The free energy of the random walk pinning model

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The free energy of the random walk pinning model Makoto Nakashima Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, Japan Received 25 September 2015; received in revised form 16 December 2016; accepted 14 April 2017 Available online xxxx

Abstract We consider the random walk pinning model. This is a random walk on Zd whose law is given as the β β Gibbs measure µ N ,Y , where the polymer measure µ N ,Y is defined by using the collision local time with another simple symmetric random walk Y on Zd up to time N . Then, at least two definitions of the phase transitions are known, described in terms of the partition function and the free energy. In this paper, we will show that the two critical points coincide and give an explicit formula for the free energy in terms of a variational representation. Also, we will prove that if β is smaller than the critical point, then X under β µ N ,Y satisfies the central limit theorem and the invariance principle PY -almost surely. c 2017 Elsevier B.V. All rights reserved. ⃝ MSC 2010: 60K35; 82B26; 82B44 Keywords: Random walk pinning model; Free energy; Delocalization; Localization; Central limit theorem; Invariance principle

We denote by (Ω , F , P) a probability space. We denote by P[X ] the expectation of a random variable X with respect to P. Let N0 = {0, 1, 2, . . .}, N = {1, 2, 3, . ∑ . .}, and Z = d d 1 {0, ±1, ±2, . . .}. For x = (x1 , . . . , x∑ ) ∈ R , |x| stands for the l -norm: |x| = d i=1 |x i |. For d d n = (n 1 , . . . , n d ) ∈ N0 , we set |n| = i=1 n i . 1. Introduction and main results 1.1. Model The random walk pinning model (RWPM) was introduced by Birkner and Sun [6]. It is known that the model is related to the parabolic Anderson model with a single moving catalyst, the E-mail address: [email protected]. http://dx.doi.org/10.1016/j.spa.2017.04.015 c 2017 Elsevier B.V. All rights reserved. 0304-4149/⃝

Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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pinning and copolymer model, and the directed polymers in random environment. We refer the readers to [6, Section 1] and [5, Section 1.2] for more details. The model is described by using two independent simple symmetric random walks X and y Y . We denote by PXx and PY the law of X and Y starting from x and y ∈ Zd , respectively. In particular, we denote by PX = PX0 and PY = PY0 . Also, we define pn (x) = PX (X n = x) for x ∈ Zd and n ∈ N. β For fixed Y , we define the Gibbs measure µ N ,Y of X by [ ( N ) ] ∑ 1 β µ N ,Y (d X ) = β PX exp β 1{X k = Yk } : d X , Z N ,Y k=1 where β ≥ 0 is the inverse temperature and [ ( N )] ∑ β Z N ,Y = PX exp β 1{X k = Yk } k=1

is the quenched partition function. Also, we define the annealed partition function by [ ( N )] [ ] ∑ β PY Z N ,Y = PX,Y exp β 1 {X k = Yk } , k=1

where PX,Y = PX ⊗ PY is the product measure of PX and PY . Let L N (X, Y ) =

N ∑

1{X k = Yk } and L(X, Y ) = lim L N (X, Y ) = N →∞

k=1

∑

1{X n = Yn }

n≥1

be the collision local time up to time N and the collision local time, respectively. The monotone convergence theorem implies that the following limit exists [ ] β β Z Y = lim Z N ,Y = PX exp (β L(X, Y )) N →∞

PY -almost surely and [ ] [ ] [ ] β β PY Z Y = lim PY Z N ,Y = PX,Y exp (β L(X, Y )) . N →∞

We set { } q β β1 (d) = sup β ≥ 0 : Z Y < ∞, PY -a.s. and { [ ] } β β1a (d) = sup β ≥ 0 : PY Z Y < ∞ . q

Then, phase transitions occur at β1 (d) and β1a (d). It is known that when X and Y are simple symmetric random walks on Zd , q

β1 (d) = β1a (d) = 0 0<

β1a (d)

<

q β1 (d)

if d = 1, 2 <∞

if d ≥ 3

[1,2,5,7]. Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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We set the quenched and annealed constrained partition function by [ ] β,pin Z N ,Y = PX exp (β L N (X, Y )) : X N = Y N [ ] [ ] β,pin PY Z N ,Y = PX,Y exp (β L N (X, Y )) : X N = Y N . We should remark that the annealed partition functions are well-known examples of the partition functions of the pinning model with the free and constrained boundary condition. Indeed, the following identifications hold: [ ( )] [ ] (1.1) PX,Y exp (β L N (X, Y )) = PX˜ exp β L N ( X˜ ) , [ ( ) ] [ ] PX,Y exp (β L N (X, Y )) : X N = Y N = PX˜ exp β L N ( X˜ ) : X˜ N = 0 , (1.2) where X˜ N = X N − Y N is a random walk induced from X and Y and L N ( X˜ ) =

N ∑

1{ X˜ k = 0}

k=1

is the local time at the origin up to time N . Then, the existence of the free energy follows from the general theory of the pinning model, that is, the following limit exists: [ ] [ ] 1 1 β,pin β F a (β) = lim log PY Z N ,Y = lim log PY Z N ,Y ≥ 0. (1.3) N →∞ N N →∞ N We call the limit (1.3) the annealed free energy. We refer the readers to [11] for details on F a (β). Also, Birkner and Sun proved the existence of the quenched free energy [6], that is, there exists a non-random constant F q (β) such that 1 1 β,pin β log Z N ,Y = lim log Z N ,Y N →∞ N N [ ] 1 β = lim PY log Z N ,Y ≥ 0, N →∞ N

F q (β) = lim

N →∞

(1.4)

where the convergence are a.s. and in L 1 w.r.t. PY . We call the limit (1.4) the quenched free energy of RWPM. We consider another pair of critical points for RWPM, defined by q

β2 (d) = sup{β ≥ 0 : F q (β) = 0} and β2a (d) = sup{β ≥ 0 : F a (β) = 0}. It is clear that β1s (d) ≤ β2s (d), for s = a, q. Also, Jensen’s inequality implies that F q (β) ≤ F a (β),

(1.5)

and hence we have q

βia (d) ≤ βi (d) for i = 1, 2. Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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Also, we know that β1a (d) = β2a (d) for d ≥ 1 [11] and q

q

β1 (d) = β2 (d) = 0 for d = 1, 2 [6]. In summary, we have that q

q

0 = β1a (d) = β2a (d) = β1 (d) = β2 (d), q q 0 < β1a (d) = β2a (d) < β1 (d) ≤ β2 (d),

d = 1, 2 d ≥ 3.

The following theorem says that equality holds in the last inequality. q

q

Theorem 1.1. For any d ≥ 1, β1 (d) = β2 (d). Remark 1.2. A similar result for random pinning model has been obtained by Mourrat [15, Theorem 2.1]. Remark 1.3. Theorem 1.1 does not give qus any information on partition function at the critical β (d)

point, that is, we do not know whether Z Y 1 < ∞ or = ∞ a.s. for d ≥ 3. On the other hand, the rigorous asymptotics of the annealed partition function at the critical point is well known [11].

For the random walk pinning model, we can give an explicit form of the quenched free energy as follows. Theorem 1.4. For any d ≥ 1, there exists a continuous, convex, and strictly decreasing function s such that ( ( )) q F q (β) = s −1 − log eβ − 1 for β ≥ β1 (d). Moreover, s has a variational representation which is given in Lemma 3.1. By looking at the variational representation of s, we will obtain bounds on the quenched free energy when β is small enough for the case d = 1, 2. Corollary 1.5. (i) (d = 1) There exist β0 (1) > 0 and c1 > 0 such that for any β ∈ (0, β0 (1)], F q (β) ≥ c1 β 2 . (ii) (d = 2) There exist β0 (2) > 0 and c2 > 0 such that for any β ∈ (0, β0 (2)], ( ) c2 F q (β) ≥ exp − . β Theorem 2.1 in [11] implies that there exist c1′ > 0 and c2′ > 0 such that F a (β) ∼ c1′ β 2 , d = 1 and log F a (β) ∼ −c2′ β −1 , d = 2 when β ↘ 0. Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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Thus, we find from the trivial bound F q (β) ≤ F a (β) and these facts that when β ↘ 0, F q (β) ≍ β 2 d = 1 log F q (β) ≍ −β −1 d = 2. q

As in the pinning model [11], β1 (d) marks the transition from the delocalized to the localized q β phase, that is, if β < β1 (d), then µ N ,Y [L N (X, Y )] = o(N ) (the delocalized regime) and if q β β > β1 (d), then µ N ,Y [L N (X, Y )] ≥ cN for some c > 0 (the localization regime). We can get a stronger result from Theorem 1.1. Corollary 1.6. q

(i) (Delocalization) If β < β1 (d), then β

lim sup µ N ,Y [L N (X, Y )] < ∞,

PY -a.s.

N →∞

q

(ii) (Localization) If β > β1 (d), then there exists a non-random constant Cβ > 0 such that ] [ 1 β L N (X, Y ) ≥ Cβ , PY -a.s. lim inf µ N ,Y N →∞ N β

Furthermore, in the delocalized phase, X under µ N ,Y satisfies the invariance principle as follows: q

Theorem 1.7. Suppose β < β1 (d). Then, we have that √ ( ) β µ N ,Y X ·(N ) ∈ · ⇒ P W (w/ d ∈ ·), PY -a.s. where for 0 ≤ t ≤ 1 ⎧ ⎨ Xk √ , X t(N ) = ⎩ N linear interpolation

k , k = 1, . . . , N N otherwise

t=

and (W, F W , P W ) is the d-dimensional Wiener-space, where we equip W = {w ∈ C([0, 1] → Rd ) : w(0) = 0} with the topology induced from the usual supremum-norm, and F W is the Borel σ -algebra and P W is the Wiener measure. 2. Preliminary 2.1. Word sequences and quenched LDP In this section, we give a minimum introduction to the word sequences and results on their annealed and quenched large deviation principles. ⋃ We refer the readers to [4] for more details. Let E be a finite set of letters. Let E˜ = n≥1 E n be the set of finite words from ( drawn ) N N E. Then, E˜ is a Polish space under the discrete topology. Let P(E ) and P E˜ be the set ˜ equipped with topology of weak of probability measures on sequences drawn from E and E, N N convergence, respectively. We define θ and(θ˜ to)be the left-shift ( ) acting on E and E˜ . Also, we ( ) ( ) denote by P inv E N , P erg E N and P inv E˜ N , P erg E˜ N for the set of probability measures which are invariant and ergodic under θ and θ˜ . Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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Let ρ be a probability measure on N with the algebraic tail property: log ρ(n) lim = −α, α ∈ [1, ∞), (2.1) n→∞ log n and τ = {τi : i ≥ 1} be i.i.d. random variables with P(τ1 = n) = ρ(n). Let ξ = {ξi : i ≥ 1} be E-valued i.i.d. random variables with law ν ∈ P(E) which are independent of {τi : i ≥ 1}. We define T0 = 0, and Ti = Ti−1 + τi , i ≥ 1. Then, we cut a letter sequence ξ at Ti (i ≥ 1), that is, we define word-valued random variables ζ = {ζi : i ≥ 1} by ζi = {ξTi−1 +1,... , ξTi }. ˜ Then, ζ are E-valued i.i.d. random variables with marginal law P (ζ1 ∈ (d x1 , . . . , d xn )) = qρ,ν (d x1 , . . . , d xn ) = ρ(n)ν(d x1 ) · · · ν(d xn ), n ∈ N, x1 , . . . , xn ∈ E. We define the empirical measure of N -tuples of words by RN =

