getdoc5051. 283KB Jun 04 2011 12:04:22 AM

in PROBABILITY

LOCALIZATION FOR (1

+

_{1)-DIMENSIONAL PINNING MODELS WITH}

(

_∇

+ ∆)

-INTERACTION

MARTIN BORECKI

TU Berlin, Institut für Mathematik, Strasse des 17. Juni 136, 10623 Berlin, Germany email: martin.borecki@alumni.tu-berlin.de

FRANCESCO CARAVENNA

Dipartimento di Matematica Pura e Applicata, Università degli Studi di Padova, via Trieste 63, 35121 Padova, Italy

email: francesco.caravenna@math.unipd.it

SubmittedJune 5, 2010, accepted in final formOctober 8, 2010 AMS 2000 Subject classification: 60K35; 82B41; 60J05

Keywords: Pinning Model; Polymer Model; Linear Chain Model; Phase Transition; Localization Phenomena; Gradient Interaction; Laplacian Interaction; Free Energy; Markov Chain

Abstract

We study the localization/delocalization phase transition in a class of directed models for a ho-mogeneous linear chain attracted to a defect line. The self-interaction of the chain is of mixed gradient and Laplacian kind, whereas the attraction to the defect line is of δ-pinning type, with strengthǫ_≥0. It is known that, when the self-interaction is purely Laplacian, such models un-dergo a non-trivial phase transition: to localize the chain at the defect line, the rewardǫ must be greater than a strictly positive critical threshold ǫ_c > 0. On the other hand, when the self-interaction is purely gradient, it is known that the transition istrivial: an arbitrarily small reward

ǫ >0 is sufficient to localize the chain at the defect line (ǫ_c=0). In this note we show that in the mixed gradient and Laplacian case, under minimal assumptions on the interaction potentials, the transition is always trivial, that isǫ_c=0.

1

Introduction

We consider a simple directed model for a homogeneous linear chain {(i,ϕ_i)_}0≤i≤N, such as a polymer, which is randomly distributed in space and is attracted to the line{(i, 0)_}0≤i≤N through a pinning interaction, see Figure 1. We will often refer to {ϕ_i}i as the field. We discuss the localization properties of the model as a function of the attraction strength ǫ _≥ 0 and of the characteristics of the chains, that are embodied in two potentialsV1andV2.

{ϕi}i

−1 0 _N

Figure 1: A sample trajectory of the modelP_ǫ,N.

1.1

The model

We first define theHamiltonian, which describes the self-interaction of the fieldϕ=_{ϕi}i:

H[−1,N+1](ϕ) = H[−1,N+1](ϕ−1, ...,ϕN+1) :=

N+_X1

i=1

V1(∇ϕi) + N

X

i=0

V2(∆ϕi), (1) whereN represents the length of the chain. The discrete gradient and Laplacian of the field are defined respectively by_∇ϕi := ϕi−ϕi−1 and∆ϕi := ∇ϕi+1− ∇ϕi =ϕi+1+ϕi−1−2ϕi. The precise assumptions on the potentialsV1andV2are stated below.

Given the strength of the pinning attractionǫ_≥0 between the chain and the defect line, we define our modelP_{ǫ,Nas the following probability measure on}RN−1_:

P_{ǫ,N(dϕ1, . . . , dϕN}

−1) :=

exp(_{−H[−1,N+1]}(ϕ))

Zǫ,N

N−1

Y

i=1

(ǫδ0(dϕi) +dϕi) (2)

where we denote byδ₀(_·)the Dirac mass at zero, by dϕ_i =Leb(dϕ_i)the Lebesgue measure on

R _{and we choose for simplicity zero boundary conditions: ϕ}

−1 = ϕ0 = ϕN = ϕN+1 = 0 (see

Figure 1). The normalization constant_Zǫ,N appearing in (2) plays an important role, as we are going to see in a moment: it is calledpartition functionand is given by

Zǫ,N =

Z

RN−1

e−H[−1,N+1](ϕ) N−1

Y

i=1

(ǫδ₀(dϕ_i) +dϕ_i). (3)

We assume that the potentialsV₁,V₂:R_→R_{appearing in (1) are measurable functions satisfying}

the following conditions:

(C1) V1 is bounded from below (infx∈RV1(x) > −∞), symmetric (V1(x) = V1(−x) for every

x_∈R_{), such that lim}

|x|→∞V1(x) = +∞and

R

Re

−2V1(x)_dx<

∞.

(C2) V2is bounded from below (infx∈RV2(x)>−∞), bounded from above in a neighborhood of

zero (sup_|x|≤γV2(x)<∞for someγ >0) and such that

R

R|x|e

−V2(x)_dx<∞.

We stress that no continuity assumption is made. The symmetry of V1 ensures that there is no

“local drift” for the gradient of the field (remarkably, no such assumption onV2is necessary; see

could be relaxed, allowing them to take the value+_∞ outside some interval(₋M,M), but we stick for simplicity to the above stated assumptions.

The modelP_{ǫ,Nis an example of arandom polymer model, more precisely a (homogeneous)pinning}

model. A lot of attention has been devoted to this class of models in the recent mathematical literature (see[8,7]for two beautiful monographs).

The main question, for models like ours, is whether the pinning rewardǫ_≥0 is strong enough to localize the field at the defect line for largeN. The case when the self-interaction of the field is of purely gradient type, i.e., when V2≡0 in (1), has been studied in depth[1,3,6,2], as well

as the purely Laplacian case whenV1≡0, cf.[4,5]. We now consider the mixed case when both

V16≡0 andV26≡0, which is especially interesting from a physical viewpoint, because of its direct

relevance in modelingsemiflexible polymers, cf.[9]. Intuitively, the gradient interaction penalizes large elongations of the chain while the Laplacian interaction penalizes curvature and bendings.

1.2

Free energy and localization properties

The standard way to capture the localization properties of models like ours is to look at the ex-ponential rate of growth (Laplace asymptotic behavior) asN_{→ ∞}of the partition function_Zǫ,N. More precisely, we define thefree energy F(ǫ)of our model as

F(ǫ):= lim N→∞

1 Nlog

‚

Zǫ,N

Z0,N

Œ

, (4)

where the limit is easily shown to exist by a standard super-additivity argument[8].

The functionǫ_{7→ Zǫ,N} is non-decreasing for fixedN (cf. (3)), henceǫ_7→F(ǫ)is non-decreasing too. Recalling thatF(0) =0, we define thecritical valueǫ_cas

ǫ_c := sup{ǫ_≥0 : F(ǫ) =0} =inf{ǫ_≥0 : F(ǫ)>0} ∈ [0,∞], (5) and we say that our model_{P_ǫ

,N}N∈Nis

• delocalizedifǫ < ǫc;

• localizedifǫ > ǫ_c.

This seemingly mysterious definition of localization and delocalization does correspond to sharply different behaviors of the typical trajectories of our model. More precisely, denoting by ℓ_N :=

#_{1_≤i_≤N₋1 :ϕ_i=0_}the number of contacts between the linear chain and the defect line, it is easily shown by convexity arguments that

• ifǫ < ǫc, for everyδ >0 there existscδ>0 such that

P

ǫ,N(ℓN/N> δ)≤e−cδN, for allN∈N; (6)

• ifǫ > ǫ_c, there existsδ_ǫ>0 andc_ǫ>0 such that

P_ǫ,N(ℓ_N/N< δ_ǫ)_≤e−cǫN_{, for allN∈N. (7)} In words: if the model is delocalized then typicallyℓ_N=o(N), while if the model is localized then typicallyℓ_N≥δ_ǫN withδ_ǫ>0. We refer, e.g., to[4, Appendix A]for the proof of these facts. We

point out that the behavior of the model at the critical point is a much more delicate issue, which is linked to the regularity of the free energy.

