)

b_n(w) < ∞, uniformly in w∈W_n.

Given z,w∈C^d+1, we use (z,w)_d+1 =^∆

d+1

X

i=1

ziw¯i and (z,w)_d=^∆

d

X

i=1

ziw¯i.

With a slight abuse of notation, we shall also write (z,w)_d for the same expression with z∈C^d.

Acknowledgement. We thank Francis Comets for pointing out the similarity between our method and the expansion used in [8] to treat the directed case.

γ4

γl

γ3

γr

γ2 γ1

LN x

0

Figure 1. The decomposition of a pathγ∈ DN into a concatenation of strongly irreducible pieces.

2. Convergence of partition functions

2.1. Irreducible decomposition of paths γ ∈ D_N. Given δ > 0, we define a positive cone along the~e=^∆~e_d+1-direction by

Y_δ =^∆

x∈R^d+1 : kx^⊥k< δx^k ,

where k · kdenotes the Euclidean norm. We say that a trajectory γ = (γ(0), . . . , γ(n)) of length|γ|=niscone-confined if

γ ⊆(γ(0) +Y_δ)∩(γ(n)− Y_δ).

Although pathsγ ∈ D_N always satisfy 0 = (γ(0),~e)_d+1 <(γ(n),~e)_d+1, evidently not all of them are cone-confined. For 16k < n=|γ|, let us say thatγ(k) is a cone-point of γ if

(γ(0),~e)_d+1 <(γ(k),~e)_d+1 <(γ(n),~e)_d+1, and, in addition, if

γ ⊆(γ(k)− Y_δ)∪(γ(k) +Y_δ).

We say that a trajectory γ is irreducible if it contains less than two cone-points. We say that it is strongly irreducible if it does not contain cone-points at all.

The following mass-separation property of irreducible trajectories, proved in [6], is cru- cial to our analysis: There exists ν >0 such that, for allN large enough,

1 D_N

X

γ∈D_N irreducible

EW_λ,β^ω (γ)6e^−νN. (2.1) On the other hand, reducible trajectories are unambiguously represented as concatenation of strongly irreducible pieces (as induced by the collection of all the cone-points of γ, see Fig. 1),

γ =γ_l∪γ₁∪ · · · ∪γ_n∪γ_r. (2.2) By construction, γ1, . . . , γn above are also cone-confined, and so is their concatenation γ₁∪ · · · ∪γ_n. Thus, (2.1) and (2.2) suggest that the asymptotics of D_N and D^ω_N should be closely related to the asymptotics of the corresponding cone-confined quantities. This intuition turns out to be correct.

Let T_N be the family of all cone-confined trajectories from 0 toL_N. Set T^ω_N(λ, β)=^∆ X

γ∈T_N

W_λ,β^ω (γ) and TN

=∆ ET^ω_N.

The following statement as well as the understanding one needs to develop for its proof are crucial: In the notation and under the conditions of Theorem A, for every β∈[0, β₀), the limit

N→∞lim T^ω_N

T_N (2.3)

exists P-a.s. and in L²(Ω). For a while we shall focus on the ensembles of cone-confined trajectories and on proving (2.3). We shall return toD_N and prove the full statement (1.4) only in Subsection 2.7.

Notation for scaled quantities. Recall the definition of the Lyapunov exponentξ in (1.3).

Given N >1 and γ ∈ T_N, we define the scaled random path weights w^ω_λ,β(γ)= e^{∆ N ξ}W_λ,β^ω (γ).