N −1 1 ∑ inv ˜ N δ ˜i ( E ), per ∈ P N i=0 θ (ζ1 ,...,ζ N )

(2.2)

where (ζ1 , . . . , ζ N )per is the periodic extension of (ζ1 , . . . , ζ N ) to an element of E˜ N . Then, R N satisfies the annealed and quenched large deviation principles. To describe the rate function, we ⊗N introduce the specific relative entropy of Q with respect to qρ,ν : ⏐ ⊗N ⏐ ⊗N ( ⏐ ⊗N ) ) ) 1 ( 1 ( = lim H Q ⏐qρ,ν h Q|F N ⏐qρ,ν |F N = sup h Q|F N ⏐qρ,ν |F N ∈ [0, ∞], (2.3) N →∞ N N ≥1 N where F N = σ (ζ1 , . . . , ζ N ) is the sigma-algebra generated by the first N words, Q|F N is the restriction of Q to F N , and h(·|·) is the relative entropy. In particular, it is known that the limit is non-decreasing in N . Theorem 2.1 (Annealed LDP [4]). The family of probability distributions P(R N ∈ ·) satisfies the LDP on P inv ( E˜ N ) with rate N and with rate function I ann : P inv ( E˜ N ) → [0, ∞] given by ( ⏐ N ) ( ⏐ ⊗N ) I ann Q ⏐qρ,ν = I ann (Q) = H Q ⏐qρ,ν . The rate function I ann is lower semi-continuous, has compact level sets, has a unique zero at ⊗N Q = qρ,ν , and is affine. ⊗N Remark 2.2. In this paper, we often specify the underlying distribution, e.g. qρ,ν , in the rate function because we will consider the large deviation principle with respect to several laws.

Also, we can formulate the quenched analogue of Theorem 2.1 as follows. Theorem 2.3 (Quenched LDP [4]). Assume (2.1). Then, for ν ⊗N -a.s. ξ , the family of the (regular) conditional probability distributions P(R N ∈ · |ξ ), N ∈ N satisfies the LDP on ( ) P inv E˜ N with rate N and with a certain deterministic rate function I que : P inv ( E˜ N ) → [0, ∞]. Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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Moreover, the rate function I que is lower semi-continuous, has compact level sets, has a unique ⊗N zero at Q = qρ,ν , and is affine. We refer the readers to [4] for the details of the rate function I que . Also, if ρ has the finite support or the exponential tail property: There exist 0 < C < ∞ and λ > 0 such that ρ(n) < Ce−λn ,

(2.4)

then we have the LDP result for word sequences. Theorem 2.4 (Quenched LDP [3]). Assume (2.4). Then, for ν ⊗N -a.s. ξ , the family of the (regular) conditional probability distributions P(R N ∈ · |ξ ), N ∈ N satisfies the LDP on ( ) P inv E˜ N with rate N and with the deterministic rate function I˜que : P inv ( E˜ N ) → [0, ∞] given by { ( ⏐ ⊗N ) ( ⏐ ⊗N ) H Q ⏐qρ,ν , Q ∈ Rν I˜que Q ⏐qρ,ν = I˜que (Q) = (2.5) ∞, Q ̸∈ Rν , where { Rν =

(

Q ∈ P inv E˜ N

)

} L−1 1∑ δθ k κ(ζ ) = ν ⊗N , Q-a.s. , : w- lim L→∞ L k=0

where κ : E˜ N → E N is the concatenation that glues a sequence of words to a sequence of letters. The rate function I˜que is lower semi-continuous, has compact level sets, has a unique zero at ⊗N Q = qρ,ν , and is affine. To give an identification of the elements in Rν , we introduce some notations. ( ) N inv ˜ E such that m Q = E Q [τ1 ] < ∞, we define Ψ Q (·) ∈ P inv (E N ) by For Q ∈ P [τ −1 ] 1 ∑ 1 ΨQ = Q δθ k κ(ζ ) (·) . mQ k=0

(2.6)

Think of Ψ Q as the shift-invariant version of the concatenation of ζ under the law Q obtained after randomizing the location of the origin. Here, we introduce the tilted version ρr ∈ P(N) of the word length ρ ∈ P(N) which often appears in this paper. Definition 2.5. Given ρ ∈ P(N), we define ρr ∈ P(N) by e−r n ρ(n) N (r ) for r ≥ 0, where ∑ N (r ) = e−r n ρ(n). ρr (n) =

n≥1

It is clear that if ρ satisfies (2.1), then ρr (r > 0) satisfies (2.4). Remark 2.6. Given ρ ∈ P(N) and ν ∈ P(E), we set qρr ,ν (d x1 , . . . , d xn ) = ρr (n)ν(d x1 ) · · · ν(d xn ), n ∈ N, x1 , . . . , xn ∈ E for r ≥ 0, where ρr ∈ P(N) is the tilted version of ρ ∈ P(N) defined in Definition 2.5. Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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˜ and we have that Then, qρr ,ν ∈ P( E) ( ⏐ ⊗N ) ( ⏐ ⊗N ) + log N (r ) + r m Q H Q ⏐qρr ,ν = H Q ⏐qρ,ν for r ≥ 0 and for Q ∈ P inv ( E˜ N ) [8, Lemma 2.1]. When we apply Eq. Theorem 2.4 to qρ⊗N , we r ,ν inv ˜ N have that for Q ∈ Rν ∩ P ( E ) ) ( ⏐ ⊗N ) ( ⏐ + log N (r ) + r m Q , (2.7) = I˜que Q ⏐qρ,ν I˜que Q ⏐qρ⊗N r ,ν ⏐ ( ) ⊗N where I˜que Q ⏐qρ,ν is the rate function defined by (2.5). ( ) We define P inv,fin E˜ N by } ( ) { P inv,fin E˜ N = Q ∈ P inv ( E˜ N ) : m Q < ∞ . Then, Birkner gave the following characterization of Q ∈ Rν ∩ P inv,fin ( E˜ N ) [3]: Ψ Q = ν ⊗N ⇔ Q ∈ Rν ∩ P inv,fin ( E˜ N ).

(2.8)

3. Proof of main theorems 3.1. Proof of Theorems 1.1 and 1.4 [ ] We consider the Laplace transformation of PX exp(β L N (X, Y )) : X N = Y N : Let r ≥ 0 and ∑ [ ] e−r N PX exp(β L N (X, Y )) : X N = Y N ∈ (0, ∞]. K (β, r ) = N ≥1

Then, (1.4) implies that F q (β) > r ⇒ K (β, r ) = ∞ F q (β) < r ⇒ K (β, r ) < ∞,

(3.1)

PY -a.s. Thus, we can obtain information of F q (β) by looking at the value K (β, r ). It is easy to see that ⎞ ⎛ N −1 N −1 ∑ ∏ ( ( ) ) 1{X j = Y j }⎠ = 1 + eβ − 1 1{X j = Y j } exp ⎝β j=1

=

N −1 ∑

(

j=1

)k eβ − 1

∑

1{X j1 = Y j1 , . . . , X jk = Y jk }

(3.2)

1≤ j1 < j2 <···< jk ≤N −1

k=0

and K (β, r ) = eβ

∑

e−r N

N ≥1

=e

β

∑( k≥1

β

N ∑ (

)k−1 eβ − 1

k=1 )k−1

e −1

∑

( ) PX X j1 = Y j1 , . . . , X jk = Y jk

1≤ j1 <···< jk =N

∑

( ) e−r jk PX X j1 = Y j1 , . . . , X jk = Y jk .

1≤ j1 <···< jk <∞

Also, the argument in [5, (3.6)] implies that ( ) β Z Y = 1 + 1 − e−β K (β, 0).

(3.3)

Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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Thus, we find that q

β > β1 (d) ⇒ K (β, 0) = ∞ q β < β1 (d) ⇒ K (β, 0) < ∞,

(3.4)

PY -a.s. Lemma 3.1. There exists the non-random limit ∑ ( ) 1 s(r ) = lim log e−r jk PX X j1 = Y j1 , . . . , X jk = Y jk k→∞ k 1≤ j <···< j <∞ 1

k

PY -a.s. for r > 0. Moreover, we have the following: (i) s(r ) has the variational representation: {∫ s(r ) = log(G r − 1) +

sup

Q∈Rν ∩P inv,fin ( E˜ N )

E˜

} ( ⏐ ⊗N ) que ⏐ ˜ (π1 Q)(dy) log f (y) − I Q qρr ,ν ,

for r > 0. Moreover, we have that s(r ) = log(G r ′ − 1) {∫ +

sup

Q∈Rν ∩P inv,fin ( E˜ N )

( ⏐ )} ⏐ ⊗N que ˜ (π1 Q)(dy) log f (y) − (r − r )m Q − I Q ⏐qρ ′ ,ν ′

E˜

r

(3.5) for 0 < r < r PY -a.s. Also, (3.5) holds for r ≥ r = 0 when d ≥ 3. (ii) s(r ) is continuous and strictly decreasing in r > 0, limr ↘0 s(r ) = ∞ and limr →∞ s(r ) = −∞ when d = 1, 2. (iii) s(r ) is continuous and strictly decreasing in r ≥ 0 and limr →∞ s(r ) = −∞ when d ≥ 3. ′

′

Proof of Theorems 1.1 and 1.4. We give the proof for the case d ≥ 3. It is clear that ( ) log eβ − 1 + s(r ) > 0 ⇒ K (β, r ) = ∞ ( ) log eβ − 1 + s(r ) < 0 ⇒ K (β, r ) < ∞ PY -a.s. Lemma 3.1 and (3.4) imply that ( q ) log eβ1 (d) − 1 + s(0) = 0.

