Fluctuation exponents for directed polymers in the intermediate disorder regime

We compute the fluctuation exponents for a solvable model of one-dimensional directed polymers in random environment in the intermediate regime. This regime corresponds to taking the inverse temperature to zero with the size of the system. The exponents satisfy the KPZ scaling relation and coincide with physical predictions. In the critical case, we recover the fluctuation exponents of the Cole-Hopf solution of the KPZ equation in equilibrium and close to equilibrium.


Main results
1.1. Introduction. The directed polymer in a random environment is a statistical physics model that assigns Boltzmann-Gibbs weights to random walk paths as a function of the environment encountered by the walk. It was originally introduced in [29] as a model of an interface in two dimensions. Here is the standard lattice formulation in d + 1 dimensions (d space dimensions, one time dimension).
The environment is a collection of i.i.d. random weights {ω(i, x) : i ∈ N, x ∈ Z d } with probability distribution P. Let P be the law of simple symmetric random walk (S t ) t∈Z + on Z d with S 0 = 0. Denote expectation under P and P by E and E, respectively. The quenched partition function of the directed polymer in environment ω and at inverse temperature β > 0 is where E[X, A] is the expectation of X restricted to the event A. This is the point-topoint partition function because the endpoint S N of the walk is constrained to be x. The version that allows S N to fluctuate freely is the point-to-line partition function. In the point-to-point setting the quenched polymer measure on paths ending at x is We restrict the discussion to the 1+1 dimensional case. Basic objects of study are the fluctuations of the free energy log Z N,N x (β) and the path (S t ) 0≤t≤N . On the crudest level the orders of magnitude of these fluctuations are described by two exponents χ and ζ: • fluctuations of log Z N,N x (β) under P have order of magnitude N χ • fluctuations of the path S t under P β N,N x have order of magnitude N ζ In the 1+1 dimensional case these exponents are expected to take the values χ = 1/3 and ζ = 2/3 independently of β, provided the i.i.d. weights ω(i, x) satisfy a moment bound. Furthermore, there are specific predictions for the limit distributions of the scaled quantities: for example, the GUE Tracy-Widom distribution for log Z N,N x (β). These properties are features of the Kardar-Parisi-Zhang (KPZ) universality class to which these models are expected to belong. See [23,44] for recent surveys. The KPZ regime should be contrasted with the diffusive regime where χ = 0, ζ = 1/2, and the path satisfies a central limit theorem. Diffusive behavior is known to happen for d ≥ 3 and small enough β [22].
There are three exactly solvable 1+1 dimensional models for which KPZ predictions have been partially proved: (a) the semidiscrete polymer in a Brownian environment [38] (b) the log-gamma polymer [42] (c) the continuum directed random polymer, in other words, the solution of the Kardar-Parisi-Zhang (KPZ) equation [2,4,31] In recent years a number of results have appeared, first for exponents and then for distributional properties. This is not a place for a thorough review, but let us cite some of the relevant papers: [4,15,16,17,10,24,37,42,43]. To do justice to history, we mention also that KPZ results appeared earlier for zero-temperature polymers (the β → ∞ limit of (1.1)-(1.2), known as last-passage percolation), beginning with the seminal papers [7,30].
Article [1] conjectured the exponents for the entire range: (1.3) χ(α) = 1 3 (1 − 4α) and ζ(α) = 2 3 (1 − α) for 0 ≤ α ≤ 1/4. In this paper we derive these intermediate disorder exponents for the semidiscrete polymer in the Brownian environment (introduced in [38], hence also called the O'Connell-Yor model). Along the way we offer some improvements to the earlier work [43] which treated the α = 0 case. This model has two versions: a stationary version with particular boundary conditions that render the process of log Z increments shift-invariant, and the point-to-point version without boundary conditions represented by (1.1)-(1.2) above. In general we have better results for the stationary version. In case the reader is encountering polymer models with boundaries for the first time but can appreciate an analogy with the totally asymmetric simple exclusion process (TASEP), then the stationary polymer corresponds to stationary TASEP with Bernoulli occupations, while the point-to-point version of the polymer is the analogue of TASEP with step initial condition.