For x∈ L_N, we define t^ω_x =^∆ X

γ:0→x γ∈T_N

w^ω_λ,β(γ), q^ω_x =^∆ X

γ:0→x γ∈T_N⁰

w_λ,β^ω (γ) and t_x=^∆ Et^ω_x, q_x=^∆ Eq^ω_x, (2.4)

where T_N⁰ denotes the set of all strongly irreducible γ ∈ T_N. Similarly, we define t^ω_N =^∆ X

x∈L_N

t^ω_x, q^ω_N =^∆ X

x∈L_N

q^ω_x and tN ∆

=Et^ω_N, qN ∆

=Eq^ω_N. We also set t^ω₀ =t₀ = 1.^∆

2.2. Renewal analysis of annealed partition function TN. With the above nota- tions, the sequence {t_N} satisfies the renewal relation

t₀= 1 and t_N =

N−1

X

M=0

t_MqN−M, N >1. (2.5)

We fix λandβ and set

µ=µ(λ, β)=^∆ X

M>1

Mq_M. (2.6)

Note that the above series converges since, by our basic mass-separation estimate for annealed quantities (2.1),

q_M 6e^−νMe^{N ξ}D_M 6e^−νM, (2.7)

where we used the fact that DM 6e^{−N ξ}, which follows from subadditivity.

Lemma 2.1. For any β>0 andλ > λ0,

N→∞lim e^{N ξ}TN = lim

N→∞tN = 1

µ(λ, β). (2.8)

Moreover, the convergence in (2.8) is exponentially fast.

Proof. This is a standard renewal argument which we shall briefly sketch for completeness.

As a consequence of our scaling, the radius of convergence of the generating function ˆt(u)=^∆ X

N>0

u^NtN

is equal to 1. On the other hand, it follows from (2.7) that the irreducible generating function

ˆ

q(u)=^∆ X

N>1

u^NqN

has radius of convergence at least 1 + ν. This implies, via standard arguments based on (2.5), that ˆq(1) = 1. Of course, µ= ˆq⁰(1). Fixρ∈(0,1). By Cauchy’s formula,

t_N − 1 µ = 1

2πi Z

Sρ

n du

u^N+1(1−q(u))ˆ − du u^N+1(1−u)µ

o

= 1 2πi

Z

Sρ

∆(u) u^N+1 du, where Sρ=∂Bρ and

∆(u) = (ˆq(u)−ˆq(1))−qˆ⁰(1)(u−1)

(ˆq(u)−ˆq(1))ˆq⁰(1)(u−1) . (2.9) Since ∆ is analytic on B1+ν⁰ for someν⁰ ∈(0, ν), the result follows.

2.3. Complex tilts and annealed diffusivity. Forδsmall enough, letP^d_2δ ⊂C^dbe the complex polydisc with all dradii equal to 2δ. By the implicit function theorem (see, e.g., [7]) and in view of the mass-gap estimate (2.7), the relations

ϕ[0] = 0 and X

M>1

X

x∈L_M

qxe−M ϕ[z]+(z,x)_d ∆

= X

M>1

qM[z] = 1

define a holomorphic function ϕ:P^d_2δ→C. We shall assume thatδ is so small that

|q_M[z]|.e^−νM/2, (2.10)

uniformly in M >1 and z∈P^d_2δ.

The analysis of the previous subsection can be readily extended to obtain the asymptotic expansion of the moment generating functions

t₀[z]= 1^∆ and, forN >1, t_N[z]=^∆ X

x∈LN

t_xe−N ϕ[z]+(z,x)_d, forz∈P^d_2δ. Indeed, tN[z] satisfies the renewal relation

t_N[z] =

N−1

X

M=0

t_M[z]qN−M[z].

Furthermore, ifδ is sufficiently small, then not only does (2.10) hold, but there also exists ν⁰ >0 such that, for all z∈P^d_2δ,u= 1 is the unique solution of the equation

X

M>1

u^MqM[z]= ˆ^∆q[z](u) = 1,

on B1+ν⁰ ⊂C. We defineµ[z] exactly as in (2.6) by µ[z]=^∆ X

N>1

Nq_N[z].

Relying on (2.10), we can choose δ so small that µ[·] is analytic and non-zero on P^d_2δ. It then follows that

Nlim→∞t_N[z] = 1 µ[z], uniformly exponentially fast on P^d_2δ.