(3.6)

Moreover, since s(r ) is continuous and strictly decreasing in r ≥ 0, s has the continuous inverse function s −1 from (−∞, s(0)] to [0, ∞). q Thus, (3.6) yields that for any β1 (d) < β there exists the unique rc (β) > 0 such that ( β ) log e − 1 + s(rc (β)) = 0. In particular, we get from (3.1) that ( ( )) F q (β) = rc (β) = s −1 − log eβ − 1 q

q

and the continuity of s −1 implies β1 (d) = β2 (d). For d = 1, 2, s has the continuous inverse function s −1 : R → (0, ∞) with limβ→∞ s −1 (β) = 0 and limβ→−∞ s −1 (β) = ∞. □ Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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In the following, we will give a part of the proof of Lemma 3.1(i). The argument is a modification of the ones by Birkner, Greven, and den Hollander [5, Theorem 1.1] and [3, Section 5]. Proof of Lemma 3.1(i). Let {ξn : n ≥ 1} be the increments of {Yn : n ≥ 1}, that is, {ξn : n ≥ 1} are i.i.d. random variables such that ξn = Yn − Yn−1 . ξ is a sequence of letters with E = {±ei : 1 ≤ i ≤ d} and ν(x) = p(x) = ei is the ith vector of the canonical basis on Zd . Let ∑ ( ) Fk (Y, r ) = e−r jk PX X j1 = Y j1 , . . . , X jk = Y jk .

1 2d

if |x| = 1, where

1≤ j1 <···< jk <∞

Then, we can write ∑ Fk (Y, r ) =

( ) e−r jk PX X j1 = Y j1 , . . . , X jk = Y jk

1≤ j1 <···< jk <∞

=

∑

e−r jk p j1 (Y j1 ) p j2 − j1 (Y j2 − Y j1 ) · · · p jk − jk−1 (Y jk − Y jk−1 )

1≤ j1 <···< jk <∞

=

∞ ∑

k ∏

e−r ℓi pℓi (Yℓ1 +···+ℓi − Yℓ1 +···+ℓi−1 )

ℓ1 ,...,ℓk =1 i=1

=

∞ ∑

k ∏

e−r ℓi pℓi (∆i (ℓ, ξ )),

ℓ1 ,...,ℓk =1 i=1

where ∆i (ℓ, ξ ) = ξℓ1 +···+ℓi−1 +1 + · · · + ξℓ1 +···+ℓi for ℓ = (ℓ1 , . . . , ℓk ) ∈ Nk and ξ = {ξn : n ≥ 1}. ∑ 2n (0) , where G = n≥0 p2n (0) for the case d ≥ 3. Then, we can construct We define ρ(n) = pG−1 the tilted version ρr ∈ P(N) and qρr ,ν ∈ ( E˜ N ) as Definition 2.5 and Remark 2.6. In particular, we have that Gr − 1 e−r n p2n (0) and N (r ) = , (3.7) ρr (n) = Gr − 1 G−1 where ∑ Gr = e−r n p2n (0). n≥0

We remark that ρr ∈ P(N) in (3.7) are defined for r > 0 even when d = 1, 2 and qρr ,ν ∈ P( E˜ N ) is also defined. Let {τn(r ) : n ≥ 1} be i.i.d. random variables with P(τk(r ) = n) = ρr (n) which is independent of Y . Let Ti(r ) = τ1(r ) + · · · + τi(r ) . Then, we have that k ( ) p (∆ (ℓ, Y )) ∏ ℓi i P τi(r ) = ℓi p 2ℓ i (0) ℓ1 ,...,ℓk =1 i=1 ⏐ ⎤ ⎡ k p (r ) (∆i (τ (r ) , Y )) ⏐ ∏ ⏐ τi k ⎣ ⏐ ξ⎦ . = (G r − 1) P ⏐ p (0) (r ) ⏐ 2τ i=1 i

Fk (Y, r ) = (G r − 1)k

∞ ∑

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{

( ) } Let ζi(r ) = ξT (r ) +1 , . . . , ξT (r ) : i ≥ 0 be the word-valued random variables induced from i i+1 ⋃ ξ and τ (r ) with E, and we define functions from E˜ = n≥1 E n to R by pn (x1 + · · · + xn ) p2n (0) ˜ for x = (x1 , . . . , xn ) ∈ E. Since E˜ carries the discrete topology, f is continuous and it is clear from the local limit theorem that log f is bounded from above. Thus, we find that log f is upper semicontinuous as a functional on P( E˜ N ). We can rewrite )⏐ ] [ ( k−1 ( ) ⏐ ∑ ⏐ (r ) k log f ζi Fk (Y, r ) = (G r − 1) P exp ⏐ξ ⏐ i=0 [ ( ∫ )⏐ ] ⏐ = (G r − 1)k P exp k log f (y)π1 Rk(r ) (dy) ⏐⏐ ξ , f (x) =

( ) where Rk(r ) ∈ P inv E˜ N is the empirical measure of words defined in (2.2) for ζ (r ) and ˜ is the projection of Rk(r ) onto the first coordinate. π1 Rk(r ) ∈ P( E) Applying Varadhan’s lemma (see [9, Lemma 4.3.6]), we can get 1 lim log Fk (Y, r ) k→∞ k } {∫ ) ( ⏐ ≤ log(G r − 1) + sup (π1 Q)(dy) log f (y) − I˜que Q ⏐qρ⊗N r ,ν E˜ Q∈Rν ∩P inv ( E˜ N ) = log(G r − 1) } {∫ ( ⏐ ⊗N ) que ⏐ ˜ (π1 Q)(dy) log f (y) − log N (r ) − r m Q − I Q qρ,ν + sup E˜ Q∈Rν ∩P inv ( E˜ N ) = log(G − 1) {∫ } ( ⏐ ⊗N ) + sup (π1 Q)(dy) log f (y) − r m Q − I˜que Q ⏐qρ,ν E˜ Q∈Rν ∩P inv,fin ( E˜ N ) PY -a.s. for r > 0, where we have used (2.7) in the first equation. We should prove 1 lim log Fk (Y, r ) k→∞ k } {∫ ( ⏐ ⊗N ) que ⏐ ˜ ≥ log(G r − 1) + sup (π1 Q)(dy) log f (y) − I Q qρr ,ν E˜ Q∈Rν ∩P inv ( E˜ N ) = log(G r − 1) {∫ } ( ⏐ ⊗N ) + sup (π1 Q)(dy) log f (y) − log N (r ) − r m Q − I˜que Q ⏐qρ,ν E˜ Q∈Rν ∩P inv ( E˜ N ) = log(G − 1) {∫ } ( ⏐ ⊗N ) + sup (π1 Q)(dy) log f (y) − r m Q − I˜que Q ⏐qρ,ν . E˜ Q∈Rν ∩P inv,fin ( E˜ N ) Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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The proof is adaptation from the proofs of [3, Proposition 2] and [8, Proposition C.1] but long, so we postpone it to Appendix A. □ Next, we give the proof of Lemma 3.1(ii) and (iii). The strict decreasingness of s(r ) follows from Lemma 3.1(i). Proof of strict decreasingness of s(r ). We put {∫ } ( ⏐ ⊗N ) que ⏐ ˜ S(r ) = sup (π1 Q)(dy) log f (y) − r m Q − I Q qρ,ν E˜ Q∈Rν ∩P inv,fin ( E˜ N ) and ∫ Mr (Q) =

E˜

( ⏐ ⊗N ) (π1 Q)(dy) log f (y) − r m Q − I˜que Q ⏐qρ,ν

′ ′ for r ≥ 0 and Q ∈ Rν ∩ P inv,fin ( E˜ N ) for d ≥ 3. Then, ( )it is clear S(r ) ≥ S(r ) if 0 ≤ r ≤ r . inv,fin N ˜ For any ε > 0, there exists Q r ∈ Rν ∩ P E such that

S(r ) < Mr (Q r ) + ε.

(3.8)

Since Mr (Q r ) = Mr ′ (Q r ) + (r ′ − r )m Qr , we have S(r ) < Mr ′ (Q r ) + ε + (r ′ − r )m Qr ≤ S(r ′ ) + ε + (r ′ − r )m Qr < S(r ′ ) if r − r ′ > m εQ . However, since m Qr ≥ 1, we can choose ε > 0 such that r − r ′ > m εQ for any r r 0 ≤ r ′ < r . Therefore, S(r ) is strictly decreasing for r ≥ 0. For the case d = 1, 2, the strict decreasingness follows by applying the same argument as above to (3.5) for 0 < r ′ < r . □ Proof of the continuity of s(r ). We will prove the convexity of s(r ) in r > 0. We can write [ ( )] k ∑ Fk (Y, r ) = µY,k exp −r Ni , i=1

where µY,k is a measure on N defined by k

µY,k ((N1 , . . . , Nk ) = (ℓ1 , . . . , ℓk )) =

k ∏

pℓi (Yℓ1 +···+ℓi − Yℓ1 +···+ℓi−1 ).

i=1

∫ { ∫ It is easy to check that } log R exp (r x) µ(d x) is convex at r in the interior of t ∈ R : R exp (t x) µ(d x) < ∞ for any general ∫ measure µ on R. Indeed, the second derivative of log R exp (r x) µ(d x) is given by ∫ 2 (∫ )2 [ ] x exp (r x) µ(d x) R R x exp (r x) µ(d x) ∫ ∫ − = µr X 2 − µr [X ]2 ≥ 0, exp x) µ(d x) exp x) µ(d x) (r (r R R where X is an R-valued random variable with law µr which is defined by ∫ exp (r x) µ(d x) µr (A) = ∫ A exp (r x) µ(d x) R for A ∈ B (R). Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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Thus, we find that s(r ) is convex in r ∈ (0, ∞). Also, since s(r ) < ∞ for r ∈ (0, ∞), s is continuous in r ∈ (0, ∞). Since it is clear that s(r ) ≤ log(G r − 1) + sup log f (y), r > 0, y∈ E˜

we find that limr →∞ s(r ) = −∞. In the rest of the proof, we will prove s(r ) is right continuous at r = 0. Suppose d ≥ 3. Then, we have for 0 < r 0 < S(0) − S(r ) < M0 (Q 0 ) + ε − S(r ) ≤ M0 (Q 0 ) − Mr (Q 0 ) + ε = r m Q 0 + ε, ( ) where Q 0 ∈ Rν ∩ P inv,fin E˜ N satisfies (3.8) for ε > 0. Since m Q 0 < ∞, we can choose r > 0 such that 0 < S(0) − S(r ) < 2ε i.e. S(r ) and s(r ) are right continuous. We will prove that limr ↘0 s(r ) = ∞ for d = 1, 2 in the proof of Corollary 1.5. □ Thus, we have obtained the variational representation of the quenched free energy F q (β). Remark 3.2. By modifying the proof of Theorem 1.4, we can get another representation of the annealed free energy F a (β). By (1.1), we have that ∑ [ ] K˜ (β, r ) = e−r N PX,Y exp (β L N (X, Y )) : X N = Y N N ≥1