We list below the precise contributions of our paper: (i) For the free energy we derive the exponent χ(α) = 1 3 (1 −4α) for the entire range 0 ≤ α ≤ 1/4 for the stationary version and for 0 ≤ α < 1/4 for the point-topoint version. For the fixed temperature case (α = 0) the lower bound χ ≥ 1/3 for the point-to-point version was not covered in [43], but is done here.
(ii) We have the path exponent ζ(α) = 2 3 (1 − α) for the stationary version, and the upper bound ζ(α) ≤ 2 3 (1 − α) for the point-to-point version. (iii) Our results refine the prediction (1.3) in the following way. We introduce a macroscopic time parameter τ > 0 and conclude that the fluctuations of log Z τ N,τ N x (β 0 N −α ) are of magnitude τ 1/3 N χ(α) while the path fluctuations are of magnitude τ 2/3 N ζ(α) . In other words, in the macroscopic variables we see again the exponents 1 3 and 2 3 . (iv) In the fixed temperature case (α = 0) the lower bound χ ≥ 1/3 was already proved in [43] for the stationary version. Here we give a considerably simpler proof of the lower bound, including the case α = 0. (v) In the critical case α = 1/4 we can connect with the KPZ equation. The macroscopic variable τ becomes the time parameter of the stochastic heat equation (SHE), and we obtain again the exponent of the stationary Hopf-Cole solution of the KPZ equation, first proved in [10]: where Z is the solution of SHE. Moreover, we prove similar bounds for solutions where the initial condition is a bounded perturbation of the stationary initial condition. Some further comments about the state of the field and the place of this work are in order. Presently one can identify the following three approaches to fluctuations of polymer models and of models in the KPZ class more broadly.
(a) The resolvent method. This is a fairly robust method used to establish superdiffusivity. It is quite general, for it can often be applied as long as a model has a tractable invariant distribution [13,33,39,40,41,46]. A drawback of the method is that often it cannot determine the exact exponents but provides only bounds on them. However, here are two exceptions. In [46] the scaling exponent of a 2d TASEP model is identified exactly. In [39,40] the method is used to give a comparison between the solvable 1d TASEP and more general 1d exclusion models to show that the scaling exponents are the same. (b) The coupling method, represented by the present work and references [8,9,10,11,19,42,43]. This approach is able to identify exact exponents, but so far has depended on the presence of special structures such as a Burke-type property. (c) Exact solvability methods. When it can be applied, this approach leads to the sharpest results, namely Tracy-Widom limit distributions. But it is the most specialized and technically very heavy. This approach became available for the semidiscrete polymers after determinantal expressions where found for the distribution of log Z [15,16,37]. For the related log-gamma polymers, see [17,24]. For the ASEP the first scaling limits were proved using Fredholm determinant formulas based on the work of [45]. The recent work of [18] uses certain duality relations to get scaling limits for the same model. Their method can be thought of as a rigorous version of the so-called 'replica trick'.
The free energy exponent χ = 1/3 in the fixed temperature case (α = 0) is also a consequence of the distributional limits for log Z in [15,16]. Presently these results cover the point-to-point case of the semidiscrete polymer for the entire fixed temperature range 0 < β < ∞. It is expected that these methods should work also in the intermediate disorder regime (personal communication from the authors). However, these works do not yet give anything on the stationary versions of the models, or on the path fluctuations in either the point-to-point or stationary version. And even in the cases covered by the distributional limits, our results give sharper bounds on moments and deviations that weak limits alone cannot provide.
The open problem that remains in the coupling approach used here is the lower bound for the path in the point-to-point case.