The annealed diffusion matrix Σ and the corresponding diffusivity constantσ² in (1.5) are defined by

Σ= D^{∆ 2}_dϕ[0] and σ^{2 ∆}= Tr(Σ), (2.11) where D_d²ϕ denotes the Hessian of ϕ. Now, since we have chosen δ sufficiently small to ensure thatµ[·] is analytic and does not vanish onP^d_2δ, the functions logtN[·] + logµ[·] are analytic and exponentially small (in N) on P^d_2δ. In particular,

Tr

D²_d logtN[z]+ logµ[z]

is also exponentially small. This shows that the leading contribution (in N) to the log- moment generating function log tN[z]e^{N ϕ[z]}

ofπ^⊥(γ) under the induced measure is given by N ϕ[z]. We have thus proved that

Lemma 2.2.

1 Nt_N

X

x∈L_N

kx^⊥k²tx−σ² . 1

N. Furthermore, π^⊥(γ)/√

N ⇒ N(0,Σ) under the sequence of annealed polymer measures µN.

2.4. Multi-dimensional renewal relation for quenched partition functions. We continue to employ the notation introduced in (2.4). It is immediate to check that the following analogues of (2.5) hold,

t^ω_x =^∆ X

y

t^ω_yq^θ_x−y^yω and t^ω_N =^∆

N−1

X

M=0

X

x∈L_M

t^ω_xq^θ_N−M^xω , (2.12)

for all x ∈ H⁺₀ =^∆

x∈Z^d+1 : x^k>0 and N >1. Set t^ω₀ = 1 and define the generating^∆ functions

ˆt^ω(u) =^∆

∞

X

N=0

u^Nt^ω_N and ˆq^ω(u) =^∆

∞

X

N=0

u^Nq^ω_N.

Since |t^ω(u)| 6 t^ω(|u|) and Eˆt^ω(ρ) 6 ˆt(ρ), the random generating function ˆt^ω(u) is P- a.s. defined and analytic in the interior of the unit disc B1 ⊂ C. Similarly, the random generating function ˆq^ω(u) isP-a.s. analytic on B1+ν for some ν >0.

We can rewrite (2.12) in terms of the generating function as ˆt^ω(u) = 1 +

∞

X

M=0

u^M X

x∈LM

t^ω_xˆq^θxω(u)

= 1 + ˆq(u)

∞

X

M=0

u^M X

x∈LM

t^ω_x

+

∞

X

M=0

u^M X

x∈L_M

t^ω_x ˆq^θxω(u)−ˆq(u)

= 1 + ˆ∆ q(u)ˆt^ω(u) + ˆΨ^ω(u). (2.13) Since |ˆq(u)|<1 whenever|u|=ρ <1, we can record the last computation as

ˆt^ω(u) = 1 + ˆΨ^ω(u) (1−q(u))ˆ . Therefore,

t^ω_N = 1 2πi

Z

Sρ

1 + ˆΨ^ω(u)

(1−q(u))uˆ ^N+1 du, (2.14) P-a.s. for all ρ∈(0,1).

2.5. Recursion under L²-weak disorder. Equation (2.14) is the starting point for proving Theorem A. In fact, we are going to develop a recursion for the limit in (2.3) whenever the conditions of the latter theorem are satisfied.

Let us decompose

t^ω_N = 1 µs^ω_N +

t^ω_N − 1

µs^ω_N

, where ²

s^ω_N =^∆ 1 2πi

Z

Sρ

1 + ˆΨ^ω(u) u^N+1(1−u)du, and, accordingly,

t^ω_N− 1

µs^ω_N = 1 2πi

Z

Sρ

(1 + ˆΨ^ω(u))∆(u)

u^N+1 du, (2.15)

with ∆(u) defined in (2.9).