=

∑(

=

∑(

)k eβ − 1

∑

e−r jk p2 j1 (0) . . . p2( jk − jk−1 ) (0)

1≤ j1 <···< jk <∞

k≥1 β

)k e − 1 (G r − 1)k .

k≥1

By (1.3), we have that the annealed free energy is the unique solution to ( ) ( ) log eβ − 1 + log G F a (β) − 1 = 0. Also, we have that {∫ log(G r − 1) = log(G r − 1) + sup ˜ q∈P( E)

E˜

} q(dy) log f (y) − h(q|qρr ,ν ) ,

˜ is defined by qr (n, x) = where the supremum is attained at q = qr , where qr ∈ P((E) )n e−r n pn (x1 +···+xn ) p2n (0) ν(x1 ) · · · ν(xn ) for n ∈ N, x = (x1 , . . . , xn ) ∈ Zd . G r −1 3.2. Proof of Corollary 1.5 Proof of Corollary 1.5. We have from Lemma 3.1 that {∫ } ( ⏐ ) s(r ) = log (G r − 1) + sup π1 Q(dy) log f (y) − I˜que Q ⏐qρ⊗N r ,ν Q∈Rν ∩P inv,fin ( E˜ N )

E˜

∫ ≥ log(G r − 1) +

E˜

qρr ,ν (dy) log f (y),

where we set Q = qρ⊗N in the second line. r ,ν Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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Thus, we have that ∑

s(r ) ≥ log(G r − 1) +

∑

n≥1 x=(x1 ,...,xn )∈E n

∑ e−r n p2n (0)

= log(G r − 1) +

n≥1

Gr − 1

e−r n p2n (0) p(x1 ) · · · p(xn ) pn (x1 + · · · + xn ) log Gr − 1 p2n (0)

(−h( pn ) − log p2n (0)),

where h( pn ) is an entropy of a probability measure pn on Zd . Since information theory gives us an upper bound (e) d lim sup (h( pn ) + log p2n (0)) ≤ log 2 2 n→∞ (see Lemma B.1), we have that s(r ) ≥ log (G r − 1) − C for some constant C > 0. The local limit theorem [14, Theorem 1.2.1] and the Tauberian theorem [14, Theorem 2.4.2] imply that ⎧ 1 ⎨r − 2 , d=1 Gr − 1 ∼ 1 1 ⎩ log , d=2 π 1 − e−r as r → 0. When we see ( ) ( ) − log eβ − 1 ≥ log G F q (β) − 1 − C, we obtain the corollary. □ 3.3. Proof of Corollary 1.6 The proof of Corollary 1.6 is simple. It is easy to see that d β β log Z N ,Y = µ N ,Y [L N (X, Y )], dβ PY -a.s. and we denote by VN ,Y (β) the term on the right hand side. β Also, log Z N ,Y is convex in β by the same argument in the proof of the continuity of s(r ) and hence, we obtain that for β ∈ R and ε > 0 β

β−ε

log Z N ,Y − log Z N ,Y

β+ε

β

log Z N ,Y − log Z N ,Y

. ε ε q q For β < β1 (d), we take ε > 0 such that β + ε < β1 (d). Then, we have β

lim

VN ,Y

lim

VN ,Y

≤ VN ,Y (β) ≤

≤

F q (β + ε) − F q (β) =0 ε

≥

F q (β) − F q (β − ε) , ε

N PY -a.s. q For β > β1 (d), we have that for any ε > 0 N →∞

β

N →∞

N

Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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q

PY -a.s. In particular, since F q (β) is strictly nondecreasing in β > β1 (d), the right hand side is strictly positive. 3.4. Proof of Theorem 1.7 In this subsection, we give the proof of Theorem 1.7. Before the proof of Theorem 1.7, we will prove the central limit theorem: q

Lemma 3.3. Suppose β < β1 (d). Then, we have that for any f ∈ Cb (Rd ) )] ∫ [ ( XN β = f (x)µ(d x), PY -a.s., lim µ N ,Y f √ N →∞ N Rd where Cb (Rd ) is the set of bounded continuous functions on Rd and µ(d x) is the normal distribution with mean zero and covariance matrix d1 I . m

m

We use the standard notation x m = x1 1 · · · xd d for m = (m 1 , . . . , m d ) ∈ Nd0 and x = (x1 , . . . , xd ) ∈ Rd . We will prove that [( ) ] ∫ XN m β lim µ N ,Y = x m µ(d x) = Mm , PY -a.s., (3.9) √ N →∞ N Rd where µ is a Gaussian measure on Rd with mean zero and covariance matrix d1 I . Then, we can complete the proof of Theorem 1.7 since the moment problem of a higher dimensional Gaussian measure is solvable [10]. The following lemma is obtained in a similar way to Lemma 2.3 in [12]. Lemma ( )n3.4. Let Φ0 be a constant and Φn = ϕn (X 1 , . . . , X n ) for n ≥ 1, where ϕn is a function on Zd . Then, for N ≥ 1 N ∑ [ ] [ ] PX Φ N exp (β L N (X, Y )) = Φ0 + PX ∆Φk exp (β L k−1 (X, Y )) k=1

+ (eβ − 1)

N ∑

[ ] PX Φk exp (β L k−1 (X, Y )) : X k = Yk ,

k=1

(3.10) where we set ∆an = an − an−1 for any sequence {an }n≥1 . Proof. We set en,X,Y = exp (β L n (X, Y )) . Then, we have that ( ) ∆ Φn en,X,Y = en−1,X,Y ∆Φn + ∆en,X,Y Φn ( ) = en−1,X,Y ∆Φn + eβ − 1 1{X n = Yn }Φn en−1,X,Y and (3.10) follows.

□

Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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When we take Φn = X nm in Lemma 3.4, we have that ) [ N ∑ d m i /2 ( ] ∑ [ m ] 1∑ mi m−2j PX X k−1 i ek−1,X,Y PX X N e N ,X,Y = d k=1 i=1 j=1 2 j N ( )∑ [ ] + eβ − 1 PX X km ek−1,X,Y : X k = Yk ,

(3.11)

k=1

where 2ji ∈ Nd0 has 2 j at the ith coordinate and 0 at the other coordinates since X is a simple symmetric random walk on Zd . q

Proposition 3.5. Suppose d ≥ 3 and β < β1 (d). Then, we have that lim

N →∞

1 N |m|/2

N ∑

[ ] PX X km ek−1,X,Y : X k = Yk = 0,

PY -a.s.

k=1

for any m ∈ Nd0 . Proof of (3.9). It is enough to show that ) ] [( XN m β e N ,X,Y = Mm Z Y , PY -a.s. lim PX √ N →∞ N for any m ∈ Nd0 . We have that for |m| = 0, [ ] 1 β PX X m N e N ,X,Y = Z Y , |m|/2 N PY -a.s. and (3.11) and Proposition 3.5 imply that for |m| = 1, [ ] 1 PX X m PY -a.s. N e N ,X,Y = 0, |m|/2 N When we assume that (3.9) holds for m ∈ Nd0 with |m| ≤ k, it follows that [ ] 1 m−2ji β lim X e = Mm−2ji Z Y , PY -a.s. P X ℓ−1,X,Y ℓ−1 ℓ→∞ (ℓ − 1)|m−2ji |/2 Then, we have from (3.11) and Proposition 3.5 that for m ∈ Nd0 with |m| = k + 1 [( ) ) ] d m i /2 ( N [ ] XN m 1 ∑ 1 ∑∑ mi m−2j lim PX e N ,X,Y = lim PX X k−1 i ek−1,X,Y √ |m|/2 N →∞ d i=1 j=1 2 j N →∞ N N k=1 ( ) d 1 ∑ mi β = Mm−21i Z Y , PY -a.s., d i=1 2 where we have used that N [ ] { β 1 ∑ Mm−2ji Z Y , m−2ji lim P X e = X k−1,X,Y k−1 N →∞ N |m|/2 0, k=1

j =1 otherwise,

PY -a.s. in the last equality. This recurrence relation is the same as the one for the moment of simple symmetric random walks X . Therefore, we obtain that [( ) ] XN m β lim PX e N ,X,Y = Mm Z Y , PY -a.s. □ √ N →∞ N Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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Proof of Proposition 3.5. The statement follows from Kronecker’s lemma if we prove that ∞ ∑ N =1

1 N |m|/2

⏐ [ m ]⏐ ⏐ PX X e N −1,X,Y : X N = Y N ⏐ < ∞, PY -a.s. N

(3.12)

Since it is obvious that for x = (x1 , . . . , xd ) ∈ Rd and for m ∈ Nd0 (|x1 | + · · · + |xd |)m 1 +···+m d ≥ |x1 |m 1 · · · |xd |m d , we have that ⏐ [ m ]⏐ ⏐ PX X e N −1,X,Y : X N = Y N ⏐ N [ ] ≤ PX |X N ||m| e N −1,X,Y : X N = Y N ⎡ ⎤ N ∑ ∑ ( ) k−1 = PX ⎣ |X N ||m| eβ − 1 : X ji = Y ji , i = 1, . . . , k ⎦ k=1 1≤ j1 <···< jk =N

≤

N ∑

∑

k

|m|−1

(

β

e −1

( k )k−1 ∑

k=1 1≤ j1 <···< jk =N

) |m|

|X ji − X ji−1 |

i=1

( ) × PX X ji = Y ji , i = 1, . . . , k for m ∈ Nd0 , where we have used (3.2) in the equality. The local limit theorem implies that for any α ∈ ( 34 , 1), there exists a constant Cα,m such that ( ) |x| |m| pn (x) ≤ Cα,m pn (x)α (3.13) √ n for x ∈ Zd and n ∈ N. Indeed, we know that )d ( ) ( 2 d|x|2 d d exp − + Cn − 2 −1 pn (x) ≤ 2 2π n 2n for |x| ≤ n [14, Theorem 1.2.1] and hence, we have that ) ( |x| |m| pn (x)1−α ≤ Cα,m,1 √ n for |x| ≤ n lim

n→∞

|m|+(d+2)(1−α) 2|m|

= n γ . Also, since we know from [9, Theorem 3.7.1] that for γ ∈

(1 2

) ,1

1 d log PX (|X n | ≥ n γ ) ≤ − , n 2γ −1 2

(3.13) follows. Thus, k ∑ ( ) |X ji − X ji−1 ||m| PX X ji = Y ji , i = 1, . . . , k |m|/2 N i=1

≤ kCα,m p j1 (Y j1 − Y j0 )α p j2 − j1 (Y j2 − Y j1 )α · · · p jk − jk−1 (Y jk − Y jk−1 )α , PY -a.s. Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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Therefore, we have that ∞ ∑ N =1

1

⏐ [ m ]⏐ ⏐ PX X e N −1,X,Y : X N = Y N ⏐ N

N |m|/2

≤ Cα,m

∞ ∑ N ∑

( )k k |m| eβ − 1

= Cα,m

( )k−1 k |m| eβ − 1

p ji − ji−1 (Y ji − Y ji−1 )α , PY -a.s.