Coming back to the critical value, it is quite easy to show thatǫ_c <_∞(it is a by-product of our main result), that is, the localized regime is non-empty. However, it is nota prioriclear whether

ǫ_c > 0, i.e. whether the delocalized regime is non-empty. For instance, in the purely Laplacian case (V_1≡0, cf.[4]), one hasǫ∆

c >0. On the other hand, in the purely gradient case (V2≡0,

cf.[2]) one hasǫ∇

c =0 and the model is said to undergo atrivial phase transition: an arbitrarily small pinning reward is able to localize the linear chain.

The main result of this note is that in the general case of mixed gradient and Laplacian interaction the phase transition is always trivial.

Theorem 1. For any choice of the potentials V1, V2satisfying assumptions(C1)and(C2)one has

ǫc=0, i.e., F(ǫ)>0for everyǫ >0.

Generally speaking, it may be expected that the gradient interaction terms should dominate over the Laplacian ones, at least when V1 andV2 are comparable functions. Therefore, having just

recalled thatǫ∇

c =0, Theorem 1 does not come as a surprise. Nevertheless, our assumptions (C1) and (C2) are very general and allow for strikingly different asymptotic behavior of the poten-tials: for instance, one could chooseV1to grow only logarithmically andV2exponentially fast (or

even more). The fact that the gradient interaction dominates even in such extreme cases is quite remarkable.

Remark 2. Our proof yields actually an explicit lower bound on the free energy, which is however quite poor. This issue is discussed in detail in Remark 9 in section 3 below.

Remark 3. Theorem 1 was first proved in the Ph.D. thesis [1] in the special case when both interaction potentials are quadratic: V1(x) = α₂x2andV2(x) =

β

2x

2_{, for anyα,β >0. We point}

out that, with such a choice for the potentials, the free modelP_0,N is a Gaussian law and several explicit computations are possible.

1.3

Organization of the paper

The rest of the paper is devoted to the proof of Theorem 1, which is organized in two parts:

• in section 2 we give a basic representation of the free model (ǫ = 0) as the bridge of an integrated Markov chain, and we study some asymptotic properties of this hidden Markov chain;

• in section 3 we give an explicit lower bound on the partition function_Zǫ,N which, together with the estimates obtained in section 2, yields the positivity of the free energy F(ǫ)for everyǫ >0, hence the proof of Theorem 1.

Some more technical points are deferred to Appendix A.

1.4

Some recurrent notation and basic results

We setR+ _{= [0,∞),} N_{:={1, 2, 3, . . .}and}N_0:=N_{∪ {0} ={0, 1, 2, . . .}. We denote by} Leb_the

Lebesgue measure onR_.

We denote by Lp(R), for p ∈[1,∞], the Banach space of (equivalence classes of) measurable functions f :R_→ R_{such thatkfkp} <_∞, wherekfkp:= (

R

R|f(x)|

p_dx)1/p _{for p∈[1,∞)and}

Given two measurable functions f,g :R _→ R+_{, their convolution is denoted as usual by} (f ∗

g)(x):=R_Rf(x−y)g(y)dy. We recall that if f ∈L1(R)andg∈L∞(R)then f ∗g is bounded and continuous, cf. Theorem D.4.3 in[11].

2

A Markov chain viewpoint

We are going to construct a Markov chain which will be the basis of our analysis. Consider the linear integral operator f _{7→ K}f defined (for a suitable class of functions f) by

(_Kf)(x):= Z

R

k(x,y)f(y)dy, where k(x,y):=e−V1(y)−V2(y−x)_{. (8)} The idea is to modifyk(x,y)with boundary terms to make_K a probability kernel.

2.1

Integrated Markov chain

By assumption (C1) we have_ke−2V1_k

1<∞. It also follows by assumption (C2) thate−V2∈L1(R),

because we can write

ke−V2k1 =

Z

R

e−V2(x)dx ≤ 2 sup x∈[−1,1]

e−V2(x)+ Z

R_\[−1,1]

|x|e−V2(x)dx < _∞.

Since we also havee−V2_∈L∞(R_{), again by (C2), it follows thate}−V2_∈Lp₍R_{)for allp∈[1,∞], in}

particular_ke−2V2_k

1<∞. We then obtain

Z

R_×R

k(x,y)2dxdy = Z

R

e−2V1(y)

‚Z

R

e−2V2(y−x)_dx

Œ

dy = _ke−2V1_k

1ke−2V2k1 < ∞.

This means that_K is Hilbert-Schmidt, hence a compact operator onL2_{(R). Sincek(x,y)≥0 for}

all x,y_∈R_{, we can then apply an infinite dimensional version of the celebrated Perron-Frobenius}

Theorem. More precisely, Theorem 1 in [13] ensures that the spectral radius λ > 0 of _K is an isolated eigenvalue, with corresponding right and left eigenfunctions v,w_∈ L2(R_)satisfying

w(x)>0 andv(x)>0 for almost every x∈R_:

v(x) = 1

λ

Z

R

k(x,y)v(y)dy, w(x) = 1

λ

Z

R

w(y)k(y,x)dy. (9) These equations give a canonical definition ofv(x)andw(x)(up to a multiplicative constant) for every x ∈R_{. Since k}(x,y)>0 for all x,y ∈R_{, it is then clear that w}(x)>0 and v(x)>0for every x∈R_{. We also point out that the symmetry assumption onV1(cf. (C1)) entails thatw}(x)is a constant multiple ofe−V1(x)v(₋x), cf. Remark 7 below.

We can now define a probability kernelP(x, dy)by setting

P(x, dy) := p(x,y)dy := 1

λ

1

v(x)k(x,y)v(y)dy. (10)

SinceP(x,R) = R_Rp(x,y)dy = 1 for every x ∈ R_{, we can define a Markov chain on}R _with

transition kernel_P(x, dy). More precisely, fora,b_∈R_{let(Ω,A, P}(a,b)_{)be a probability space on}

which is defined a Markov chainY =_{Y_i}i∈N₀ onRsuch that

and we define the correspondingintegrated Markov chain W=_{Wi}i∈N₀ setting

W0 = b, Wn = b+Y1+. . .+Yn. (12) The reason for introducing such processes is that they are closely related to our model, as we show in Proposition 5 below. We first need to compute explicitly the finite dimensional distributions of the processW.