After examining the definition of ˆΨ^ω in (2.13), we arrive at the following expression for s^ω_N,

s^ω_N =1 + ˆΨ^ω(u) 1−u

N = 1 +

N−1

X

M=0

X

x∈L_M

t^ω_x q^θ_1,N−M^xω −q1,N−M

= 1 + X

x∈H⁻_N,y∈H⁻_N+1

t^ω_x q^θ_y−x^xω −qy−x

, (2.16) where

q^ω_1,l=^∆

l

X

k=1

q^ω_k and q_1,l =^∆Eq^ω_1,l,

2Note that the definition does not depend on the particular choice ofρ∈(0,1).

and we used the standard notationP

k>0aku^k

N =aN for expansion coefficients.

The following theorem is proved in Subsection3.2 and Subsection3.3.

Theorem 2.3. For every λ > λ0, there exist β0 = β0(λ, d) and p∞ > 0, such that if Assumption (A) holds with p6p∞, then, for everyβ ∈[0, β₀):

(1) The sequence t^ω_N −s^ω_N/µconverges to zero P-a.s. and inL²(Ω).

(2) The sequence s^ω_N converges P-a.s. and in L²(Ω)to s^ω = 1 +^∆ X

x

t^ω_x q^θ_1,∞^xω −1

, (2.17)

the latter sum also converging inL²(Ω).

Theorem 2.3 implies that the limit in (2.3) indeed exists and, furthermore, that it is equal to the random variable s^ω:

Nlim→∞

T^ω_N

T_N = lim

N→∞

t^ω_N

t_N = lim

N→∞s^ω_N =s^ω.

Note that if 0 6∈ Cl∞(V), then t^ω_N = 0 for all N sufficiently large, say N > N₀(ω).

Consequently, in this case s^ω is a difference of two convergent series, s^ω= 1 + X

x,y∈H⁻_N

0

t^ω_xq^θ_y−x^xω − X

∈H⁻_N

0

t^ω_x = 0.

Positivity of s^ω (or rather of the full limit d^ω in (1.4)) on the event {0∈Cl∞(V)} is established in the concluding Subsection 4.5of the paper.

2.6. Relation with Sinai’s representation. Our representation (2.16) can be seen as an effective random walk version of the high temperature expansion employed by Sinai in [8]. Indeed, let x∈ L_N. Then

t^ω_x = X

n>0

X

x1,...,xn

n

Y

k=0

q^ω_x

k,xk+1

= X

n>0

X

x1,...,xn

n

Y

k=0

qxk,xk+1Φ^ω(x0, . . . ,xn+1).

where we have set x0 = 0, xn+1=x, and Φ^ω(x₀, . . . ,x_n+1)=^∆

n

Y

k=0

q^ω_x

k,xk+1

qxk,xk+1

=∆ n

Y

k=0

1 +φ^ω(x_k,x_k+1) .

Using the expansion

Φ^ω(x₀, . . . ,x_n+1) = X

A⊂{0,...,n}

Y

k∈A

φ^ω(x_k,x_k+1), we obtain the representation

t^ω_x = X

n>0

X

x1,...,xn

n

Y

k=0

q_xk,xk+1 X

n>0

X

A⊂{0,...,n}

Y

k∈A

φ^ω(X_k,X_k+1).

Given n, x1, . . . ,xn and ∅ 6= A ⊂ {0, . . . , n}, let us say that (x_k^∗,xk^∗+1) is the last per- turbed segment if k^∗ = max{k : k∈ A}. Keeping the last perturbed segment fixed and resumming all the rest, we arrive at

t^ω_x =t_x+X

y,z

t^ω_y

q^θ_z−y^yω −qz−y

tx−z. (2.18)

Similarly, keeping the first perturbed segment fixed and resumming all the rest, we arrive at

t^ω_x =tx+X

y,z

ty

q^θ_z−y^yω −qz−y

t^θ_x−z^zω. (2.19)

It would have been possible to work directly with the above representations oft-quantities.

In fact, Theorem2.3(1) can be considered as the first step along these lines: it enables us to substitute and control the t-quantities by the more tractable s-quantities, as appears in (2.16).