( 34 , ∞) ∑

F˜k (Y, α) =

k ∏

∑

1≤ j1 <···< jk <∞ i=1

k=1

We define for α ∈

p ji − ji−1 (Y ji − Y ji−1 )α

1≤ j1 <···< jk =N i=1

N =1 k=1 ∞ ∑

k ∏

∑

k ∏

p ji − ji−1 (Y ji − Y ji−1 )α .

1≤ j1 <···< jk <∞ i=1

We will prove that for α ∈ ( 34 , ∞) 1 log F˜k (Y, α) k→∞ k exists, is non-random constant PY -a.s., and continuous at α = 1. We remark that s˜ (1) = s(0). It q implies that if β < β1 , then (3.12) holds. ∑ α , where G˜ (α) = n≥0 p2n (0)α . Let ρ˜ (α) (n) = Gp˜2n(α)(0) −1 ( ) { } Let τn(α) : n ≥ 1 be i.i.d. random variables with P τk(α) = n = ρ˜ (α) (n) which are indepens˜ (α) = lim

dent of Y . Let Ti(α) = τ1(α) + · · · + τi(α) . Then, we have that ∞ k ( )k ∑ ( ) ( p (∆ (ℓ, Y )) )α ∏ ℓi i (α) (α) ˜ ˜ Fk (Y, α) = G − 1 P τi = ℓi p 2ℓ i (0) ℓ1 ,...,ℓk =1 i=1 ⎡ ⎛ ( ( (α) )) ⎞α ⏐⏐ ⎤ k ( )k p (α) ∆i τ , Y ∏ ⏐ (α) ˜ ⎠ ⏐ ξ⎦ . ⎣ ⎝ τi = G −1 P ⏐ p2τ (α) (0) ⏐ i=1 i ) } { ( Let ζi(α) = ξT (α) +1 , . . . , ξT (α) : i ≥ 0 be the word-valued random variables induced from i i+1 ξ and τ (α) with E, and the functions from E˜ to R, f (α) be ( ) pn (x1 + · · · + xn ) α (α) f (x) = = f (x)α . p2n (0) Then, we have by the same argument as the proof of Theorem 1.2 in [5] that s˜ (α) = lim

k→∞

1 log F˜k (Y, α) k

= log(G˜ (α) − 1) +

sup

Q∈P inv,fin ( E˜ N )

{ ∫ ( )} ⊗N que α (π1 Q)(dy) log f (y) − I Q|ρ˜ (α) ,ν , E˜

PY -a.s. Thus, it is enough to show that s˜ (α) is continuous at α = 1. To prove it, we will show that log F˜k (Y, α) is convex in α ∈ ( 43 , ∞). Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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We write [

(

F˜k (Y, α) = µ˜ Y,k exp α

k ∑

)] (

log p Ni Y N1 +···+Ni − Y N1 +···+Ni−1

)

,

i=1

where µ˜ Y,k is the counting measure on Nk , that is µ˜ Y,k ((N1 , . . . , Nk ) = (ℓ1 , . . . , ℓk )) = 1, for (ℓ1 , . . . , ℓk ) ∈ Nk . Thus, we can apply the same argument in the proof of the continuity of s(r ) and log F˜k (Y, α) is convex. Hence, s˜ (α) is convex. Also, s˜ (α) < ∞ in α ∈ ( 34 , ∞). Indeed, it is trivial that ( ) s˜ (α) ≤ log G˜ (α) (0) − 1 + α max log f (y) < ∞. y∈ E˜

Hence, s˜ (α) is continuous at α = 1 and we complete the proof. □ Now, we are ready to prove the invariance principle for the random walk pinning model. In a manner similar to the usual proof of invariance principle, the proof of Theorem 1.7 is divided into two parts: Step 1. Convergence of the finite-dimensional distributions. Step 2. Tightness. q

Proof of Step 1. Since for β < β1 (d) β

β

β

lim Z N ,Y = Z Y , and Z Y < ∞, PY -a.s.,

N →∞

it is enough to show that for h 1 , . . . , h k ∈ Cb1 (Rd ) and 0 = t0 < t1 < · · · < tk ≤ 1, [ k ] )] [ ( k ) ∏ ( (N ) ∏ wti −ti−1 β (N ) , lim PX h i X ti − X ti−1 e N ,X,Y = Z Y P W hi √ N →∞ d i=1 i=1 PY -a.s., where Cb1 (Rd ) is the set of the bounded differentiable continuous functions on Rd . Since we have 1 |X t(N ) − X s(N ) | ≤ √ N for |t − s| ≤

1 N

for any sample path of X , it is enough to show that [ k ] [ ( )] k ∏ ( X ⌊N ti ⌋ − X ⌊N ti−1 ⌋ ) ∏ wti −ti−1 β W lim PX hi e N ,X,Y = Z Y P hi , √ √ N →∞ N d i=1 i=1

(3.14)

PY -a.s. When k = 1, we have that ⏐ [ ( ) ] [ ( ) ]⏐ ⏐ ⏐ X ⌊N t1 ⌋ ⌊N t1 ⌋ ⏐ PX h 1 X√ e N ,X,Y − PX h 1 √ e⌊N t1 ⌋,X,Y ⏐⏐ ⏐ N (N ) β β ≤ C(h 1 ) Z N ,Y − Z ⌊N t1 ⌋,Y → 0, N → ∞ Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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PY -a.s., where C(h 1 ) is a non-random constant depending only on h 1 . Hence, Lemma 3.3 implies that [ ( [ ( ) ] )] wt1 X ⌊N t1 ⌋ β W h1 √ lim PX h 1 √ e N ,X,Y = Z Y P N →∞ N d PY -a.s. and (3.14) holds for k = 1. Suppose that (3.14) holds for k ≥ 1. Then, we have that ⏐ [k+1 ( ] ) ⏐ ∏ X ⌊N ti ⌋ − X ⌊N ti−1 ⌋ ⏐ hi e N ,X,Y √ ⏐ PX ⏐ N i=1 ]⏐ [k+1 ( ) ⏐ ∏ X ⌊N ti ⌋ − X ⌊N ti−1 ⌋ ⏐ e⌊N tk ⌋,X,Y ⏐ −PX hi √ ⏐ N i=1 ( ) β β ≤ C(h 1 , . . . , h k+1 ) Z N ,Y − Z ⌊N tk ⌋,Y → 0, N → ∞ PY -a.s., where C(h 1 , . . . , h k+1 ) is a non-random constant depending only on h 1 , . . . , h k+1 . It is easy to see from the Markov property that [k+1 ( ] ) ∏ X ⌊N ti ⌋ − X ⌊N ti−1 ⌋ PX hi e⌊N tk ⌋,X,Y √ N i=1 [ k ] ∑ ∏ ( X ⌊N ti ⌋ − X ⌊N ti−1 ⌋ ) = PX hi e⌊N tk ⌋,X,Y : X ⌊N tk ⌋ = x √ N i=1 x∈Zd [ ( )] X ⌊N tk+1 ⌋−⌊N tk ⌋ − x x × PX h k+1 √ N [ k ] )] [ ( ∏ ( X ⌊N ti ⌋ − X ⌊N ti−1 ⌋ ) X ⌊N tk+1 ⌋−⌊N tk ⌋ = PX hi e⌊N tk ⌋,X,Y PX h k+1 √ √ N N i=1 and the last term converges to [ ( )] k+1 ∏ wti −ti−1 β ZY P W hi , √ d i=1 PY -a.s. by induction and the central limit theorem. Thus, we have completed the proof of Step 1. □ Proof of Step 2. By Kolmogorov’s tightness criterion, it is enough to show that for any ε > 0 [ ] ⏐ ⏐ ⏐ (N ) (N ) ⏐ lim lim PX e N ,X,Y : max ⏐X t − X s ⏐ ≥ ε = 0, δ→0 N →∞

0≤s,t≤1 |t−s|≤δ

PY -a.s. Let m = m(δ) > 2 be a positive integer that satisfies 1 1 <δ≤ . m m−1 Then, we have that lim

N →∞

N +1 1 =
Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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so that we may assume N + 1 < m(⌊N δ⌋ + 1) for N large enough. It is easy to see that } { ⏐ ⏐ ⏐ ⏐ (N ) max ⏐X t − X s(N ) ⏐ ≥ ε 0≤s,t≤1 |t−s|≤δ

⎧ ⎨

⎫ √ ⎬ − Xk| ≥ ε N ⎭

max |X j+k ⎩ 1≤ j≤⌊N δ⌋+1 1≤k≤N } m { ⋃ 1 √ max |X j+ p(⌊N δ⌋+1) − X p(⌊N δ⌋+1) | ≥ ε N ⊂ 1≤ j≤⌊N δ⌋+1 3 p=0

⊂

(see the proof of [13, Lemma 4.19, Theorem 4.20 in Chapter 2]). We define { } 1 √ A p (δ, N ) = max |X j+ p(⌊N δ⌋+1) − X p(⌊N δ⌋+1) | ≥ ε N . 1≤ j≤⌊N δ⌋+1 3 Then, we have by a similar argument to the proof of Step 1 that for p ≥ 1 [ ] PX e N ,X,Y : A p (δ, N ) − PX [e p(⌊N δ⌋+1),X,Y : A p (δ, N )] → 0 PY -a.s. as N → ∞. So we obtain from the Markov property that lim PX [e N ,X,Y : A p (δ, N )] = lim PX [e p(⌊N δ⌋+1),X,Y : A p (δ, N )]

N →∞

=

N →∞ β Z Y lim N →∞

PX (A0 (δ, N )).