Proposition 4. For every n∈N_{, setting w}

−1:=b−a and w0:=b, we have

P(a,b) (W₁, ...,W_n)_∈(dw₁, ..., dw_n) = v(wn−wn−1)

λn_v(a) e−H[−1,n]

(w₋₁,...,wn)

n

Y

i=1

dw_i. (13) Proof. SinceY_i = W_i−W_i−1 for all i _≥ 1, the law of (W₁, ...,W_n) is determined by the law of

(Y₁, ...,Y_n). If we set y_i:=w_i−w_i−1fori_≥2 and y₁:=w₁₋b, it then suffices to show that the right hand side of equation (13) is a probability measure under which the variables(y_i)i=1,...,nare distributed like the firstnsteps of a Markov chain starting atawith transition kernelp(x,y). To this purpose, the Hamiltonian can be rewritten as

H[−1,n](w−1, ...,wn) = V1(y1) +V2(y1−a) +

n

X

i=2

V1(yi) +V2(yi−yi−1)

. Therefore, recalling the definitions (8) ofk(x,y)and (10) ofp(x,y), we can write

v(w_n−w_n−1)

λn_v(a) e−H

[−1,n](w−1,...,wn) ₌ v(yn)

λn_v(a)k(a,y1) n

Y

i=2

k(yi−1,yi)

= p(a,y1)

n

Y

i=2

p(yi−1,yi),

which is precisely the density of (Y1, ...,Yn) under P(a,b) with respect to the Lebesgue measure dy1· · ·dyn. Since the map from(wi)i=1,...,nto(yi)i=1,...,nis linear with determinant one, the proof is completed.

Forn_≥2 we denote byϕ(a_n,b)(_·,_·)the density of the random vector(Wn−1,Wn):

ϕ_n(a,b)(w₁,w₂) := P

(a,b) _(W

n−1,Wn)∈(dw1, dw2)

dw1dw2

, forw₁,w_2∈R_{. (14)}

We can now show that our modelP_ǫ,Nin the free case, that is forǫ=0, is nothing but a bridge of the integrated Markov chainW.

Proposition 5. For every N_∈N_{the following relations hold:} P

0,N(.) = P(0,0) (W1, ...,WN−1)∈ ·

W_N=WN+1=0

, (15)

Z0,N = λN+1ϕ (0,0)

N+1(0, 0). (16)

Proof. By Proposition 4, for every measurable subsetA_⊆RN−1_{we can write}

P(0,0) (W1, ...,WN−1)∈A

WN=WN+1=0

= 1

λN+1_ϕ(0,0) N+1(0, 0)

Z

A

e−H[−1,N+1](w−1,...,wN+1) N−1

Y

i=1

dwi,

where we setw₋1=w0=wN=wN+1=0. ChoosingA=RN−1and recalling the definition (3) of

the partition functionZǫ,N, we obtain relation (16). Recalling the definition (2) of our modelPǫ,N forǫ=0, we then see that (17) is nothing but (15).

2.2

Some asymptotic properties

We now discuss some basic properties of the Markov chainY =_{Y_i}i∈N₀, defined in (11). We recall

that the underlying probability measure is denoted by P(a,b)and we havea=Y0. The parameter

bdenotes the starting pointW0of the integrated Markov chainW={Wi}i∈N₀ and is irrelevant for

the study ofY, hence we mainly work under P(a,0)_.

Since p(x,y)> 0 for all x,y ∈R, cf. (10) and (8), the Markov chain Y isϕ-irreduciblewith

ϕ=Leb: this means (cf.[11, §4.2]) that for every measurable subsetA⊆R_withLeb(A)>0 and for everya_∈R_{there existsn∈}N_{, possibly depending onaandA, such that P}(a,0)_(Y

n∈A)>0. In our case we can taken=1, hence the chainY is alsoaperiodic.

Next we observe thatR_Rv(x)w(x)dx ≤ kvk2kwk2<∞, because v,w∈ L2(R)by construction.

Therefore we can define the probability measureπonR_by

π(dx) := 1

c v(x)w(x)dx, where c:=

Z

R

v(x)w(x)dx. (18) The crucial observation is that πis an invariant probability for the transition kernel _P(x, dy): from (10) and (9) we have

Z

x∈R

π(dx)_P(x, dy) = Z

x∈R

v(x)w(x)

c dx

k(x,y)v(y)

λv(x) dy = w(y)v(y)

c dy = π(dy).

(19)

Beingϕ-irreducible and admitting an invariant probability measure, the Markov chainY=_{Y_i}i∈N₀

ispositive recurrent. For completeness, we point out thatY is also Harris recurrent, hence it is a positive Harris chain, cf. [11, §10.1], as we prove in Appendix A (where we also show thatLeb_is

a maximal irreducibility measure forY).

Next we observe that the right eigenfunctionv is bounded and continuous: in fact, spelling out the first relation in (9), we have

v(x) = 1

λ

Z

R

e−V2(y−x)e−V1(y)v(y)dy = 1

λ e

−V2_∗(e−V1_{v)(x). (20)} By construction v∈L2(R)and by assumption (C1) e−V1 ∈ L2(R), hence(e−V1v)_∈L1(R). Since e−V2∈L∞(R)by assumption (C2), it follows by (20) thatv, being the convolution of a function in L∞₍R_{)with a function inL}1₍R_{), is bounded and continuous. In particular, inf}

|x|≤Mv(x)>0 for everyM >0, because v(x)>0 for every x_∈R_{, as we have already remarked (and as it is clear}

from (20)).

Next we prove a suitabledrift conditionon the kernel_P. Consider the function U(x) := |x|e

V1(x)

and note that

(_PU)(x) = Z

R

p(x,y)U(y)dy = 1

λv(x) Z

R

e−V2(y−x)|y|dy

= 1

λv(x) Z

R

e−V2(z)_|z+x_|dz _≤ c0+c1|x|

λv(x) ,

(22)

wherec0:=

R

R|z|e

−V2(z)_dz<∞_andc

1:=

R

Re

−V2(z)_dz<∞_{by our assumption (C2). Then we fix} M∈(0,∞)such that

U(x)₋(_PU)(x) = |x|e

V1(x)

v(x) −

c1|x|+ c0

λv(x) ≥

1+_|x_|

v(x) , for|x|>M.

This is possible becauseV1(x)→ ∞as|x| → ∞, by assumption (C1). Next we observe that

b := sup |x|≤M

(_PU)(x)₋U(x) < _∞,

as it follows from (21) and (22) recalling thatvis bounded and inf_|x|≤Mv(x)>0 for all M>0. Putting together these estimates, we have shown in particular that

(_PU)(x)₋U(x) _{≤ −}1+|x|

v(x) + b1[−M,M](x). (23)

This relation is interesting because it allows to prove the following result. Proposition 6. There exists a constant C_∈(0,_∞)such that for all n_∈N_{we have}

E(0,0) _|Y_n|≤ C, E(0,0)

₁

v(Yn)

≤ C. (24)

Proof. In Appendix A we prove that Y = _{Y_i}i∈N₀ is a T-chain (see Chapter 6 in [11] for the

definition of T-chains). It follows by Theorem 6.0.1 in[11]that for irreducible T-chains every compact set is petite (see §5.5.2 in[11]for the definition of petiteness). We can therefore apply Theorem 14.0.1 in[11]: relation (23) shows that condition (iii) in that theorem is satisfied by the functionU. Since U(x)<_∞for everyx_∈R_{, this implies thatfor every starting point x}

0∈Rand

for every measurable function g:R_→R_{with|g(x)| ≤(const.)(1+|x|)/v(x)we have}

lim n→∞E

(x0,0) _g(Y n)

=

Z

R

g(z)π(dz) < _∞. (25) The relations in (24) are obtained by takingx0=0 andg(x) =|x|org(x) =1/v(x).