Notice though that it is not clear how to prove the almost sure convergence in TheoremA without having recourse to martingale arguments as developed in Subsection3.1.

2.7. Extension to the full D_N-ensemble. Let us go back to Theorem A. In view of (2.1), there is no loss in redefining D_N as the set of all reducible paths from 0 to L_N. Thus, any γ ∈ D_N automatically satisfies (2.2). By construction (decomposition with respect to all cone-points), none of the paths γ_l, γ₁, . . . , γ_n, γ_r in (2.2) has cone-points.

Recall that we use the notation T⁰ for cone-confined paths without cone-points. Thus, γ₁, . . . , γ_n∈ T⁰.

Paths γl = (γl(0), . . . , γl(m)) satisfy γl ⊆ γl(m) − Y_δ, and, similarly, paths γr = (γ_r(0), . . . , γ_r(k)) satisfy γ_r ⊆ γ_r(0) +Y_δ. We denote by T_l⁰ and T_r⁰ the sets of such paths; in this way, T⁰ =T_l⁰∩ T_r⁰.

Following (2.4), define

l^ω_x =^∆ X

γ:07→x γ∈T_l⁰

w^ω_λ,β(γ) and r^ω_x =^∆ X

γ:07→x γ∈T_r⁰

w_λ,β^ω (γ).

As usual, we denote the corresponding annealed quantities by lx and rx. The scaled full D_N partition function satisfies

d^ω_N = e^{∆ N ξ}D^ω_N = X

x∈L_N

X

γ:07→x γ∈D_N

w_λ,β^ω (γ) = X

06Ml<Mr6N

X

x∈L_Ml y∈L_Mr

l^ω_xt^θ_y−x^xωr^θ_N^yω_−M

r

= X

06M_l<Mr6N

X

x∈L_Ml

l^ω_xt^θ_M^xω

r−M_lrN−M_r

+ X

06Ml<Mr6N

X

x∈L_Ml y∈L_Mr

l^ω_xt^θ_y−x^xω r^θ_N^yω_−M

r−r_N−Mr

.

(2.20)

Define cr ∆

=P

MrM. The following theorem is proved in Subsection 3.4.

Theorem 2.4. For every λ > λ0, there exist β0 = β0(λ, d) and p∞ > 0 such that, if Assumption (A) holds with p6p∞, then, for everyβ ∈[0, β₀):

(1) The second term on the right-hand side of (2.20) converges to zero P-a.s. and in L²(Ω).

(2) The first term on the right-hand side of (2.20) converges to c_r

µ X

x

l^ω_xs^θxω, (2.21)

P-a.s. and inL²(Ω).

Consequently, (1.4) of Theorem Afollows with d^ω= lim

N→∞

D^ω_N D^ω_N =cr

X

x

l^ω_xs^θxω.

Positivity ofd^ωon the event{0∈Cl∞(V)}is established in the concluding Subsection4.5.

3. Proofs

3.1. The key computation. Below, we formulate the key statement, essential for all our results in this paper. It heavily relies on the assumptions of weak disorder. We relegate the proof of Proposition 3.1to the concluding Section 4.

Proposition 3.1. For every λ > λ0, there exist β0 =β0(λ, d) and p∞ >0 such that, if Assumption (A) holds with p6p∞, then, for everyβ ∈[0, β₀),

sup

N>1E h X

x∈H⁻_N

X

y∈H⁺_K

t^ω_x(q^θ_y−x^xω −qy−x)g(y)i2

.(K+ 1)^1−d/2kgk²_∞, (3.1) uniformly in K >0 and in bounded functions g on Z^d+1.

Furthermore, E

h X

x∈H⁻_K

X

y∈H⁺_K

t^ω_x(q^θ_y−x^xω −qy−x)g(y) i2

.(K+ 1)^−d/2kgk²_∞, (3.2) uniformly in K >0 and in bounded functions g on Z^d+1. Similarly,

E h X

x,y∈H⁻_K

X

z∈H⁺_K

l^ω_xt^θ_y−x^xω

r^θ_z−y^yω −rz−y

g(z)

i2

.(K+ 1)^−d/2kgk²_∞, (3.3) also uniformly in K >0 and in bounded functions g onZ^d+1.