(3.15)

Thus, we can prove [ lim lim PX e N ,X,Y

δ→0 N →∞

⏐ ⏐ ⏐ ⏐ : max ⏐X t(N ) − X s(N ) ⏐ ≥ ε

]

0≤s,t≤1 |t−s|≤δ

≤ lim lim PX [e N ,X,Y : A0 (δ, N )] + lim lim δ→0 N →∞

δ→0 N →∞

m ∑

PX [e N ,X,Y : A p (δ, N )] = 0

p=1

PY -a.s., provided lim lim PX [e N ,X,Y : A0 (δ, N )] = 0

(3.16)

δ→0 N →∞

since we have from [13, Lemma 4.18 in Chapter 2] that 1 lim lim PX (A0 (δ, N )) = 0. δ→0 N →∞ δ However, H¨older’s inequality implies that [ ]1 1 PX [e N ,X,Y : A0 ] ≤ PX exp( pβ L N (X, Y )) p PX (A0 ) q where 1 < p, q < ∞ with

q

+ q1 = 1 such that βp < β1 (d) and therefore we can show that ( )1 1 βp p lim PX [R N : A0 ] ≤ Z Y lim PX (A0 (δ, N )) q → 0

N →∞

1 p

N →∞

PY -a.s. as δ → 0. Thus, (3.16) holds and we have completed the proof of Step 2. □ Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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Acknowledgments The author would like to thank Prof. Francis Comets, Prof. Nobuo Yoshida and Ryoki Fukushima for their comments on this research. Also, the author would like to express gratitude to the referees and the associated editors for their comments and advices. This research was supported by JSPS Grant-in-Aid for Young Scientists (B) 26800051. Appendix A. Proof of Lemma 3.1(i) We should prove that 1 (a) lim log Fk (Y, r ) ≥ log(G r − 1) k k→∞ } {∫ ) ( ⏐ (π1 Q)(dy) log f (y) − I˜que Q ⏐qρ⊗N + sup r ,ν E˜ Q∈Rν ∩P inv,fin ( E˜ N ) for r > 0 and (b)

(3.5) holds at r ≥ r ′ = 0 when d ≥ 3.

To start the proof, we prepare a lemma and notation given in [4]. 2 Notation 1. Let E be a discrete set. For (U, V ) and (U˜ , V˜ ), independent E -valued random ˜ we write variables with law P and P,

˜ ) = P( ˜ U˜ = U ) and P(U

˜ |V ) = P( ˜ U˜ = U |V˜ = V ). P(U

˜ ) and P(U ˜ |V ) are R-valued random variables. We remark P(U ⏐ ⏐ ( ) Lemma A.1 ([4, Lemma 3, Lemma 4]). Let ζ = ζ i i∈N have distribution Q: τ i = ⏐ζ i ⏐, ( ) K N = κ ζ 1 , . . . , ζ N . Assume Q ∈ P erg ( E˜ N ) satisfies m Q = Q[τ 1 ] < ∞. Then, we have 1 log Q(K N ) = m Q H (Ψ Q ), N →∞ N 1 lim − log Q(τ 1 , . . . , τ N |K N ) = HτK (Q) N →∞ N exist Q-a.s., and the limit HτK (Q) is a constant. In particular, the specific entropy and specific ⊗N relative entropy of Q with respect to qρ,ν can be represented as lim −

H (Q) = m Q H (Ψ Q ) + HτK (Q) ⏐ ⏐ ⊗N ) ( ) [ ] H Q ⏐qρ,ν = m Q H Ψ Q ⏐ν ⊗N − Q log ρ(τ 1 ) − HτK (Q). (

Also, we define a new probability measure Qˆ on P inv ( E˜ N ) by {( (( ) ) [ )}] 1 Qˆ ζ 1 , . . . , ζ k ∈ Bk = Q τ 1 1 Bk ζ 1 , . . . , ζ k mQ for each Bk ∈ E˜ k for some k ∈ N and Q ∈ P inv,fin ( E˜ N ). Then, Qˆ ≪ Q. ˆ let V be a random variable with uniform distribution on {0, . . . , τˆ1 − For given ζˆ with law Q, 1}. Then, Ψ Q is given as the law of ( ) Z = θ V κ(ζˆ ) (see [3, Section 3]). Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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Let R N (x) =

N −1 1 ∑ inv δi (E N ) per ∈ P N i=0 θ ((x1 ,...,xn ) )

be the corresponding nth empirical letter process measure for x ∈ E n , n ∈ {N + 1, N + 2, . . .} ∪ {∞} and we define N −1 1 ∑ R˜ Nj1 ,..., j N (x) = δ ˜i per N i=0 θ ((x|(0, j1 ] ,x|( j1 , j2 ] ,...,x|( j N −1 , j N ] ) )

for x ∈ E M and 1 ≤ j1 < · · · < j N < M, where we set x|(n,m] = (xn+1 , . . . , xm ) for x ∈ E M and 0 ≤ n < m < M. We remark that R (rN ) = R˜ T (r ) ,...,T (r ) (ξ ). N

1

Proof of (a). It is enough to show that for any Q ∈ Rν ∩ P inv,fin ( E˜ N ), [ ( ∫ )⏐ ] ⏐ 1 (r ) lim log P exp k log f (y)π1 Rk (dy) ⏐⏐ ξ k→∞ k ∫ ) ( ⏐ ≥ (π1 Q)(dy) log f (y) − I˜que Q ⏐qρNr ,ν .

(A.1)

E˜

We first consider a shift ergodic Q. We note that Ψ Q = ν ⊗N . Since we know that w- lim R N (ξ ) = ν ⊗N N →∞

ν

⊗N

-a.s., we have that for ε > 0 ν ⊗N (R N (ξ ) ∈ U for any sufficiently large N ) = 1

where U is an open neighbor set of ν ⊗N given in Lemma A.2 for ε > 0. Hence, we have that for N large enough [ ( ∫ ) { }⏐⏐ ] (r ) (r ) P exp N log f (y)π1 R N (dy) 1 R N ∈ O Q ⏐⏐ ξ ∑

≥

N ∏

(∫ exp

) N ˜ log f (y)π1 R j1 ,..., j N (dy) ρr ( ji − ji−1 )

1≤ j1 <···< j N =⌊m Q N ⌋ i=1 R˜ N j1 ,..., j N (ξ )∈O Q

( ≥ exp N

(∫

⏐ ) log f (y)π1 Q(dy) − H Q ⏐qρ⊗N −ε r ,ν (

))

ν ⊗N -a.s., where O Q is an open neighborhood of Q considered in Lemma A.2 and we proved (A.1) for Q ∈ Rν ∩ P erg,fin ( E˜ N ). Also, we can apply the same strategy in [3, Proposition 2] to the case Q ∈ Rν ∩ P inv,fin ( E˜ N ) and we have completed the proof of (a). □ In the following, we take an open neighborhood O˜ Q of Q ∈ Rν ∩ P erg,fin ( E˜ N ) as the form with O˜ Q = {Q ′ ∈ P inv ( E˜ N ) : |Q(yi ) − Q ′ (yi )| < ε˜ i , i = 1, . . . , K } for some yi ∈ E˜ L i , i = 1, . . . , K and ε˜ i > 0. Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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The following lemma is almost the same as Lemma 9 in [3]. Lemma A.2. Suppose Q ∈ Rν ∩ P erg,fin ( E˜ N ). Let O˜ Q be an open neighborhood of Q. Then, for any ε > 0 small enough, there exist an open neighborhood of ν ⊗N , U and N0 such that N ≥ N0

and

x ∈ E ⌊N m Q ⌋ with R⌊N m Q ⌋ (x) ∈ U

implies ∑

N ∏

(∫ exp

) log f (y)π1 R˜ Nj1 ,..., j N (dy) ρr ( ji − ji−1 )

1≤ j1 <···< j N =⌊m Q N ⌋ i=1 ˜ R˜ N j1 ,..., j N (x)∈O Q

)) ( (∫ ) ( ⏐ − ε . ≥ exp N log f (y)π1 Q(dy) − H Q ⏐qρ⊗N r ,ν Proof. In this proof, we present a way to choose an open neighborhood of ν ⊗N . The construction is almost the same as Step 1 and Step 2 in the proof of Lemma 9 in [3], but we need take care when we deal with log f . Since the argument after the construction of U is the same as Step 3 in the proof of Lemma 9 in [3], we will omit the rigorous proof and give an idea here. We take O˜ Q′ = {Q ′ ∈ P inv ( E˜ N ) : |Q(yi ) − Q ′ (yi )| < ε˜ i /2, i = 1, . . . , K }. Let ζ = (ζ i )i∈N be a random sequence of words with the law Q and we write τ i = |ζ i |. By ergodicity of Q and Lemma A.1, we have Q-a.s. 1 lim |κ(ζ 1 , . . . , ζ N )| = m Q N →∞ N ( ) 1 lim log Q κ(ζ 1 , . . . , ζ N ) = −m Q H (Ψ Q ) N →∞ N 1 lim log Q(ζ 1 , . . . , ζ N ) = −H (Q) N →∞ N N 1 ∑ lim log ρr (τ i ) = Q[log ρr (τ 1 )] N →∞ N i=1 w- lim R N = Q N →∞ ∫ ∫ 1 log f (y)π1 R N (dy) = log f (y)π1 Q(dy) = F(Q), lim N →∞ N E˜ E˜ where the third one follows from Lemma A.1 and the last one from Birkhoff’s ergodic theorem. Indeed, it is easy to see that − n log(2d) ≤ f (y) ≤ C n

(A.2) 1

for y ∈ E , where C is an upper bound of log f (y) and hence, log f ∈ L (π1 Q). Also, they hold ˆ Q-a.s. since Qˆ ≪ Q. Hence, for ε1 > 0, ε2 > 0, there exist a large N and A pairwise different words z a ∈ E˜ and Bk different decompositions of z a , ya,b = (ya,b,1 , . . . , ya,b,N ) ∈ E˜ N (b = 1, . . . , Ba ), for each a = 1, . . . , A, where κ(ya,b,1 , . . . , ya,b,N ) = z a Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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such that |z a | ∈ [N (m Q − ε1 ), N (m Q + ε1 )] (A.3) ) [ −N (m H (Ψ )+ε ) −N (m H (Ψ )−ε ) ] Q Q Q Q 1 1 Q K = za ∈ e ,e (A.4) ) ) [ −N (H (Q)+ε ) −N (H (Q)−ε ) ] (( 1 1 Q ζ 1 , . . . , ζ N = (ya,b,1 , . . . , ya,b,N ) ∈ e (A.5) ,e N ∑ [ ] 1 log ρr (|ya,b,i |) ∈ Q[log ρr (τ 1 )] − ε1 , Q[log ρr (τ 1 )] + ε1 N i=1 (

N

(A.6) 1 N

N ∑

log f (ya,b,i ) ∈ [F(Q) − ε1 , F(Q) + ε1 ]

(A.7)

i=1

N −1 1 ∑ ˜′ δ ˜i per ∈ O Q N i=0 θ ((ya,b,1 ,...,ya,b,N ) )

(A.8)

) ) ζ 1 , . . . , ζ N = (ya,b,1 , . . . , ya,b,N ) ≥ 1 − ε2 ,

(A.9)

(( ) ) ζˆ1 , . . . , ζˆ N = (ya,b,1 , . . . , ya,b,N ) ≥ 1 − ε2 .

