Free energy of directed polymers in random environment in $1+1$-dimension at high temperature

We consider the free energy $F(\beta)$ of the directed polymers in random environment in $1+1$-dimension. It is known that $F(\beta)$ is of order $-\beta^4$ as $\beta\to 0$. In this paper, we will prove that under a certain condition of the potential, \begin{align*} \lim_{\beta\to 0}\frac{F(\beta)}{\beta^4}=\lim_{T\to\infty}\frac{1}{T}P_{\mathcal{Z}}\left[\log \mathcal{Z}_{\sqrt{2}}(T)\right] =-\frac{1}{6}, \end{align*} where $\{\mathcal{Z}_\beta(t,x):t\geq 0,x\in\mathbb{R}\}$ is the unique mild solution to the stochastic heat equation \begin{align*} \frac{\partial}{\partial t}\mathcal{Z}=\frac{1}{2}\Delta \mathcal{Z}+\beta \mathcal{Z}{\dot{\mathcal W}},\ \ \lim_{t\to 0}\mathcal{Z}(t,x)dx=\delta_{0}(dx), \end{align*} where $\mathcal{W}$ is a time-space white noise and \begin{align*} \mathcal{Z}_\beta(t)=\int_\mathbb{R}\mathcal{Z}_\beta(t,x)dx. \end{align*}

where {Z β (t, x) : t ≥ 0, x ∈ R} is the unique mild solution to the stochastic heat equation where W is a time-space white noise and Z β (t) =

Introduction and main result
Directed polymers in random environment was introduced by Henly and Huse in the physical literature to study the influence by impurity of media to polymer chain [20]. In particular, random media is given as i.i.d. time-space random variables and the shape of polymer is achieved as time-space path of walk whose law is given by Gibbs measure with the inverse temperature β ≥ 0, that is, time-space trajectory s up to time n appears as a realization of a polymer by the probability where H N (s) is a Hamiltonian of the trajectory s, (S, P 0 S ) is the simple random walk on Z d starting from x ∈ Z d , S [0,N] = (S 0 , S 1 , · · · , S N ) ∈ Z d N+1 , and Z β ,N is the normalized constant which is called the quenched partition function.
There exists β 1 such that if β < β 1 , then the effects by random environment are weak and if β > β 1 , then environment has a meaningful influence. This phase transition is characterized by the uniform integrability of the normalized partition functions. Also, we have another phase transition characterized by the non-triviality of the free energy, i.e. there exists β 2 such that if β < β 2 , then the free energy is trivial and if β > β 2 , then the free energy is non-trivial. The former phase transition is referred to weak versus strong disorder phase transition and the latter one is referred to strong versus very strong disorder phase transition. We have some known results on the phase transitions: β 1 = β 2 = 0 when d = 1, 2 [17,22] and β 2 ≥ β 1 > 0 when d ≥ 3 [9,15]. In particular, the best lower bound of β 1 is obtained by Birkner et.al. by using size-biased directed polymers and random walk pinning model [7,8,28].

Model and main result
To define the model precisely, we introduce some random variables.
• (Simple random walk) Let (S, P x S ) be a simple random walk on Z d starting from x ∈ Z d . We write P S = P 0 S for simplicity.

Then, the Hamiltonian H(s) is given by
It is clear that for any β ∈ R.
In particular, it is conjectured that when d = 1, where 1 24 appears in the literature of stochastic heat equation or KZP equation [6,4]. Our main result answers this conjecture in some sense. Theorem 1.1. Suppose d = 1. We assume the following concentration inequality: There exist γ ≥ 1, C 1 ,C 2 ∈ (0, ∞) such that for any n ∈ N and for any convex and 1-Lipschitz function f : R n → R, where 1-Lipschitz means | f (x) − f (y)| ≤ |x − y| for any x, y ∈ R n and ω 1 , · · · , ω n are i.i.d. random variables with the marginal law Q(η(n, x) ∈ dy). Then, we have The constant − 1 6 appears as the limit of the free energy of the continuum directed polymers (see Lemma 2.3): where Z x β (t, y) is the unique mild solution to the stochastic heat equation with the initial condition lim t→0 Z (t, y)dy = δ x (dy) and W is a time-space white noise and Z β (t) = Z 0 β (t) for simplicity. − 1 6 seems to be different from the value − 1 24 in the conjecture. However, it has the relation and √ 2 appears from the periodicity of simple random walk. Thus, the conjecture it true essentially. (1.3) are given in [10] for pinning model. Under this assumption, {η(n, x) : n ∈ N, x ∈ Z} satisfies a good concentration property (see Lemma 3.1). It is known that the following distribution satisfies (1.3