3.2. Proof of Theorem 2.3(1). Recall that t^ω_N− 1

µs^ω_N = 1 2πi

Z

Sρ

(1 + ˆΨ^ω(u))∆(u)

u^N+1 du, (3.4)

for each ρ∈(0,1). We are going to show that

Lemma 3.2. (3.4) still holds at ρ= 1 and Ψ^ω(e^iθ)∈L2(Ω×[0,2π]).

In particular, Ψ^ω(e^iθ)∈L₂([0,2π]) P-a.s. Consequently, the right-hand side of (2.15) is P-a.s. equal to theN-th Fourier coefficient of (1 + ˆΨ^ω(e^iθ))∆(e^iθ). Therefore, by Parseval’s theorem,

E X

N

t^ω_N − 1 µs^ω_N2

= 1 2πE

Z 2π

0

(1 + ˆΨ^ω(e^iθ))∆(e^iθ) ²dθ <∞.

It thus follows from Fubini’s theorem that

N→∞lim t^ω_N− 1 µs^ω_N

= 0, P-a.s. and in L²(Ω).

It remains to prove Lemma 3.2. First of all, ˆΨ^ω(e^iθ) can be rewritten as Ψˆ^ω(e^iθ) =X

x,y

t^ω_x(q^θ_y−x^xω −qy−x)e^{iθ(~ed+1,y)d+1}.

Applying Proposition3.1 withK = 0,N =∞ andg(y) = e^{iθ(~ed+1,y)d+1}, we conclude that sup

θ

E Ψˆ^ω(e^iθ)2

.1, and hence Ψ^ω(e^iθ)∈L2(Ω×[0,2π]) indeed.

In a completely similar fashion, one concludes from Proposition3.1 that

K→∞lim sup

|u|61

E X^∞

M=K

u^Mψ^ω_M2

= 0, (3.5)

where {ψ_N^ω}are the expansion coefficients of Ψ^ω(u) =P

Mu^Mψ_M^ω , i.e., explicitly, ψ^ω_M = X

y∈L_M

X

x

t^ω_x(q^θ_y−x^xω −qy−x).

Obviously, for eachK fixed,

ρ→1lim

K

X

N=1

ρe^iθN

ψ^ω_N =

K

X

N=1

e^iθN

ψ_N^ω

inL2(Ω×[0,2π])). In view of (3.5), the latter implies that

ρ→1limE Z 2π

0

Ψ^ω(ρe^iθ)−Ψ^ω(e^iθ)2

dθ= 0.

As a result one can indeed pass to the limitρ→1 in (3.4).

3.3. Proof of Theorem 2.3(2). LetF_N be theσ-algebra generated by{V_x}_x∈H⁻

N+1, and let us introduce

A^ω_N = 1 +^∆

N−1

X

M=0

X

x∈L_M

t^ω_x q^θ_1,∞^xω −1 ,

B_N^ω =^∆

N−1

X

M=0

X

x∈L_M

t^ω_x q^θ_N−M^xω _+1,∞−qN−M+1,∞

, C_N^ω =^∆E A^ω_N

F_N . We can then express s^ω_N as

s^ω_N =C_N^ω + (A^ω_N− C_N^ω)− B_N^ω.

The P-a.s. andL²(Ω) convergence in (2.17) follows from the next two lemmas, since they imply that, P-a.s. and in L²(Ω),B_N^ω and A^ω_N− C_N^ω tend to 0, whileC_N^ω converges to s^ω.

Lemma 3.3. For every λ > λ0, there existβ0 >0andpˆ∞>0such that, ifAssumption (A) holds with p 6 pˆ∞,then, for each β ∈ [0, β₀], the sequence {C_N} is an L²-bounded martingale.