One more expected universal feature of polymer exponents worth highlighting here is the scaling relation χ = 2ζ − 1. This is expected to hold very generally across models and dimensions. The exponents we derive satisfy this identity. There is important recent work on this identity that goes beyond exactly solvable models: first [20], and then [5] with a simplified proof, derived this relation for first passage percolation under strong assumptions on the existence of the exponents. These results are extended to positive temperature directed polymers in [6].
Finally, we point out that the coupling method applied to directed polymers first appeared in the work [42] in the context of discrete polymers in a log-gamma environment. Most of the results of [43] have discrete analogues in [42]. The intermediate regime can also be investigated for the polymers in log-gamma environment. Although this model is formulated for β = 1, the parameters of the environment can be tuned to emulate the situation β → 0. We have obtained proofs for the fluctuation exponents of the log-gamma model in the intermediate scaling regime. The methods are very similar to the ones used here for the semidiscrete polymer model, but involve considerably heavier asymptotics so we decided not to include them in the present paper.  0 s θ−1 e −s ds and the Gamma(θ, r) distribution has density f (x) = r θ Γ(θ) −1 x θ−1 e −rx for 0 < x < ∞. If r = 1 then we drop it from the notation, i.e. Gamma(θ) is the same as Gamma(θ, 1). We use the notations Ψ 0 = Γ ′ /Γ and Ψ 1 = Ψ ′ 0 are the digamma and trigamma functions. Ψ −1 1 is the inverse function of Ψ 1 . See Section 4 for more on polygamma functions and their relations to the Gamma(θ, r) distribution. The environment distribution P has expectation symbol E. Generically expectation under a probability measure Q is denoted by E Q . To simplify notation we drop integer parts. A real value s in a position that takes an integer should be interpreted as the integer part ⌊s⌋.
Aknowledgements. The authors thank Michael Damron for the decomposition idea in the proof of Theorem 2.4.

1.2.
The semi-discrete polymer in a Brownian environment. We begin with the results for the semi-discrete polymer in a Brownian environment. This is a semidiscrete version of the generic polymer model described in (1.1). As already mentioned, the model has two versions: a point-to-point and a stationary version.
In the integral ds 1,n−1 is short for ds 1 · · · ds n−1 . The limiting free energy density was computed for a fixed β in [36]: Our first result identifies the free energy fluctuation exponent χ = 1 3 (1 − 4α) for the point-to-point semi-discrete polymer in the intermediate disorder regime. In the fixed temperature case (α = 0) the upper bound was proved in [43] but a lower bound proof with coupling methods is new even in this case. (To clarify, the correct exponent in the α = 0 case has of course been identified in the weak convergence results [15,16] with exact solvability methods.) Note that we see the intermediate regime exponent on the scaling parameter n, but for the macroscopic variable τ we see the exponent 1 3 corresponding to the KPZ scaling. Theorem 1.1. Fix α ∈ [0, 1/4) and 0 < β 0 < ∞. Let β = β 0 n −α . There exist finite positive constants C, n 0 , b 0 , τ 0 that depend on (α, β 0 ) such that the following bounds hold. For τ ≥ τ 0 , n ≥ n 0 and b ≥ b 0 , We turn to the fluctuations of the polymer path. The quenched polymer measure Q n,t,β on paths is defined, in terms of the expectation of a bounded Borel function f : R n−1 → R, by × exp β B 1 (0, s 1 ) + · · · + B n (s n−1 , t) ds 1,n−1 .
The jump times as functions of the path are denoted by σ i . Averaged (or annealed ) probability and expectation are denoted by P n,t,β (·) = EQ n,t,β (·) and E n,t,β (·) = EE Q n,t,β (·).