Organization of this paper
This paper is structured as follows: • We first give the strategy of the proof of our main result in section 2.
• Section 3 is devoted to prove the statements mentioned in section 2 related to discrete directed polymers.
• Section 4 is also devoted to prove the statement mentioned in section 2 related to continuum directed polymers.

Proof of Theorem 1.1
In this section, we give the proof of Theorem 1.1.

Proof of limit inferior
The idea is simple. Alberts, Khanin and Quastel proved the following limit theorem.
Taking the limit in T , we have that Therefore, it is enough to show the following lemmas.
We should take n = ⌊β −4 n ⌋ in general. However, we may consider the case β n = n − 1 4 without loss of generality.

Proof of limit superior
We use the coarse graining argument to prove the limit superior. We divide Z into the blocks with size of order n 1/2 : For y ∈ Z, we set B n y = (2y − 1)⌊n 1/2 ⌋ + y, (2y + 1)⌊n 1/2 ⌋ + y .
For each ℓ ∈ N, we denote by B n y (ℓ) the set of lattice z ∈ Z such that that is the set of lattices in B n y which can be reached by random walk (S, P S ) at time ℓ. We will give an idea of the proof. It is clear by Jensen's inequality that for each θ ∈ (0, 1), T ∈ N, and N ∈ N, (2.4) We will take the limit superior of both sides in N → ∞, n → ∞, T → ∞, and then θ → 0 in this order. Then, it is clear that We would like to estimate the right hand side.
Then, we have that The following result gives us an upper bound of the limit superior:

Remark 2.7. (1.3)
is not assumed in lemmas in this subsection. Thus, we find that the limit superior in Theorem 1.1 is true for general environment.
In the rest of the paper, we will prove the above lemmas.
We take R m as R E n in the proof of Lemma 2.2 with E n = {1, · · · , T n} × {−T n, · · · , T n} which contains all lattices simple random walk can each up to time T n.

Proof of Lemma 2.2. When we look at
Thus, we can apply Lemma 3.1 to logW β n ,T n (η). Since where µ η β ,n is the probability measure on the simple random walk paths defined by is the product probability measure of µ η β n ,T n , and S and S ′ are paths of independent directed path with the law µ η β n ,T n . We write for s = (s 1 , · · · , s n ) and s ′ = (s ′ 1 , · · · , s ′ n ) ∈ Z n . We define the event A n on the environment by for some C 4 > 0 which we will take large enough. We claim that for C 4 > 0 large enough, there exists a constant δ > 0 such that for all n ≥ 1. If (3.1) holds, then Lemma 2.2 follows. Indeed, applying Lemma 3.1, we have that Thus, we find the L 2 -boundedness of logW β n ,T n (η) and hence uniform integrability. We will complete the proof of Lemma 2.2 by showing that (3.1). We observe that is the simple random walk on Z starting from the origin and P S,S ′ is the product measure of P S and P S ′ . Paley-Zygmund's inequality yields that Also, we have that as n → ∞ for r > 0, there exists a constant r > 0 such that for any n large enough. The L 2 -boundedness of W β n ,T n (η) (see Theorem 2.1) implies that there exist C 5 > 0 and C 6 > 0 such that We conclude that and we obtain (3.1) by taking C 4 > 0 large enough.