Lemma 3.4. For every λ > λ₀, there existβ₀ >0andpˆ∞>0such that, ifAssumption (A) holds withp6pˆ∞, then

X

N

E(B^ω_N)² <∞ and X

N

E(A^ω_N − C_N^ω)² <∞, for each β ∈[0, β0].

Proof of Lemma 3.3. The fact thatC_N^ω is a martingale is straightforward: For anyN and each x∈ L_N,

E(C_N^ω | F_N) =E(A^ω_N| F_N) + X

x∈L_N

E t^ω_x(q^θ_1,∞^xω −1) F_N

, and

E t^ω_x(q^θ_1,∞^xω −1) F_N

=t^ω_xE(q^θ_1,∞^xω −1) = 0, sincex∈ L_N.

It remains to check thatC_N^ω isL²(Ω)-bounded. We first deduce from Jensen’s inequality that E(C_N^ω)² 6 E(A^ω_N)². However, uniform L²(Ω)-boundedness of the latter quantities follows immediately from Proposition 3.1by takingK = 0 and g≡1.

Proof of Lemma 3.4. Note that A^ω_N− C_N^ω = X

x∈H⁻_N

X

y∈H⁺_N

t^ω_x

q^θ_y−x^xω −E q^θ_y−x^xω F_N

and B_N^ω = X

x∈H⁻_N

X

y∈H⁺_N

t^ω_x q^θ_y−x^xω −qy−x .

Thus,A^ω_N − C_N^ω andB_N^ω have very similar forms. In fact, E(A^ω_N − C_N^ω)² 64E(B_N^ω)².

On the other hand, taking g ≡1 in the second of the statements of Proposition 3.1, we readily conclude that P

NE(B^ω_N)² <∞.

3.4. Proof of Theorem 2.4. The claim (2)of the theorem follows from the P-a.s. and L²(Ω) convergence to s^ω in (2.17) and from the fact that

El^ω_x =l_x6e^−ν|x|1_{x∈Yδ}.

The first claim(1) follows by an application of (3.3) withK =N andg(z) =1_{z∈LN}. 3.5. Proof of TheoremB. Letf be a bounded continuous function onR^d. Using (2.18), we can write, for any K>0,

X

z∈L_N

t^ω_zf z^⊥

√ N

= X

x∈H⁻_N

X

y

t^ω_x(q^θ_y−x^xω −qy−x) X

z∈L_N

tz−yf z^⊥

√ N

= X

x∈H⁻_N

X

y∈H⁻_K

t^ω_x(q^θ_y−x^xω −qy−x) X

z∈L_N

tz−yf z^⊥

√N

+ X

x∈H⁻_N

X

y∈H⁺_K−1

t^ω_x(q^θ_y−x^xω −qy−x) X

z∈LN

tz−yf z^⊥

√ N

.

(3.6)

Choosing K=K(N) =N for some <1/2 and setting g(y) =gN(y) = X

z∈L_N

tz−yf z^⊥

√ N

,

we can infer from Proposition 3.1 that the second sum on the right-hand side of (3.6) converges to zero in L²(Ω). As for the first sum on the right-hand side of (3.6), it follows from the annealed central limit theorem (and the continuity off) that

Nlim→∞ max

y∈H⁻_N ∩Yδ

X

z∈LN

tz−yf z^⊥

√ N

− 1 µ

1 pdet(2πΣ)

Z

R^d

f(x)e^{−12(Σ−1x,x)}dx = 0.

By another application of Proposition3.1, this time with K= 0 and g(y) = X

z∈L_N

tz−yf z^⊥

√N − 1

µ 1 pdet(2πΣ)

Z

R^d

f(x)e^{−12(Σ−1x,x)}dx 1_{y∈H⁻

N ∩Y_δ}, we conclude, in view of Theorem 2.3, that the first sum in (3.6) converges inL²(Ω) to

s^ω µ

1 pdet(2πΣ)

Z

R^d

f(x)e^{−12(Σ−1x,x)}dx= t^ω pdet(2πΣ)

Z

R^d

f(x)e^{−12(Σ−1x,x)}dx.