In the point-to-point setting the path exponent ζ describes the order of magnitude of the deviations of the path from the diagonal. A path close to the diagonal in the rectangle {1, . . . , n} × [0, t] would have σ i ≈ it/n. The next theorem shows that the path exponent ζ is bounded above by its conjectured value 2 3 (1 − α). Theorem 1.2. Fix α ∈ [0, 1/4) and 0 < β 0 < ∞. Let β = β 0 n −α . There exist finite positive constants C, n 0 , b 0 , τ 0 that depend on (α, β 0 ) such that the following bound holds. For all 0 < γ < 1, τ ≥ τ 0 , b ≥ b 0 , and n ≥ n 0 , Stationary semi-discrete polymer. The proofs of the above theorems rely on comparison with a stationary version of the model. Enlarge the environment by adding another Brownian motion B independent of {B i } i≥1 . Introduce a parameter θ ∈ (0, ∞) and restrict to β = 1 for a moment. The stationary partition function is, for n ∈ N and t ∈ R, This model has a useful stationary structure described by [38]. Let Y 0 (t) = B(t) and, for k ≥ 1, define inductively For each fixed t ≥ 0, {r k (t)} k≥1 are i.i.d. and e −r k (t) has Gamma(θ) distribution [38]. Thus the law of Z θ n,t e B(t)−θt is independent of t. This stationarity is part of a broader Burke-type property (see [43,Section 3.1] for more details).
The following theorem identifies the fluctuation exponent χ for the stationary model. A key difference between the point-to-point and stationary versions is that KPZ fluctuations appear in the stationary version only in a particular characteristic direction (n, t) determined by the parameters. In other directions the diffusive fluctuations of the boundaries dominate (see [42], Corollary 2.2, in the context of discrete polymers in a log-gamma environment). Once we choose β = β 0 n −α , to make the diagonal a characteristic direction we are forced to pick θ = βΨ −1 1 (β 2 ) ∼ β −1 . To simplify notation we suppress the n-dependence of the parameters β and θ.
(Upper bound on the tail.) There is a constant (Bounds on the first absolute moment.) as a model of a randomly growing interface in 1 + 1 dimension: if we let h(t, x) denote the height of the interface at site x ∈ R and time t ≥ 0, then the evolution of the interface can be represented by the stochastic partial differential equation where W is a space-time white noise. We will be interested in initial conditions of the form B + ϕ where B is a double-sided one-dimensional Brownian motion and ϕ is a bounded function. It is well known that the meaning of a solution to equation (1.19) is a delicate matter. We will always consider the so called Cole-Hopf solution: we take h = − log Z, where Z solves the stochastic heat equation The relation between (1.19) and (1.20)-(1.21) can be seen by a formal application of Itô's formula. For a more detailed overview of the KPZ equation, we refer the reader to the review [23] and its references. See also [28] for recent rigorous work on solving (1.19).
It is expected that, for a wide family of initial conditions, the fluctuations of log Z(t, x) are of order t 1/3 . This was first proved in [10] in the stationary case with ϕ = 0. The proof was based on the convergence of the rescaled height function of the weakly asymmetric exclusion process to the Cole-Hopf solution of the KPZ equation [14], together with non-asymptotic fluctuation bounds on the current of the asymmetric simple exclusion process [11]. It is not clear that this approach can be extended to the case of non-zero ϕ.
When the initial condition is Z(0, x) = δ 0 (x), the asymptotic distribution of the fluctuations of log Z is identified in [4] as the Tracy-Widom distribution. The proof is based on heavy asymptotic analysis of exact formulas for the weakly asymmetric simple exclusion process.
We will extend the result of [10] to the case of a bounded perturbation ϕ. Our approach is different as we use an approximation of Z by partition functions of the Brownian semidiscrete directed polymer in the critical case α = 1 4 rather than by particle systems.
Building on the techniques of [1], it is shown in [35] that a suitable renormalization of the partition function of the semi-discrete model with α = 1 4 converges to Z ϕ . More precisely, let ϕ n (x) = ϕ − x √ n and let The renormalized partition function is n Z θ,β,ϕ τ n,τ n . When ϕ = 0, we simply denote this by Z N (τ ). . Then there exist constants C 1 , C 2 , τ 0 > 0 such that We note that our results for the path of the stationary polymer could in principle have a meaning in the context of the SHE. In [2], Z is identified as the partition function of a continuum directed polymer. Theorem 1.4 strongly suggests that the fluctuations of the path of the continuum polymer are of order t 2/3 , in agreement with the KPZ scaling.