As a particular case of (25), we observe that for every measurable subsetA_⊆ R _{and for every}

x_0∈R_{we have}

lim n→∞P

(x0,0)_(Y

n∈A) = π(A) = 1 c

Z

A

v(x)w(x)dx. (26) This is actually a consequence of the classical ergodic theorem for aperiodic Harris recurrent Markov chains, cf. Theorem 113.0.1 in[11].

Remark 7. Although we do not use this fact explicitly, it is interesting to observe that the invariant probabilityπis symmetric. To show this, we setev(x):=e−V1(x)_v(−_x)_{and we note that by the first} relation in (9), with the change of variables y7→ −y, we can write

e

v(x) = 1

λ

Z

R

e−V1(x)k(₋x,y)v(y)dy = 1

λ

Z

R

e−V1(x)k(₋x,₋y)eV1(y)

e

v(y)dy.

Howevere−V1(x)k(₋x,−y)eV1(y)=k(y,x), as it follows by (8) and the symmetry ofV1(recall our

assumption (C1)). Therefore_evsatisfies the same functional equation_ev(x) =1

λ

R

Rev(y)k(y,x)dy

as the right eigenfunction w, cf. the second relation in (9). Since the right eigenfunction is uniquely determined up to constant multiples, there must existC>0 such thatw(x) =C_ev(x)for all x_∈R_{. Recalling (18), we can then write}

π(dx) = 1 ece

−V1(x)

v(x)v(₋x)dx, ec:= c

C, (27)

from which the symmetry ofπis evident. From the symmetry ofπand (25) it follows in particular that E(0,0)(Yn)→0 as n→ ∞, whence the integrated Markov chainW = {Wi}i∈N₀ is somewhat

close to a random walk with zero-mean increments.

We stress that the symmetry ofπfollows just by the symmetry ofV1, with no need of an analogous

requirement onV2. Let us give a more intuitive explanation of this fact. WhenV1is symmetric,

one can easily check from (10) and (8) that the transition densityp(x,y)(or equivalentlyk(x,y)) is invariant under the joint application oftime reversal and space reflection: by this we mean that for alln∈Nandx1, . . . ,xn∈R

p(x1,x2)· · ·p(xn−1,xn)·p(xn,x1) = p(−xn,−xn−1)· · ·p(−x2,−x1)·p(−x1,−xn). (28) Note thatV2plays no role for the validity of (28). The point is that, whenever relation (28) holds,

the invariant measure of the kernel p(x,y)is symmetric. In fact, (28) implies that the function h(x):=p(x,x)/p(₋x,−x), wherex∈R_{is an arbitrary fixed point, satisfies}

h(x)p(x,y) = h(y)p(₋y,₋x), _∀x,y_∈R_{. (29)}

It is then an immediate consequence of (29) thath(₋x) =h(x)for allx_∈R_{and that the measure}

h(x)dx is invariant. (For our model one computes easily h(x) = (const.)e−V1(x)v(x)v(₋x), in accordance with (27).)

2.3

Some bounds on the density

We close this section with some bounds on the behavior of the density ϕ(_n0,0)(x,y)at (x,y) = (0, 0).

Proposition 8. There exist positive constants C₁,C₂such that for all odd N_∈N

C₁ N ≤ ϕ

(0,0)

N (0, 0) ≤ C2. (30)

The restriction to odd values ofN is just for technical convenience. We point out that neither of the bounds in (30) is sharp, as the conjectured behavior (in analogy with the pure gradient case, cf.[3]) isϕ(_N0,0)(0, 0)_∼(const.)N−1/2.

Proof of Proposition 8. We start with the lower bound. By Proposition 5 and equation (3), we have

ϕ₂(0,0_N+)₁(0, 0) = 1

λ2N+1Z0,2N = 1

λ2N+1

Z

R2N−1

e−P2i=N1+1V1(∇ϕi)−

P2N i=0V2(∆ϕi)

2_YN−1

i=1

dϕ_i,

where we recall that the boundary conditions areϕ₋₁=ϕ₀=ϕ_2N =ϕ_2N+1=0. To get a lower bound, we restrict the integration on the set

C_N1 := §

(ϕ₁, . . . ,ϕ_2N−1)_∈R2N−1_{: |ϕ}

N−ϕN−1|<

γ

2 , |ϕN−ϕN+1|<

γ

2

ª

,

whereγ >0 is the same as in assumption (C2). OnC_N1 we have_|∇ϕ_N+1|< γ/2 and_|∆ϕ_N|< γ, therefore V₂(∆ϕ_N) _≤ M_γ := sup_|x|≤γV₂(x) < _∞. Also note that V₁(_∇ϕ_2N+1) = V₁(0) due to the boundary conditions. By the symmetry of V₁ (recall assumption (C1)), setting C2

N(ϕN) :=

{(ϕ₁, . . . ,ϕ_N−1)_∈RN−1_{: |ϕ}

N−ϕN−1|< γ/2}, we can write

ϕ₂(0,0_N+)₁(0, 0)

≥ e

−(Mγ+V1(0)) λ2N+1

Z

C1

N

e−PNi=1V1(∇ϕi)−

PN−1

i=0 V2(∆ϕi)_e−

P2N+1

i=N+1V1(∇ϕi)−

P2N

i=N+1V2(∆ϕi)

2_YN−1

i=1

dϕ_i

= e

−(Mγ+V1(0))

λ2N+1

Z

R

dϕ_N

  Z

C2

N(ϕN)

e−PNi=1V1(∇ϕi)−

PN−1

i=0 V2(∆ϕi)

N−1

Y

i=1

dϕ_i

 

2

.

For a givenc_N >0, we restrict the integration overϕ_N∈[₋c_N,c_N]and we apply Jensen’s inequal-ity, getting

ϕ₂(0,0_N+)₁(0, 0) _≥ e

−(Mγ+V1(0)) λ_·2cN

  1

λN

ZcN

−cN

dϕ_N

Z

C2

N(ϕN)

e−PNi=1V1(∇ϕi)−

PN−1

i=0 V2(∆ϕi)

N−1

Y

i=1

dϕ_i

 

2

≥ e

−(Mγ+V1(0)) λ_·2c_N

v(0)2

kv_k2

∞

  1

λN

Z cN

−cN

dϕ_N

Z

C2

N(ϕN)

v(ϕ_N−ϕ_N−1)

v(0)

·e−PNi=1V1(∇ϕi)−

PN−1

i=0 V2(∆ϕi)

N−1

Y

i=1

dϕ_i

 

2

= e

−(Mγ+V1(0)) λ_·2c_N

v(0)2

kvk2

∞

”

P(0,0)(_|WN| ≤cN , |WN−WN−1| ≤γ/2)

—2

, (31)

where in the last equality we have used Proposition 4. Now we observe that

P(0,0)(_|WN| ≤cN , |YN| ≤γ/2) ≥ 1−P(0,0)(|WN|>cN)−P(0,0)(|YN|> γ/2)

≥ 1− 1

c_N E (0,0)_[

|WN|]−P(0,0)(|YN|> γ/2).