Extension to the fullD_N-ensemble. Proceeding as in Subsection2.7, we conclude that, for any bounded continuous f, the series

X

z∈L_N

d^ω_zf z^⊥

√ N

converges in L²(Ω) to

crP

xl^ω_xt^θxω pdet(2πΣ)

Z

R^d

f(x)e^{−12(Σ−1x,x)}dx.

Together with (2.21), this implies (1.6).

Local limit description. As in [8], equations (2.18) and (2.19) suggest the following quenched Ornstein-Zernike asymptotics fort^ω_x (as inherited from the annealed OZ-asymptotics oftx

in [6]): Givenx∈Z^d+1, let ˆθxω be the reflection with respect to the hyperplaneL_N of the shifted environment θ_xω. In other words, ˆθ_xω is the environment as seen backwards from x. Of course, the reflected environment has the very same averaged polymer connectivity functions. We conjecture that

t^ω_x

t_x = (1 +s^ω)(1 +s^θˆxω)(1 +o(1)). (3.7) Clearly, the strength of the above conjecture depends on what is meant byo(1) in (3.7). A P-a.s. statement would be a refinement of aP-a.s. CLT, which is, as we already mentioned, an open problem by itself. Weaker statements, on the other hand, are feasible via an appropriate refinement of Proposition 3.1.

y v

x

u x− Yδ

y− Y_δ

Figure 2. Withx,y,u,vas in the picture, the paths contributing tot^ω_xt^ω_y(q^ω_y,v− qy,v) lie inside the blue region, while the paths contributing to q^ω_x,u−qx,u lie inside the red region. These two quantities are thus independent (w.r.t. the disorder).

4. L²(Ω)estimates at weak disorder

4.1. Preliminaries. Our proof of Proposition3.1 is based on a comparison with weakly interacting random walks on Z^d. The bottom line is that, under Assumption (A), tran- sience wins over attraction. From a technical point of view the approach is similar to [2].

Since, in all the estimates below, only the supremum norm ofgin Proposition3.1would matter, we can assume, without loss of generality, that g≡1. It is convenient to use the alternative notations

q^ω_x,u=^∆ X

γ∈T_x,u⁰

W_λ,β^ω (γ) = X

γ∈T_u−x⁰

W_λ,β^θxω(γ), and q_x,u =qu−x ∆

= Eq^ω_x,u. Above, T_u⁰ is the set of irreducible cone-confined paths from 0 tou and T_x,u⁰ =^∆x+T_u−x⁰ .

Given xandu, we define the diamond shape

D(x,u)= (x^∆ +Y_δ)∩(u− Y_δ).

By construction, any pathγ ∈ T_x,u satisfiesγ ⊂D(x,u). Hence,q^ω_x,uonly depends on the environment inside D(x,u).

Here is a useful observation (see Fig. 2): If D(x,u)∩D(y,v) =∅, then E

t^ω_x q^ω_x,u−q_x,u

t^ω_y q^ω_y,v−q_y,v = 0.

Indeed, unless x = y, it is always true that either D(x,u)∩(y− Y_δ) = ∅ or D(y,v)∩ (x− Y_δ) = ∅. If, in addition, the diamond shapes do not intersect, then in the former case q^ω_x,u−q_x,u

is independent oft^ω_xt^ω_y q^ω_y,v−q_y,v

, and similarly for the latter case.

Consequently, neglecting non-positive terms, we obtain E

n X

x∈H⁻_N

X

y∈H⁺_K

t^ω_x(q^θ_y−x^xω −qy−x)o2

6 X

x,y∈H⁻_N u,v∈H⁺_K

E

t^ω_xq^ω_x,ut^ω_yq^ω_y,v+t^ω_xqx,ut^ω_yqy,v 1{D(x,u)∩D(y,v)6=∅}.