Proofs for the semi-discrete polymer model
The proofs of our Theorems 1.1, 1.2, 1.3 and 1.4 are given in this section. We first prove the results for the stationary model. The results for the point-to-point model are then done by comparison.

Preliminaries.
We recall some facts from [43]. Throughout this section we take β = 1 as we can reduce the situation to this by Brownian scaling (see Section 2.2). The stationary model can be written as where the r k processes are defined recursively in (1.10). Recall that the random variables r k (0) are i.i.d. and e −r k (0) has Gamma(θ) distribution.
The processes r k and Y k give space and time increments of the partition function: The appearance of polygamma functions in our results is natural because of the identities The variance of log Z θ n,t was computed in Theorem 3.6 in [43]: We will also need the following lemma from [43]: Finally, we note a shift-invariance of the stationary model (Remark 3.1 of [43]): This follows from the stationarity of Z θ n (t) exp(B(t) − θt) by observing that the density of (σ 0 , . . . , σ n−1 ) under Q θ n,t can also be written as where B(u) = B(u) − θu/2 (and similarly for B k ) and Z θ n (t) = Z θ n (t) exp(B(t) − θt). Using the same ideas one can also show a shift-invariance property in n (see the proof of Theorem 6.1 in [43]): n,t f (σ k , σ k+1 , . . . , σ n−1 ) d = E Q θ n−k,t f (σ 0 , σ 1 , . . . , σ n−k−1 ). (2.8) 2.2. Rescaled models and characteristic direction. For the proofs we scale β away via the following identity in law which is obtained by Brownian scaling: We drop β = 1 from the notation and write Z (1,n),(0,t) = Z (1,n),(0,t) (1). The regime β = β 0 n −α corresponds to studying Z (1,n),(0,β 2 0 n 1−2α ) . Similarly we scale β away from the stationary partition function (1.14): As we take (n, t) to infinity, we have to follow approximately a characteristic direction determined by θ. The characteristic direction is found by minimizing the right-hand side of (2.4) with respect to θ, or equivalently, by arranging the cancellation of the first two terms on the right of (2.5). The following condition on the triples (n, t, θ) expresses the fact that (n, t) is close to the characteristic direction: By the scaling relation (2.10), we can see that the choice of parameters in Theorem 1.3 corresponds to the characteristic direction.

2.3.
Upper bounds for the stationary model. The main tool for our upper bounds is the following lemma. The proof of the upper bound in Theorem 1.3 will follow by a particular choice of the parameters and can be found at the end of this section.

2.4.
Lower bound for the stationary model. In this section we prove the lower bound in Theorem 1.3. Again, the proof will follow by a particular choice of the parameters in the next proposition and can be found at the end of the section.
A similar shifting argument gives and the upper bound (2.12) can be applied with λ andt. Hence, we can fix constants 0 < c 1 < c 2 < ∞ such that, for a given ε 0 > 0 and large n, Observe that c 1 can be taken as close to 0 as we wish.