(32)

By (26), asN_{→ ∞}we have P(0,0)₍

|Y_N|> γ/2)_→π(R_\(−γ 2,

γ

2)) =: 1−3η, withη >0, therefore

P(0,0)(_|YN|> γ/2)≤1−2ηforN large enough. On the other hand, by Proposition 6 we have E(0,0)[_|W_N|] _≤

N

X

n=1

If we choosecN:=C N/η, from (31), (32) and (33) we obtain

ϕ(₂0,0_N+)₁(0, 0) _≥ e

−(Mγ+V1(0)) 2λC

v(0)2

kv_k2

∞

η3 1

N =

(const.)

N ,

which is the desired lower bound in (30).

The upper bound is easier. By assumptions (C1) and (C2) bothV₁andV₂are bounded from below, therefore we can replace V₁(_∇ϕ_2N+1), V₁(_∇ϕ_2N), V₂(∆ϕ_2N) and V₂(∆ϕ_2N−1) by the constant

ec:=inf_x∈Rmin{V₁(x),V₂(x)} ∈Rgetting the upper bound:

ϕ(₂0,0_N+)₁(0, 0) = 1

λ2N+1

Z

R2N−1

e−H[−1,2N+1](ϕ)

2_YN−1

i=1

dϕ_i

≤ e

−4ec

λ2N+1

Z

R2N−1

e−H[−1,2N−1](ϕ)

2_YN−1

i=1

dϕ_i.

Recalling Proposition 4 and Proposition 6, we obtain

ϕ(₂0,0_N+)₁(0, 0) _≤ e

−4ec

λ2

Z

R2

v(0)

v(ϕ_2N−1−ϕ_2N−2) P

(0,0)_(W

2N−2∈dϕ2N−2, W2N−1∈dϕ2N−1)

= v(0)

λ2 e− 4ec_E(0,0)

₁

v(Y2N−1)

≤ v_λ(02)e−

4ec_{C = (const.),} which completes the proof of (30).

3

A lower bound on the partition function

We are going to give an explicit lower bound on the partition function in terms of a suitable renewal process. First of all, we rewrite equation (3) as

Zǫ,N= N−_X1

k=0

ǫk X

A⊆{1,...,N−1} |A|=k

Z

e−H[−1,N+1](ϕ)Y m∈A

δ0(dϕm)

Y

n∈Ac

dϕn, (34)

where we setAc:=_{1, . . . ,N−1} \Afor convenience.

3.1

A renewal process lower bound

We restrict the summation over Ain (34) to the class of subsets B

2k consisting of 2k points organized inkconsecutive couples:

B

2k :=

Plainly,B

2k=;fork>(N−1)/2. We then obtain from (34)

Zǫ,N ≥

⌊(N−_X1)/2⌋ k=0

ǫ2k X A∈B_2k

Z

e−H[−1,N+1](ϕ)Y m∈A

δ₀(dϕ_m)Y

n∈Ac

dϕ_n

=

⌊(N_X−1)/2⌋ k=0

ǫ2k X

0=t0<t1<...<tk<tk+1=N+1 ti−ti−1≥2∀i≤k+1

k+1

Y

j=1

e

K(tj−tj−1), (35)

where we have set forn_∈N

e

K(n) :=             

0 ifn=1

e−H[−1,2](0,0,0,0)_=e−2V1(0)−2V2(0)) _ifn=2

Z

Rn−2

e−H[−1,n](w−1,...,wn)_dw

1· · ·dwn−2

withw₋1=0,w0=0,wn−1=0,wn=0

 

 ifn≥3

. (36)

We stress that a factorization of the form (35) is possible because the Hamiltonian_H[−1,N+1](ϕ) consists of two- and three-body terms and we have restricted the sum over subsets inB₂

k, that consist of consecutive couples of zeros. We also note that the conditionti−ti−1≥2 is immaterial,

because by definitionKe(1) =0.

We now give a probabilistic interpretation to the right hand side of (35) in terms of a renewal process. To this purpose, for everyǫ >0 and forn∈N_{we define}

Kǫ(1) := 0 , Kǫ(n) :=

ǫ2

λnKe(n)e−

µǫn _{= ǫ}2_ϕ(0,0)

n (0, 0)e−

µǫn_{, ∀n≥2 .}

where the second equality follows recalling (36), Proposition 4 and the definition (14) of the densityϕ_n. The constantµ_ǫis chosen to makeKǫa probability onN:

X

n∈N

Kǫ(n) = 1 , that is ∞

X

n=2

ϕ(0,0)

n (0, 0)e−

µǫn ₌ 1

ǫ2. (37)

It follows from Proposition 8 that 0 < µ_ǫ < _∞ for every ǫ > 0. We can therefore define a renewal process(_{η_n}n≥0,Pǫ)onN0 with inter-arrival law Kǫ(·). More explicitly, η0 := 0 and

the increments {η_k+1−η_k}k≥0 are independent, identically distributed random variables with marginal law_Pǫ(η_k+1−η_k=n) =Kǫ(n). Coming back to (35), we can write

Zǫ,N ≥

λN+1_e(N+1)µǫ ǫ2

⌊(N_X−1)/2⌋ k=0

X

0=t0<t1<...<tk<tk+1=N+1 k+1

Y

j=1

Kǫ(tj−tj−1)

= λ

N+1_e(N+1)µǫ ǫ2

⌊(N−_X1)/2⌋ k=0

X

0=t0<t1<...<tk<tk+1=N+1

Pǫ η1=t1, . . . ,ηk+1=tk+1

= λ

N+1_e(N+1)µǫ ǫ2

⌊(N_X−1)/2⌋ k=0

Pǫ ηk+1=N+1

= λ

N+1_e(N+1)µǫ

ǫ2 Pǫ N+1∈η

, (38) where in the last equality we look atη=_{ηk}k≥0as a random subset ofN0, so that{N+1∈η}=

S∞

m=1{ηm=N+1}(note thatPǫ(ηk+1=N+1) =0 fork>⌊(N−1)/2⌋).

We have thus obtained a lower bound on the partition functionZǫ,N of our model in terms of the renewal mass function (or Green function) of the renewal process(_{η_n}n≥0},Pǫ).

3.2

Proof of Theorem 1

Recall the free energy from definition 4

F(ǫ) = lim N→∞

1 Nlog

Zǫ,N

Z0,N .

From now on, the limitsN→ ∞will be implicitly taken along the odd numbers. Observe that by Proposition 5 and both bounds in Proposition 8

lim N→∞

1

NlogZ0,N = Nlim→∞ 1 N

(N+1)logλ+logϕ_N+(0,0₁)(0, 0) = logλ. Therefore for everyǫ >0 by (38) we obtain

lim N→∞

1 N log

Zǫ,N

Z0,N ≥

lim sup N→∞

1 Nlog

–

λN+1_eµǫ(N+1)

ǫ2 Pǫ(N+1∈η)

™

−logλ

≥ µ_ǫ +lim sup N→∞

1

NlogPǫ(N+1∈η). (39)

Since_Pǫ(η1=n)>0 for alln∈Nwithn≥2, the renewal process({ηk}k≥0,Pǫ)is aperiodic and by the classical renewal theorem_Pǫ(N+1∈η)→ _m1

ǫ asN→ ∞, where mǫ =

X

n≥2

n Kǫ(n) = ǫ2

X

n≥2

nϕ(_n0,0)(0, 0)e−µǫn _{< ∞.}

by Proposition 8. Therefore from (39) we get F(ǫ)_≥µǫ. As we already mentioned above, we have µǫ >0, hence F(ǫ)>0, for all ǫ >0. This shows that our model exhibit a trivial phase transition.