Proof of Lemma 2.4
Since the finite dimensional distributions We will use Garsia-Rodemich-Rumsey's lemma [29, Lemma A.3.1] a lot of times in the proof for limit superior.
is an open ball in R d centered at x with radius r, then for all s,t ∈ B r (x), where λ d is a universal constant depending only on d.
Then, we have that We will show that for some p ≥ 1, q > 0 with pq > 2, there exist C p,T,θ > 0 and η p,θ − pq > −1 such that β n ,T n (η) θ for p > 1 and therefore Lemma 2.4 follows.
Proof of (3.3). We remark that , for x, y ∈ B n 0 (0). When we define i.i.d. random variables by we find that Then, we can write where p n (y) = P S (S n = y) for (n, y) ∈ N × Z, x 0 = x, x = (x 1 , · · · , x k ), and Then, it is easy to see that Thus, we have that Since we know that for k ≥ 1 where Γ(s) is a Gamma function at s > 0 [2, Section 3.4 and Lemma A.1], y)).

Proof of Lemma 2.5
The idea is the same as the proof of Lemma 2.4.
Proof of Lemma 2.5. We set Then, we know that By the same argument as the proof of Lemma 2.4 that We write Then, we have that Hence, as the proof of Lemma 2.4. We obtain by Hölder's inequality that for p = 5 where we have used the hypercontractivity as the proof of Lemma 2.4, C 3,T is independent of the choice of z and Also, we know that

Continuum directed polymers
To prove Lemma 2.3 and Lemma 2.6, we recall the property of continuum directed polymers.
Also, we define the four parameter field by Then, we have the following fact[1, Theorem 3.1]: There exists a version of the field Z β (s, x;t, y) which is jointly continuous in all four variables and have the following properties:

Proof of Lemma 2.6
We first show a weak statementt: Proof. We will show that there exists a K > 0 such that for |θ | ∈ (0, 1). For fixed T ∈ N, we define σ -field Then, we write are martingale differences. Here, we introduce new random variableŝ Since it is clear that we have Also, we consider a new probability measure on R 2 by µ (i) Then, it is clear that and Jensen's inequality implies from Theorem 4.1 (ii) and (iv) that where we have used that (see Corollary 4.3). Thus, we have from Jensen's inequality that Also, Jensen's inequality implies that where we have used that Thus, we have confirmed conditions in [25, Theorem 2.1] so that we have proved 4.1.
We can find that the above proof is true when we replace Z √ 2 (T ) by Z √ 2 (T, 0). Therefore, we have the following corollary from (4.1).

Corollary 4.5. We have
In particular, we have Proof of Lemma 2.6. The proof is similar to the proofs of Lemma 2.4 and Lemma 2.5. Also, we will often use the equations in Appendix to compute integrals of functions of heat kernels.
We write where A(T ) = [−a(T ), a(T )] is a segment with length of order T 3 . Hereafter, we will look at I 1 (T, x) and I 2 (T, x).
We will show in the lemmas below that Thus, we complete the proof.
Lemma 4.6. We have that Lemma 4.7. We have that for any θ ∈ (0, 1) Proof of Lemma 4.7. It is easy to see from Lemma 4.1 (i) that Thus, it is enough to show that Applying (3.2) to the continuous function I 2 (T, y) θ with d = 1, x = 0, Thus, we will show that for θ ∈ (0, 1), there exist p ≥ 1 and q > 0 with pq > 2 such that We remark that I 2 (T, x) have the following Wiener chaos representation: for 0 < t 1 < · · · < t ℓ ≤ 1 < t ℓ+1 < · · · < t k ≤ T.
We will estimate for k ≥ 0. It is easy to see that Also, we have Also, we have For k ≥ 3, Thus, we have that where C is a constant independent of k. By hypercontractivity of Wiener chaos [21, Theorem 5.10], we have that for p ≥ 2 Thus, (4.2) holds with p = 10 θ and q = θ 4 .
Proof of Lemma 4.6. It is clear that where L ∈ N is taken large later. Thus, we have that If there exists a constant C > 0 such that for k ∈ Z, then we have Also, we know that where ν (1,L) (u) is the probability density function on R given by Then, we have from Lemma 4.4 that and we can complete the proof of Lemma 2.6. We will prove (4.3).
We consider a function on .
Finally, we need to prove the free energy F Z ( √ 2) = − 1 6 . The proof is a modification of the proof of Lemma 2.6.
Proof of Lemma 2.3. It is easy to see that for a ′ (T ) ∈ [0, ∞) If lim T →∞ a ′ (T )