Introduce temporary notations s 1 = c 1 v and s 2 = c 2 v. Assumptions (2.11) and θ ≥ θ 0 > 0 guarantee that s 2 < t for large enough n. Hence Using (2.11), (2.30) Bound probability (2.31) with Chebyshev: Var(log Z λ n,t ). The variance is estimated by (2.1) and (2.28): This implies that (2.31) ≤ Cb −3 . Let A denote the event in probability (2.30): We wish to replace λ by θ inside the integral to match it with the parameter in E(log Z θ n,t ) on the right-hand side. For this we use the Cameron-Martin-Girsanov formula to add a drift λ − θ to the Brownian motion {B(s 1 , s) : s 1 ≤ s ≤ s 2 }. Note that the other random objects in the event A, namely {B(s 1 ); B i (·) : 1 ≤ i ≤ n}, are independent of B(s 1 , · ). Let whereB is a standard Brownian motion. By Cauchy-Schwarz The first expectation is finite: where C depends only on θ 0 . We bound the probability P(A) as follows: recall c 0 from (2.20), where the last line follows after choosing c 1 small enough. Put the estimates back on lines (2.30)-(2.31) to conclude that for a constant C that depends only on (κ, θ 0 ). This completes the proof of the proposition.

2.6.
Bounds for the point-to-point model. This section derives bounds on the path and free energy fluctuations in the point-to-point case without boundaries, with β = 1, uniformly in (n, t, θ). Theorems 1.1 and 1.2 follow after a Brownian scaling step. For n ∈ N, t > 0 and events D on the paths write Z θ n,t (D) = Z θ n,t Q θ n,t (D) for the unnormalized quenched measure.
Then there exist constants b 0 , C that depend only on θ 0 so that for n ≥ n 0 , b ≥ b 0 we have Remark 2.5. The basic strategy of our proofs is the following. If we consider Z θ n,t with θ defined according to the theorem then E θ n,t σ 0 = 0. Since we expect σ 0 to be fairly close to its mean, this would suggest that Z θ n,t is fairly close to Z (1,n),(0,t) . The main components of the proofs will rely on comparisons between the partition functions of the two models and on the results proved about the stationary model.
Note that the centering θt − nΨ 0 (θ) inside probability (2.33) is right choice, as it can be seen from the exact expression for the free energy (1.5) and Brownian scaling.
Proof of the upper bounds in (2.33) and (2.34). Inequality (2.37) gives where the last step comes from Markov's inequality and e −r 1 (0) ∼ Gamma(θ). Estimate (2.16) gives the Chebyshev bound The bound on the other tail will be proved in two steps. In Lemma 2.9 below we will show that there is a c 0 > 0 depending on θ 0 so that if b > c 0 n 2/3 θ 2/3 then In Lemma 2.8 below we will show that if b ≤ c 0 n 2/3 θ 2/3 then there are constants C and n 0 depending on c 0 and θ 0 so that for all n > n 0 . Using the Chebyshev bound again with (2.40) and then combining it with (2.39) we get The estimates (2.38) and (2.41) together establish (2.33).
Integrating out b in (2.33) gives Combining the above with verifies the upper bound of (2.34).
Except for the technical estimates postponed to Section 2.7, this completes the proof of Theorem 2.4.
We now turn to the path fluctuations for the model without boundaries.
On the right hand side, the first term is bounded by Cb −3 by Markov's inequality, since e −r 1 (0) ∼ Gamma(θ). The second term is bounded by Cr −3/2 ≤ Cb −3 by (2.40) above. To see that we can actually apply Lemma 2.8 note that by (2.45) we have with a constant depending on θ 0 and ε 1 which was the condition needed for the lemma. Finally, the shift invariance (2.8) and Lemma 2.2 give, for large enough n and b and uniformly for h ∈ (b −3 , 1), It is above that we need θ ≤ θ −1 0 n 1/2−ε 0 for ε 0 > 0, for otherwise the right-hand side e −rn 1 3 θ − 2 3 θ −1 b −6 cannot be bounded below by e −δθ 2 u 2 /(n−ℓ) . Collecting the estimates gives P Q (1,n),(0,t) (|σ ℓ − t ′ | > u) > h ≤ Cb −3 and from this This completes the proof.
2.7. The tail estimates. In this section we prove the missing components of the proofs of Theorem 1.4 and Theorem 2.4. We begin with some definitions.