Remark 9. We have just shown thatF(ǫ)_≥µ_ǫ. Recalling the definition (37) ofµ_ǫ, it is clear that the lower bound in (30) onϕ(_N0,0)(0, 0)yields a corresponding lower bound onµ_ǫ, hence onF(ǫ). Unfortunately, this lower bound is very poor: in fact, by standard Tauberian theorems, from (30) we getµ_ǫ≥exp(₋(const.)/ǫ2_{), which vanishes asǫ↓0 faster than any polynomial. On the other}

hand, the conjectured correct behavior of the free energy, in analogy with the purely gradient case, should beF(ǫ)_∼(const.)ǫ2_.

One could hope to sharpen the lower bound on µǫ by improving the one on ϕN(0,0)(0, 0). This is possible, but only to a certain extent: even the conjectured sharp lower bound ϕ_N(0,0)(0, 0)_≥ (const.)/pN (in analogy with the purely gradient case) would yield onlyµ_ǫ≥(const.)ǫ4_{. This}

discrepancy is a limitation of our lower bound technique: in order to have a genuine renewal structure, the chain is forced to visit the defect line at couples of neighboring points, which are rewardedǫ2_{instead ofǫ. If one could replace 1/ǫ}2_{by 1/ǫin (37), the lower boundϕ}(0,0)

N (0, 0)≥

(const.)/pNwould yieldµǫ≥(const.′)ǫ2, as expected.

A

Some recurrence properties

We have already remarked thatY =_{Yi}i∈N₀ isLeb-irreducible, hence it is alsoπ-irreducible, see

(18), becauseπis absolutely continuous with respect toLeb_{. By Proposition 4.2.2 in}[11], a max-imal irreducibility measure for Y isψ(dx):=P∞_n=0 1

2n+1(πP

R

z∈Rπ(dz)Q(z, dx) for any kernelQ and we use the standard notation P

0_{(z, dx) := δ}

z(dx),

P1_{=P (we recall (10)) and forn≥1}

Pn+1(z, dx) := Z

y∈R

Pn(z, dy)_P(y, dx).

Since the lawπis invariant for the kernel_P, see (19), we haveπ_Pn_{=πfor alln≥0, therefore} the maximal irreducibility measureψis nothing butπitself. Since a maximal irreducibility mea-sure is only defined up to equivalent meamea-sures (in the sense of Radon-Nikodym), it follows that

Leb_{, which is equivalent to}π, is a maximal irreducibility measure.

(As a matter of fact, it is always true that if a ϕ-irreducible Markov chain admits an invariant measureπ, thenπis a maximal irreducibility measure, cf. Theorem 5.2 in[12].)

Next we prove thatY is a T-chain, as it is defined in Chapter 6 of[11]. To this purpose, we first show thatY is a Feller chain, that is, for every bounded and continuous function f :R_→R_the

function(_Pf)(x):=R_RP(x, dy)f(y)is bounded and continuous. We recall that the functionv is continuous, as we have shown in §2.2. We then write

(_Pf)(x) := Z

R

P(x, dy)f(y) = 1

λv(x) Z

R

e−V1(y)−V2(y−x)v(y)f(y)dy

= 1

λv(x) e

−V2

∗(e−V1v f)(x),

from which the continuity of Pf follows, because e−V2 ∈ _L∞(R) and (e−V1_{v f}) ∈ _L1(R) and we recall that the convolution of a function in L∞(R)with a function in L1(R)is bounded and continuous. SinceY is aLeb_{-irreducible Feller chain, it follows from Theorem 6.0.1 (iii) in}[11]

thatY is aLeb-irreducibleT-chain.

Finally, we observe that from the drift condition (23) it follows thatY is a Harris recurrent chain. For this it suffices to apply Theorem 9.1.8 in[11], observing that the functionU defined in (21) iscoercive, i.e. lim_|x|→∞U(x) = +∞, hence it is “unbounded off petite sets” (cf. [11, §8.4.2]) because every compact set is petite for irreducibleT-chains, by Theorem 6.0.1 (ii) in[11].

Acknowledgements

We thank Jean-Dominique Deuschel for fruitful discussions. F.C. gratefully acknowledges the sup-port of the University of Padova under grant CPDA082105/08.

References

[1] M. Borecki,Pinning and Wetting Models for Polymers with(_∇+ ∆)-Interaction, Ph.D. Thesis, TU-Berlin. Available athttp://opus.kobv.de/tuberlin/volltexte/2010/2663/.

[2] E. Bolthausen, T. Funaki and T. Otobe,Concentration under scaling limits for weakly pinned Gaussian random walks, Probab. Theory Relat. Fields143(2009), 441-480. MR2475669

[3] F. Caravenna, G. Giacomin and L. Zambotti,Sharp Asymptotic Behavior for Wetting Models in (1+_{1)-dimension, Elect. J. Probab.11(2006), 345-362. MR2217821}

[4] F. Caravenna and J.-D. Deuschel,Pinning and Wetting Transition for (1+_{1)-dimensional Fields}

with Laplacian Interaction, Ann. Probab.36(2008), 2388-2433. MR2478687

[5] F. Caravenna and J.-D. Deuschel, Scaling limits of (1+_{1)-dimensional pinning models with}

Laplacian interaction, Ann. Probab.37(2009), 903-945. MR2537545

[6] J.-D. Deuschel, G. Giacomin and L. Zambotti,Scaling limits of equilibrium wetting models in (1+1)-dimension, Probab. Theory Related Fields132(2005), 471-500. MR2198199

[7] F. den Hollander,Random Polymers, École d’Été de Probabilités de Saint-Flour XXXVII-2007, Lecture Notes in Mathematics 1974, Springer (2009). MR2504175

[8] G. Giacomin, Random Polymer Models, Imperial College Press, World Scientific (2007). MR2380992

[9] P. Gutjahr, J. Kierfeld and R. Lipowsky,Persistence length of semiflexible polymers and bending rigidity renormalization, Europhys. Lett.76(2006), 994-1000.

[1] Y. Isozaki and N. Yoshida,Weakly Pinned Random Walk on the Wall: Pathwise Descriptions of the Phase Transitions, Stoch. Proc. Appl.96(2001), 261-284. MR1865758

[11] S. Meyn and R.L. Tweedie,Markov chains and stochastic stability, Second Edition, Cambridge University Press (2009). MR2509253

[12] E. Nummelin,General irreducible Markov chains and non-negative operators, Cambridge Uni-versity Press (1984). MR0776608

[13] M. Zerner,Quelques propriétés spectrales des opérateurs positifs, J. Funct. Anal.72(1987), 381-417. MR0886819

Proof of Proposition 8. We start with the lower bound. By Proposition 5 and equation (3), we have ϕ₂(0,0_N+)₁(0, 0) = 1

λ2N+1Z0,2N =

1 λ2N+1

Z R2N−1

e−P2i=N1+1V1(∇ϕi)−

P2N i=0V2(∆ϕi)

2_YN−1

i=1 dϕ_i,

where we recall that the boundary conditions areϕ₋₁=ϕ₀=ϕ_2N =ϕ_2N+1=0. To get a lower bound, we restrict the integration on the set