(A.10)

and Ba A ∑ ∑ a=1 b=1 Ba A ∑ ∑

Q

Qˆ

((

a=1 b=1

Then, by retaking {z a : a = 1, . . . , A}, we can assume that for each a ( ) Ba ∑ ) ) (( ε2′ Q ζ 1 , . . . , ζ N = ya,b ≥ 1 − Q(K N = z a ) 2 b=1 Ba A ∑ ∑ a=1 b=1 Ba A ∑ ∑

) ) ζ 1 , . . . , ζ N = ya,b ≥ 1 − ε2′

(A.12)

(( ) ) ζˆ1 , . . . , ζˆ N = ya,b ≥ 1 − ε2′

(A.13)

Q

Qˆ

(A.11)

((

a=1 b=1

for any sufficiently small ε2′ > 0. To see this, we set A(ε) to be the subset of {z a : a = 1, . . . , A} such that Ba ∑

Q

((

) ) ζ 1 , . . . , ζ N = ya,b < (1 − ε) Q(K N = z a ).

b=1

Then, we have that Ba A ∑ ∑

Q

((

Ba ∑ ∑ ) ) (( ) ) ζ 1 , . . . , ζ N = ya,b = Q ζ 1 , . . . , ζ N = ya,b a∈A(ε) b=1

a=1 b=1

+

Ba ∑ ∑

Q

((

) ) ζ 1 , . . . , ζ N = ya,b

a∈Ac (ε) b=1

< (1 − ε)

∑ a∈A(ε)

∑ ( ) ( ) Q K N = za + 1 − Q K N = za . a∈A(ε)

Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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√ When we take ε = ε2 and assume that ∑ ( ) Q K N = z a > ε, a∈A(ε)

it contradicts (A.9). Thus, we have ∑ ( ) √ Q K N = z a ≤ ε2 a∈A(ε)

and we can construct {z a′ : a = 1, . . . , A′ } by removing z a , a ∈ A(ε2′ ). We write them {z a : a = 1, . . . , A} again. Now, (A.4), (A.5) and (A.11) imply that ( ) ( ( )) Ba ≥ 1 − ε2′ exp M H (Q) − m Q H (Ψ Q ) − 2ε1 ( ) = (1 − ε2′ ) exp M(HτK (Q) − 2ε1 ) , where we have used Lemma A.1 in the last equation. Thus, when we set for 1 ≤ M ≤ N , x ∈ E N , ε > 0, O˜

Q (x) I M,N ⎧ ,ε ⎫ ˜Q ˜M ⎪ ⎪ 1 ≤ j < · · · < j = N , R (x) ∈ O 1 M ⎪ ⎪ j ,..., j M 1 ⎪ ⎪ ⎪ ⎪ M ⎪ ⎪ ∑ ⎪ ⎪ 1 ⎪ ⎪ ⎨ log ρr ( ji − ji−1 ) ∈ [Q[log ρr (τ1 )] − ε, Q[log ρr (τ1 )] + ε]⎬ , = ( j1 , . . . , j M ) : M i=1 ⎪ ⎪ ⎪ ⎪ M ⎪ ⎪ ⎪ ⎪ 1 ∑ ⎪ ⎪ ⎪ ⎪ log f (ya,b,i ) ∈ [F(Q) − ε, F(Q) + ε] ⎪ ⎪ ⎩ ⎭ M i=1

we have ⏐ ⏐ ⏐ ⏐ O˜ Q′ ( ) ′ K ⏐ ⏐I (z ) ⏐ N ,|za |,ε1 a ⏐ ≥ (1 − ε2 ) exp M(Hτ (Q) − 2ε1 ) for each a = 1, . . . , A. Thus, we can construct “Q-typical letters” with length about N m Q . Next, we construct “Q-typical letters” with length ⌊N m Q ⌋. We shall consider the set of letters { ( ) {˜zr : r = 1, . . . , R} = θ i κ(ya,b,1 , . . . , ya,b,Ba ) : 0 ≤ i < |ya,b,1 |, j = 1, . . . , Ba , a = 1, . . . , A} and truncate them at ⌊N (m Q − 2ε1 )⌋-letters. Then, the definition of Ψ Q mentioned after Lemma A.1 and (A.13) implies that R ∑

( ) ν ⊗N |[1,...,⌊N (m Q −2ε1 )⌋] z˜r |[1,...,⌊N (m Q −2ε1 )⌋] ≥ 1 − ε2′ .

r =1

We generate the set of letters with length ⌊N m Q ⌋, {z ℓ , ℓ = 1, . . . , L}, by considering all possible extension of {˜zr |⌊M(m Q −2ε1 )⌋ : r = 1, . . . , R}. Since we have taken N large enough, we have that for each z ℓ (ℓ = 1, . . . , L) ⏐ ⏐ ⏐ O˜ Q ⏐ ( ) ′ K ⏐I ⏐ (z ) (A.14) ⏐ N ,⌊N m Q ⌋,2ε1 ℓ ⏐ ≥ (1 − ε2 ) exp M(Hτ (Q) − 2ε1 ) and L ∑

ν ⊗N |[1,...,⌊N (m Q )⌋] (z ℓ ) ≥ 1 − ε2′ .

ℓ=1 Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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Let A = {z 1 , . . . , z L } be the set of letters constructed as above for fixed ε1 > 0, ε2 > 0 small enough and for L , M large enough. When we take an open neighborhood of ν ⊗N , U , for ε3 > 0 as { } ⏐ ⏐ ε3 U = Ψ ∈ P inv (E N ) : ⏐ν ⊗N |[1,...,⌊N m Q ⌋] (z ℓ ) − Ψ[1,...,⌊N m Q ⌋] (z ℓ )⏐ < , ℓ = 1, . . . , L , 2L we can prove the statement. The idea is as follows: For each x ∈ E ⌊Mm Q ⌋ (M ≫ N ) such that R⌊N m Q ⌋ (x) ∈ U , we can find about M patterns chosen from A without overlapping and with small gaps. Influences N of such small gaps are negligible and we find that for each good decompositions of A -letters, y (1) , . . . , y (M/N ) , we have that M ∏

M

log f (y (i) )ρr (|y (i) |)“ ≥ ”(exp (N (F(Q) + Q[log ρr (τ1 )] − 2ε1 ))) N exp(−C N ε˜ )

i=1

and (A.14) implies that M ∏

∑

( ( ( ) )) log f (y (i) )ρr (|y (i) |)“ ≥ ” exp M F(Q) − H Q|qρ⊗N −ε . r ,ν

□

1≤ j1 ≤···≤ j M =⌊Mm Q ⌋ i=1 ˜ R˜ M j1 ,..., j M (x)∈O Q

Theorem 1.2 in [5] says that for d ≥ 3 {∫ s(0) = log(G 0 − 1) +

sup

log f (y)π1 Q(dy) − I

Q∈P erg,fin ( E˜ N )

que

(

} ⏐ ⊗N ) ⏐ Q qρ,ν .

Therefore, (b) immediately follows when we show the following lemma. Lemma A.3. For each Q ∈ P erg,fin ( E˜ N ), there exists {Q n ∈ Rν ∩ P inv,fin ( E˜ N ) : n ≥ 1} such that w- lim Q n = Q ∫ n→∞ ∫ ( ⏐ ⊗N ) ( ⏐ ⊗N ) log f (y)π1 Q n (dy) − I˜que Q n ⏐qρ,ν → log f (y)π1 Q(dy) − I que Q ⏐qρ,ν . The proof is a modification of the one of Proposition C.1 in [8]. Proof of Lemma A.3. We may assume Q ̸∈ Rν . Let ζ = {ζ i : i ≥ 1} be a random sequence of words with the law Q. Let τ i = |ζ i | be the length of ith word ζ i . We define T 0 = 0, T i = T i−1 + τ i , i ≥ 1 and write ζ i as ζ i = (ξ T i−1 +1 , . . . , ξ T i ), where ξ n is nth letter in the concatenating sequence κ(ζ ). Since Q is ergodic, we have that |K | N τj 1 ∑ 1 ∑∑ lim log ν(ξ i ) = lim log ν(ξ τ ) T j−1 +i N →∞ N N →∞ N i=1 j=1 i=1 N

= m Q Ψ Q [log ν(ξ 1 )], Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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where we have used (2.6). Recall the proof of Lemma A.2. Then, for ε1 > 0, ε2′ > 0 and sufficiently large N , we can construct a set of letters with length ⌊N m Q ⌋ and their decompositions to N -words, ⌊N m Q ⌋

A = {z a = (z a1 , . . . , z a

) : a = 1, . . . , A} ⊂ E ⌊N m Q ⌋

and Ba = {ya,b = (ya,b,1 , . . . , ya,b,N ) : κ(ya,b ) = z a , b = 1, . . . , Ba }, a = 1, . . . , A which satisfy (A.4)–(A.8) and (A.11)–(A.12). Additionally, we may assume that ⌊N m Q ⌋ ] [ 1 ∑ log ν(z ai ) ∈ m Q Ψ Q [log ν(ξ 1 )] − ε1 , m Q Ψ Q [log ν(ξ 1 )] + ε1 . N i=1

Then, it is clear from (A.4) that [ ( ) ( )] A ∈ (1 − ε2′ ) exp N (m Q H (Ψ Q ) − ε1 ) , exp N (m Q H (Ψ Q ) + ε1 ) .