Augment the family Z (j,k),(s,t) = Z (j,k),(s,t) (1) defined for j ≥ 1 in (1.13) by introducing, for k ∈ N and t ∈ R + , (2.46) It is also convenient to set, for A ⊆ R, The following bounds are proved in Lemma 3.8 of [43] .
The second inequality of (2.49) makes sense only for n ≥ 1.
We will also define a reversed system: construct a new environmentω with Quantities that use environmentω are marked with a tilde. From the definitions one checks that (2.50) Z 1,n (s, t) =Z 0,n−1 (0, t − s) for any t > 0 and s ∈ (−∞, t).
Lemma 2.8. Let θ = Ψ −1 1 (t/n) and assume that θ 0 ≤ θ ≤ θ −1 0 √ n with a fixed θ 0 > 0. Proof. Note that once we prove (2.51) for b > b 0 with a constant b 0 depending on θ 0 , c 0 then we can get it for all b by adjusting the constant C. Thus we may assume that b is big enough compared to θ 0 and c 0 .
Begin with (2.50) and then apply comparison (2.49): Z (1,n),(s,t) Z (1,n),(0,t) =Z (0,n−1),(0,t−s) where we used (2.2) for the reversed system. Specializing the above to our context and substituting it in the probability that is to be bounded: (2.55) To treat probability (2.54) set where we used our assumptions on θ, the bounds (4.2), and took b large enough in relation to θ 0 . Use invariance (2.6) of Q and upper bound (2.12): (2.56) To justify our use of the upper bound, note that so the upper bound (2.12) is valid forū.
For probability (2.55), observe first that s →Ỹ n−1 (t − s, t) is a standard Brownian motion which is independent of B by construction. By introducing B † (s) = 1 where the upper bound follows by choosing ε small enough (for fixed b 0 , n 0 , θ 0 ). We will show that The first probability is P( For the second probability we note that by Dufresne's identity [26] the integral has the same distribution as the reciprocal of a Gamma(ν) random variable. Thus the second term is which proves the estimate (2.57).
Collecting everything we get that The case of −u < σ < 0 goes similarly, with small alterations. Now λ = θ +ν ≤ 3θ/2. Utilizing (2.50) and comparison (2.49) the ratio is developed as follows: The rest follows along the same lines as above. This proves (2.51).

Proofs for the KPZ equation
Proof of Theorem 1.6. We first start with the case ϕ = 0. From (2.5) one can verify that for some finite F (τ ). This implies that {log Z N (τ ) : N ≥ 1} is uniformly integrable, and hence E log Z N (τ ) → E log Z(τ, 0). Note that this could have been obtained by exact computations for the stochastic heat equation as well. Together with Theorem 1.5, this gives the convergence in law Fatou's lemma and Theorem 1.3 then give , for some C > 0. As for the lower bound, P log Z(τ ) − E log Z(τ ) ≥ cτ 1/3 = lim n P log Z n (τ ) − E log Z n (τ ) ≥ cτ 1/3 ≥ δ, for some constants c, δ > 0, and τ large enough, thanks to the lower bound in Proposition 2.3. We have proved that there exist some constant C > 0 such that 1 C τ 2 3 ≤ Var log Z(τ ) ≤ Cτ for τ large enough. We now turn to the case |ϕ| ≤ K for some 0 < K < +∞. From (1.22), we can verify that e −K Z ϕ N (τ ) ≤ Z N (τ ) ≤ e K Z ϕ N (τ ). This implies that Var log Z ϕ N (τ ) ≤ 8K 2 + 2Var log Z N (τ ), which in turns implies the uniform integrability of {log Z ϕ N (τ ) : N ≥ 1}. Fatou's lemma and the upper bound on Var log Z N (τ ) show that for some C ′ > 0 and τ > 0 large enough. The lower bound follows from Proposition 2.3 for suitable c, δ > 0 and all N and τ large enough. This completes the proof of the theorem.

Facts about the gamma function and Gamma distribution
We collect here some basic facts about the gamma function and the Gamma distribution.