C_N1 :=

§

(ϕ₁, . . . ,ϕ_2N−1)_∈R2N−1_{: |ϕ}

N−ϕN−1|<

γ

2 , |ϕN−ϕN+1|< γ 2

ª

,

whereγ >0 is the same as in assumption (C2). OnC_N1 we have_|∇ϕ_N+1|< γ/2 and_|∆ϕ_N|< γ, therefore V₂(∆ϕ_N) _≤ M_γ := sup_|x|≤γV₂(x) < _∞. Also note that V₁(_∇ϕ_2N+1) = V₁(0) due to the boundary conditions. By the symmetry of V₁ (recall assumption (C1)), setting C2

N(ϕN) :=

{(ϕ₁, . . . ,ϕ_N−1)_∈RN−1_{: |ϕ}

N−ϕN−1|< γ/2}, we can write

ϕ₂(0,0_N+)₁(0, 0) ≥ e

−(Mγ+V1(0))

λ2N+1

Z C1

N

e−PNi=1V1(∇ϕi)−

PN−1

i=0 V2(∆ϕi)_e−

P2N+1

i=N+1V1(∇ϕi)−

P2N

i=N+1V2(∆ϕi)

2_YN−1

i=1 dϕ_i

= e

−(Mγ+V1(0))

λ2N+1

Z R dϕ_N   Z C2

N(ϕN)

e−PNi=1V1(∇ϕi)−

PN−1

i=0 V2(∆ϕi)

N−1

Y i=1 dϕ_i   2 .

For a givenc_N >0, we restrict the integration overϕ_N∈[₋c_N,c_N]and we apply Jensen’s inequal-ity, getting

ϕ₂(0,0_N+)₁(0, 0) _≥ e

−(Mγ+V1(0))

λ_·2cN   1

λN ZcN

−cN

dϕ_N

Z C2

N(ϕN)

e−PNi=1V1(∇ϕi)−

PN−1

i=0 V2(∆ϕi)

N−1

Y i=1 dϕ_i   2 ≥ e

−(Mγ+V1(0))

λ_·2c_N

v(0)2 kv_k2

∞

  1

λN Z cN

−cN

dϕ_N

Z C2

N(ϕN)

v(ϕ_N−ϕ_N−1)

v(0) ·e−PNi=1V1(∇ϕi)−

PN−1

i=0 V2(∆ϕi)

N−1

Y i=1 dϕ_i   2 = e

−(Mγ+V1(0))

λ_·2cN

v(0)2 kvk2

∞

”

P(0,0)(_|WN| ≤cN , |WN−WN−1| ≤γ/2)

—2

, (31)

where in the last equality we have used Proposition 4. Now we observe that

P(0,0)(_|WN| ≤cN , |YN| ≤γ/2) ≥ 1−P(0,0)(|WN|>cN)−P(0,0)(|YN|> γ/2)

≥ 1− 1

c_N E (0,0)_[

|WN|]−P(0,0)(|YN|> γ/2).

(32)

By (26), asN_{→ ∞}we have P(0,0)₍

|Y_N|> γ/2)_→π(R_\(−γ

2, γ

2)) =: 1−3η, withη >0, therefore P(0,0)(_|YN|> γ/2)≤1−2ηforN large enough. On the other hand, by Proposition 6 we have

E(0,0)[_|W_N|] _≤

N X n=1

If we choosecN:=C N/η, from (31), (32) and (33) we obtain

ϕ(₂0,0_N+)₁(0, 0) _≥ e

−(Mγ+V1(0))

2λC

v(0)2 kv_k2

∞

η3 1

N =

(const.)

N ,

which is the desired lower bound in (30).

The upper bound is easier. By assumptions (C1) and (C2) bothV₁andV₂are bounded from below, therefore we can replace V₁(_∇ϕ_2N+1), V₁(_∇ϕ_2N), V₂(∆ϕ_2N) and V₂(∆ϕ_2N−1) by the constant

ec:=inf_x∈Rmin{V₁(x),V₂(x)} ∈Rgetting the upper bound:

ϕ(₂0,0_N+)₁(0, 0) = 1 λ2N+1

Z R2N−1

e−H[−1,2N+1](ϕ) 2_YN−1

i=1 dϕ_i

≤ e

−4ec

λ2N+1

Z R2N−1

e−H[−1,2N−1](ϕ) 2_YN−1

i=1 dϕ_i. Recalling Proposition 4 and Proposition 6, we obtain

ϕ(₂0,0_N+)₁(0, 0) _≤ e

−4ec

λ2

Z R2

v(0)

v(ϕ_2N−1−ϕ_2N−2) P

(0,0)_(W

2N−2∈dϕ2N−2, W2N−1∈dϕ2N−1) = v(0)

λ2 e− 4ec_E(0,0)

₁

v(Y2N−1)

≤ v_λ(0)2 e−

4ec_{C = (const.),}

which completes the proof of (30).

3

A lower bound on the partition function

We are going to give an explicit lower bound on the partition function in terms of a suitable renewal process. First of all, we rewrite equation (3) as

Zǫ,N= N_X−1

k=0

ǫk X

A⊆{1,...,N−1} |A|=k

Z

e−H[−1,N+1](ϕ)Y

m∈A

δ0(dϕm) Y n∈Ac

dϕn, (34)

where we setAc:=_{1, . . . ,N−1} \Afor convenience.

3.1

A renewal process lower bound

We restrict the summation over Ain (34) to the class of subsets B

2k consisting of 2k points

organized inkconsecutive couples:

B

2k :=

Plainly,B

2k=;fork>(N−1)/2. We then obtain from (34)

Zǫ,N ≥

⌊(N_X−1)/2⌋

k=0

ǫ2k X A∈B_2k

Z

e−H[−1,N+1](ϕ)Y

m∈A

δ₀(dϕ_m)Y

n∈Ac

dϕ_n

=

⌊(N_X−1)/2⌋

k=0

ǫ2k X

0=t0<t1<...<tk<tk+1=N+1

ti−ti−1≥2∀i≤k+1

k+1

Y j=1

e

K(tj−tj−1), (35)

where we have set forn_∈N

e K(n) :=

            

0 ifn=1

e−H[−1,2](0,0,0,0)_=e−2V1(0)−2V2(0)) _ifn=2

Z Rn−2

e−H[−1,n](w−1,...,wn)_dw

1· · ·dwn−2 withw₋1=0,w0=0,wn−1=0,wn=0

 

 ifn≥3

. (36)

We stress that a factorization of the form (35) is possible because the Hamiltonian_H[−1,N+1](ϕ)

consists of two- and three-body terms and we have restricted the sum over subsets inB₂

k, that

consist of consecutive couples of zeros. We also note that the conditionti−ti−1≥2 is immaterial, because by definitionKe(1) =0.