(A.15)

(A.16)

We cut ξ = {ξi : i ≥ 1} into blocks with length M = ⌊N m Q ⌋ as Ξ j = (ξ( j−1)M+1 , . . . , ξ j M ) and let G j = 1{an element of A appears in Ξ j }. Then, we can see from (A.15) and (A.16) that [ ] p = p(N , ε1 , ε2′ ) = ν ⊗N G j =

A ∏ M ∑

ν(z ai )

a=1 i=1

[ ( ( )) ∈ (1 − ε2′ ) exp N m Q H (Ψ Q ) + m Q Ψ Q [log ν(ξ 1 )] − 2ε1 , ))] ( ( exp N m Q H (Ψ Q ) + m Q Ψ Q [log ν(ξ 1 )] + 2ε1 . Thus, we know that ⏐ ) )) ( ( ( p(N , ε1 , ε2′ ) = exp −N m Q H Ψ Q ⏐ν ⊗N + o(1) . Since {G j : j ≥ 1} are i.i.d. random variables, σ1 = inf{n ≥ 1 : G n = 1} is geometrically distributed with success probability p(N , ε1 , ε2′ ). Now, we construct words sequence ζˇ . For given ξ and σ1 , we define ( ) ζˇ (1) = ξ1 , . . . , ξ(σ1 −1)M ( ) ζˇ (2) , . . . , ζˇ (N +1) = (ya,b,1 , . . . , ya,b,N ), where (ya,b,1 , . . . , ya,b,N ) is chosen uniformly from Ba when Ξσ1 = z a . If σ1 = 1, then ζˇ (1) , . . . , ζˇ (N ) is constructed in the same manner as ζˇ (2) , . . . , ζˇ (N +1) for σ1 ̸= 1. Repeating this construction, we have a random word sequence ζˇ = {ζˇ (n) : n ≥ 1}. We denote by Qˇ N ,ε1 ,ε2′ the law of ζˇ . It is clear that κ(ζˇ ) = ξ and Qˇ N ,ε1 ,ε2′ ∈ Rν , but Qˇ N ,ε1 ,ε2′ is not θ˜ -shift invariant. For given σ1 , let U be a uniform random variable on {0, . . . , N + 1{σ1 ̸= 1}}. We define another random Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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words sequence by ζ˘ = θ˜ U ζˇ and denote by Q˘ N ,ε1 ,ε2′ its law. Then, by construction, we have that Q˘ N ,ε1 ,ε2′ ∈ Rν ∩ P inv,fin ( E˜ N ) and Q˘ N ,ε1 ,ε2′ → Q weakly as N → ∞ and ε1 → 0, ε2′ → 0. We know that ( ⏐ ⊗N ) ( ⏐ ⊗N ) I˜que Q˘ N ,ε ,ε′ ⏐qρ,ν → I que Q ⏐qρ,ν 1 2

by the same argument as the proof of Proposition C.1 in [8]. Thus, we need to show that ∫ ∫ log f (y)π1 Q˘ N ,ε1 ,ε2′ (dy) → log f (y)π1 Q(dy) as N → ∞ and ε1 → 0, ε2′ → 0. When we apply Fatou’s lemma to log f (y) which is bounded from above and upper semicontinuous function in weak topology, we obtain that ∫ ∫ lim′ lim log f (y)π1 Q˘ N ,ε1 ,ε2′ (dy) ≤ log f (y)π1 Q(dy). ε1 →0,ε2 →0 N →∞

Thus, it is enough to see that ∫ ∫ ˘ lim lim log f (y)π1 Q N ,ε1 ,ε2′ (dy) ≥ log f (y)π1 Q(dy). ε1 →0,ε2′ →0 N →∞

By construction of Q˘ N ,ε1 ,ε2′ , we can write ∫

log f (y)π1 Q˘ N ,ε1 ,ε2′ (dy) = +

N [ ] 1 ∑ ˇ Q N ,ε1 ,ε2′ log f (ζˇ (i) ) : σ1 = 1 N i=1

N +1 [ ] 1 ∑ ˇ Q N ,ε1 ,ε2′ log f (ζˇ (i) ) : σ1 ̸= 1 N + 1 i=1

≥ p(N , ε1 , ε2 )(F(Q) − ε1 ) [ ] ) 1 (ˇ Q N ,ε1 ,ε2′ log f (ζˇ (1) ) : σ1 ̸= 1 + N (F(Q) − ε1 ) . + N +1 Moreover, we know that [ ] Qˇ N ,ε1 ,ε′ log f (ζˇ (1) ) : σ1 ̸= 1 2

=

≥

∞ ∑ n=1 ∞ ∑ n=1

p(1 − p)n

∑ Ξ1 ,...,Ξn ̸∈A

p(1 − p)n

∑ Ξ1 ,...,Ξn ̸∈A

ν ⊗n M ((Ξ1 , . . . , Ξn )) (log pn M (x1 + · · · + xn M ) − log p2n M (0)) (1 − p)n ν ⊗n M ((Ξ1 , . . . , Ξn )) pn M (x1 + · · · + xn M ) log − log p2n M (0) (1 − p)n (1 − p)n

log(1 − p) + p ∞ ∑ log(1 − p) ≥ p(1 − p)n (−h( p˜ n M ) − log p2n (0)) + , p n=1

Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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where we set 1 ν ⊗n M p˜ n M (x) = (1 − p)n

({ (Ξ1 , . . . , Ξn ) :

nM ∑

}) ξi = x, Ξi ̸∈ A , i = 1, . . . , n

i=1

and h( p˜ n M ) is the entropy of p˜ n M (see Appendix B). Since we know that ∑

p˜ n M (x)

x∈Dn M

d ∑

xi2 ≤

i=1

d ∑ ∑ 1 nM p (x) , xi2 ≤ n M n M (1 − p) x∈D (1 − p)n M i=1 nM

we can apply (B.1) and obtain that [ ] Qˇ N ,ε1 ,ε2′ log f (ζˇ (1) ) : σ1 ̸= 1 ≥ −C for some constant C < ∞ and we have completed the proof of Lemma A.3.

□

Appendix B. Upper bound of entropy of simple symmetric random walk Let {X n : n ≥ 0} be a simple symmetric random walk on Zd starting from the origin. We denote by P the probability measure of X . We define P(X n = x) = pn (x), n ∈ N, x ∈ Zd . Then, the entropy of pn (·), h( pn ), is given by ∑ h( pn ) = − pn (x) log pn (x). x∈Zd

We will look at an upper bound of h( pn ). Lemma B.1. For any d ≥ 1, (e) d log . 2 2 n→∞ Let Dn be the set of points x with |x| + n ∈ 2N. Then, it is sufficient to prove that for λ > 0 ( ) ( ) d eλ max h(µ) + log p2n (0) ≤ log , (B.1) lim sup µ∈P(Dn ,λn) 2 2 n→∞ where P(Dn , m) is the set of probability measure which is defined by ⎧ ⎫ d ⎬ ⎨ ∑ ∑ P(Dn , m) = µ : µ is a probability measure on Dn and µ(x) xi2 ≤ m . ⎩ ⎭ lim sup (h( pn ) + log p2n (0)) ≤

x∈Dn

i=1

Actually, we have the following bound from Theorem 11 in [16]. Theorem B.2. There exists the unique probability measure µ∗n,λ ∈ P(Dn , λn) such that max

µ∈P(Dn ,λn)

h(µ) = h(µ∗n,λ ).

In particular, µ∗ is given by ( ) d ∑ 1 2 µn (x) = exp −γn,λ xi , Z γn,λ i=1 Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.

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for x = (x1 , . . . , xd ) ∈ Dn , where Z γn,λ is a normalized constant such that µn becomes a probability measure on Dn and γn,λ > 0 is a constant such that µn satisfies ∑

µn (x)

x∈Dn

d ∑

xi2 = λn.

i=1

Proof of Lemma B.1. It follows from Theorem B.2 that max

µ∈P(Dn ,λn)

h(µ) = log Z γn,λ + γn,λ λn.

It is easy to see that d γn,λ λn → , as n → ∞. 2 Moreover, we have that ( )d/2 d lim 2 Z γn,λ = 1. n→∞ 2π nλ By combining this with the local limit theorem, (B.1) is obtained. □ References [1] Quentin Berger, Hubert Lacoin, The effect of disorder on the free-energy for the random walk pinning model: smoothing of the phase transition and low temperature asymptotics, J. Stat. Phys. 142 (2) (2011) 322–341. [2] Quentin Berger, Fabio Lucio Toninelli, On the critical point of the random walk pinning model in dimension d = 3, Electron. J. Probab. 15 (21) (2010) 654–683. [3] Matthias Birkner, Conditional large deviations for a sequence of words, Stochastic Process. Appl. 118 (5) (2008) 703–729. [4] Matthias Birkner, Andreas Greven, Frank den Hollander, Quenched large deviation principle for words in a letter sequence, Probab. Theory Related Fields 148 (3–4) (2010) 403–456. [5] Matthias Birkner, Andreas Greven, Frank den Hollander, Collision local time of transient random walks and intermediate phases in interacting stochastic systems, Electron. J. Probab. 16 (20) (2011) 552–586. [6] Matthias Birkner, Rongfeng Sun, Annealed vs quenched critical points for a random walk pinning model, Ann. Inst. Henri Poincaré Probab. Stat. 46 (2) (2010) 414–441. [7] Matthias Birkner, Rongfeng Sun, Disorder relevance for the random walk pinning model in dimension 3, Ann. Inst. Henri Poincaré Probab. Stat. 47 (1) (2011) 259–293. [8] Erwin Bolthausen, Frank den Hollander, Alex A. Opoku, A copolymer near a selective interface: variational characterization of the free energy, Ann. Probab. 43 (2) (2015) 875–933. [9] Amir Dembo, Ofer Zeitouni, Large Deviations Techniques and Applications, second ed., in: Applications of Mathematics, vol. 38, Springer-Verlag, New York, New York, 1998. [10] Bent Fuglede, The multidimensional moment problem, Expo. Math. 1 (1) (1983) 47–65. [11] Giambattista Giacomin, Random Polymer Models, Imperial College Press, London, 2007. [12] Yasuki Isozaki, Nobuo Yoshida, Weakly pinned random walk on the wall: pathwise descriptions of the phase transition, Stochastic Process. Appl. 96 (2) (2001) 261–284. [13] Ioannis Karatzas, Steven E. Shreve, Brownian Motion and Stochastic Calculus, second ed., in: Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. [14] Gregory F. Lawler, Intersections of Random Walks, in: Probability and Its Applications, Birkhäuser Boston, Inc., Boston, MA, 1991. [15] Jean-Christophe Mourrat, On the Delocalized Phase of the Random Pinning Model, in: Séminaire De Probabilités XLIV, in: Lecture Notes in Math., vol. 2046, Springer, Heidelberg, 2012, pp. 401–407. [16] Flemming Topsøe, Information-theoretical optimization techniques, Kybernetika (Prague) 15 (1) (1979) 8–27.

Please cite this article in press as: M. Nakashima, The free energy of the random walk pinning model, Stochastic Processes and their Applications (2017), http://dx.doi.org/10.1016/j.spa.2017.04.015.