We now give a probabilistic interpretation to the right hand side of (35) in terms of a renewal process. To this purpose, for everyǫ >0 and forn∈N_{we define}

Kǫ(1) := 0 , Kǫ(n) := ǫ2

λnKe(n)e−

µǫn _{= ǫ}2_ϕ(0,0)

n (0, 0)e−

µǫn_{, ∀n≥2 .}

where the second equality follows recalling (36), Proposition 4 and the definition (14) of the densityϕ_n. The constantµ_ǫis chosen to makeKǫa probability onN:

X n∈N

Kǫ(n) = 1 , that is

∞

X n=2

ϕ(0,0)

n (0, 0)e−

µǫn ₌ 1

ǫ2. (37)

It follows from Proposition 8 that 0 < µ_ǫ < _∞ for every ǫ > 0. We can therefore define a renewal process(_{η_n}n≥0,Pǫ)onN_{0 with inter-arrival law Kǫ}(_·). More explicitly, η₀ := 0 and the increments {η_k+1−η_k}k≥0 are independent, identically distributed random variables with marginal law_Pǫ(η_k+1−η_k=n) =Kǫ(n). Coming back to (35), we can write

Zǫ,N ≥

λN+1_e(N+1)µǫ ǫ2

⌊(N_X−1)/2⌋

k=0

X

0=t0<t1<...<tk<tk+1=N+1

k+1

Y j=1

Kǫ(tj−tj−1)

= λ

N+1_e(N+1)µǫ ǫ2

⌊(N_X−1)/2⌋

k=0

X

0=t0<t1<...<tk<tk+1=N+1

Pǫ η₁=t₁, . . . ,η_k+1=t_k+1

= λ

N+1_e(N+1)µǫ ǫ2

⌊(N_X−1)/2⌋

k=0

Pǫ ηk+1=N+1

= λ

N+1_e(N+1)µǫ

ǫ2 Pǫ N+1∈η

, (38) where in the last equality we look atη=_{ηk}k≥0as a random subset ofN0, so that{N+1∈η}=

S∞

m=1{ηm=N+1}(note thatPǫ(ηk+1=N+1) =0 fork>⌊(N−1)/2⌋).

We have thus obtained a lower bound on the partition functionZǫ,N of our model in terms of the

3.2

Proof of Theorem 1

Recall the free energy from definition 4

F(ǫ) = lim

N→∞

1

Nlog

Zǫ,N

Z0,N

.

From now on, the limitsN→ ∞will be implicitly taken along the odd numbers. Observe that by Proposition 5 and both bounds in Proposition 8

lim

N→∞

1

NlogZ0,N = Nlim→∞

1

N

(N+1)logλ+logϕ_N+(0,0₁)(0, 0) = logλ. Therefore for everyǫ >0 by (38) we obtain

lim

N→∞

1

N log

Zǫ,N

Z0,N ≥

lim sup

N→∞

1

Nlog –

λN+1_eµǫ(N+1)

ǫ2 Pǫ(N+1∈η)

™

−logλ ≥ µ_ǫ +lim sup

N→∞

1

NlogPǫ(N+1∈η). (39)

Since_Pǫ(η1=n)>0 for alln∈Nwithn≥2, the renewal process({ηk}k≥0,Pǫ)is aperiodic and by the classical renewal theorem_Pǫ(N+1_∈η)_→ 1

mǫ asN→ ∞, where

mǫ =

X n≥2

n Kǫ(n) = ǫ2

X n≥2

nϕ(_n0,0)(0, 0)e−µǫn < _∞.

by Proposition 8. Therefore from (39) we get F(ǫ)_≥µǫ. As we already mentioned above, we have µǫ >0, hence F(ǫ)>0, for all ǫ >0. This shows that our model exhibit a trivial phase transition.

Remark 9. We have just shown thatF(ǫ)_≥µ_ǫ. Recalling the definition (37) ofµ_ǫ, it is clear that the lower bound in (30) onϕ(_N0,0)(0, 0)yields a corresponding lower bound onµ_ǫ, hence onF(ǫ). Unfortunately, this lower bound is very poor: in fact, by standard Tauberian theorems, from (30) we getµ_ǫ≥exp(₋(const.)/ǫ2_{), which vanishes asǫ↓0 faster than any polynomial. On the other} hand, the conjectured correct behavior of the free energy, in analogy with the purely gradient case, should beF(ǫ)_∼(const.)ǫ2_.

One could hope to sharpen the lower bound on µǫ by improving the one on ϕN(0,0)(0, 0). This

is possible, but only to a certain extent: even the conjectured sharp lower bound ϕ_N(0,0)(0, 0)_≥ (const.)/pN (in analogy with the purely gradient case) would yield onlyµ_ǫ≥(const.)ǫ4_{. This} discrepancy is a limitation of our lower bound technique: in order to have a genuine renewal structure, the chain is forced to visit the defect line at couples of neighboring points, which are rewardedǫ2_{instead ofǫ. If one could replace 1/ǫ}2_{by 1/ǫin (37), the lower boundϕ}(0,0)

N (0, 0)≥

(const.)/pNwould yieldµǫ≥(const.′)ǫ2, as expected.

A

Some recurrence properties

We have already remarked thatY =_{Yi}i∈N₀ isLeb-irreducible, hence it is alsoπ-irreducible, see

(18), becauseπis absolutely continuous with respect toLeb_{. By Proposition 4.2.2 in}[11], a max-imal irreducibility measure for Y isψ(dx):=P∞_n=0 1

2n+1(πP

R

z∈Rπ(dz)Q(z, dx) for any kernelQ and we use the standard notation P

0_{(z, dx) := δ}

z(dx),

P1_{=P (we recall (10)) and forn≥1} Pn+1(z, dx) :=

Z y∈R

Pn(z, dy)_P(y, dx).

Since the lawπis invariant for the kernel_P, see (19), we haveπ_Pn_{=πfor alln≥0, therefore}

the maximal irreducibility measureψis nothing butπitself. Since a maximal irreducibility mea-sure is only defined up to equivalent meamea-sures (in the sense of Radon-Nikodym), it follows that Leb_{, which is equivalent to}π, is a maximal irreducibility measure.

(As a matter of fact, it is always true that if a ϕ-irreducible Markov chain admits an invariant measureπ, thenπis a maximal irreducibility measure, cf. Theorem 5.2 in[12].)

Next we prove thatY is a T-chain, as it is defined in Chapter 6 of[11]. To this purpose, we first show thatY is a Feller chain, that is, for every bounded and continuous function f :R_→R_the

function(_Pf)(x):=R_RP(x, dy)f(y)is bounded and continuous. We recall that the functionv

is continuous, as we have shown in §2.2. We then write (_Pf)(x) :=

Z R

P(x, dy)f(y) = 1 λv(x)

Z R

e−V1(y)−V2(y−x)_v(y)f(y)dy

= 1

λv(x) e

−V2_∗(e−V1_{v f)(x),}

from which the continuity of Pf follows, because e−V2 ∈ _L∞(R) and (e−V1_{v f}) ∈ _L1(R) and we recall that the convolution of a function in L∞(R)with a function in L1(R)is bounded and continuous. SinceY is aLeb_{-irreducible Feller chain, it follows from Theorem 6.0.1 (iii) in}[11]

thatY is aLeb-irreducibleT-chain.

Finally, we observe that from the drift condition (23) it follows thatY is a Harris recurrent chain. For this it suffices to apply Theorem 9.1.8 in[11], observing that the functionU defined in (21) iscoercive, i.e. lim_|x|→∞U(x) = +∞, hence it is “unbounded off petite sets” (cf. [11, §8.4.2])

because every compact set is petite for irreducibleT-chains, by Theorem 6.0.1 (ii) in[11].

Acknowledgements

We thank Jean-Dominique Deuschel for fruitful discussions. F.C. gratefully acknowledges the sup-port of the University of Padova under grant CPDA082105/08.

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