On a Factorization Formula for the Partition Function of Directed Polymers

We prove a factorization formula for the point-to-point partition function associated with a model of directed polymers on the space-time lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}^{d+1}$$\end{document}Zd+1. The polymers are subject to a random potential induced by independent identically distributed random variables and we consider the regime of weak disorder, where polymers behave diffusively. We show that when writing the quotient of the point-to-point partition function and the transition probability for the underlying random walk as the product of two point-to-line partition functions plus an error term, then, for large time intervals [0, t], the error term is small uniformly over starting points x and endpoints y in the sub-ballistic regime \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert x - y \Vert \le t^{\sigma }$$\end{document}‖x-y‖≤tσ, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma < 1$$\end{document}σ<1 can be arbitrarily close to 1. This extends a result of Sinai, who proved smallness of the error term in the diffusive regime \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert x - y \Vert \le t^{1/2}$$\end{document}‖x-y‖≤t1/2. We also derive asymptotics for spatial and temporal correlations of the field of limiting partition functions.


Introduction
The theory of directed polymers has been actively studied in the mathematical and physical literature in the last 30 years.From the point of view of probability theory and statistical mechanics, directed polymers are random walks in a random potential.The probability distribution for a random path γ of length t is given by the Gibbs distribution , where β is the inverse temperature, H t ω (γ ) is the total energy of the interaction between the path γ and a fixed realization of the external random potential, and the normalizing factor Z t ω is the partition function.The random potential is a functional defined on some probability space, and a point ω in this probability space completely characterizes a fixed realization of the potential.In this paper we are interested only in the case of non-stationary time-dependent random potentials.The simplest setting corresponds to the discrete space-time lattice Z d+1 , where d is the spatial dimension.In this case the random potential normally is assumed to be given by the i.i.d.field ω = {ξ(x, i) : x ∈ Z d , i ∈ Z}, and H t ω = − t i=0 ξ(γ i , i).As usual one is interested in the asymptotic behavior of directed polymers as t → ∞.
The first rigorous results for directed polymers were obtained by Imbrie and Spencer [1], Bolthausen [2], and Sinai [3].It was proved that in the case of weak disorder, namely when d ≥ 3 and |β| is small, the polymer almost surely has diffusive behavior with a non-random covariance matrix.It was later proved by Carmona and Hu [4], and Comets et al. [5] that in the cases d = 1, 2, and d ≥ 3 with |β| large, the asymptotic behavior is very different.In this regime, called strong disorder, the directed polymers are not spreading as t → ∞ but remain concentrated in certain random places.
Sinai's approach in [3] is based on the study of asymptotic properties of partition functions Z t ω as t → ∞.It turns out that if the polymer starts at a point x at time s, then in the limit t → ∞ the properly normalized partition function converges almost surely to a random variable Z ∞ x,s .Here, in order to simplify notation, we are not indicating the dependence on ω.In a similar way one can consider backward in time partition functions, and prove that after the same normalization they also converge to limiting partition functions Z y,t −∞ , where (y, t) is the endpoint of the polymer.The proof of the diffusive behavior follows from a factorization formula proved by Sinai.Namely, a bridging partition function Z y,t x,s corresponding to the random-walk bridge between points (x, s), (y, t), t > s, satisfies the following asymptotic relation: where q y−x t−s is the transition probability of the simple symmetric random walk, and a small error term δ y,t x,s tends to zero as t − s → ∞, provided y − x belongs to the diffusive region: Later, Sinai's formula was extended by Kifer [6] to the continuous setting.
The interest in the asymptotic behavior of directed polymers is largely motivated by the connection between directed polymers and the theory of the stochastic heat equation and the random Hamilton-Jacobi equation which is related to the stochastic heat equation through the Hopf-Cole transformation (x, t) = − ln Z (x, t).The connection between directed polymers and the stochastic heat equation is a direct consequence of the Feynman-Kac formula, see, e.g., [7].
The main conjecture about the asymptotic behavior of the solutions to the random Hamilton-Jacobi equation can be formulated in the following way.For a fixed value of the average velocity b = ∇ (x, •) , which is preserved by the equation, with probability one there exists a unique (up to an additive constant) global solution.This means that solutions starting from two different initial conditions 1 (x, 0) = b • x + 1 (x, 0), 2 (x, 0) = b • x + 2 (x, 0) approach each other up to an additive constant as t → ∞, provided 1 (x, 0) and 2 (x, 0) are functions of sublinear growth in x [7].
In terms of the stochastic heat equation, a similar uniqueness statement up to a multiplicative constant conjecturally holds for two initial conditions of the form In order to be able to prove the above conjecture in the weak-disorder case, one has to extend the factorization formula (1) to a much larger scale.This is the purpose of the present article: We prove that the factorization formula holds for x − y < (t − s) σ , where σ can be taken arbitrarily close to 1. Compared to [3], such an extension of the factorization formula requires very different analytical methods.
In this paper we restrict ourselves to the simplest discrete case, i.e., polymers live on the discrete space-time lattice Z d+1 and the potential is induced by an i.i.d.field of random variables.This allows us to make the exposition more transparent.However, in order to prove the uniqueness conjecture for the stochastic heat equation in the weak-disorder regime, one needs to consider the parabolic Anderson model, which is discrete in space and continuous in time.The proof of the factorization formula in this semi-discrete setting, which is based on similar ideas but technically more involved, will be published elsewhere.The proof of the uniqueness conjecture itself will be published separately as well.
We conclude the introduction with several remarks: 1.The factorization formula can be extended to the full sub-ballistic regime x − y = o(t − s).We are considering a smaller region x − y < (t − s) σ , which allows for effective estimates of the smallness of the error term δ y,t x,s .2. We believe that a similar factorization formula can be proved in the fully continuous case.
For this, one would need to assume that the correlations of the disorder field {ξ(x, t) : x ∈ R d , t ∈ R} are decaying sufficiently fast.3. It is interesting to study the probability distribution for the limiting partition function Z := Z ∞ x,s and for = − ln Z .Although these probability distributions are not universal, we believe that the tail distributions have many universal features.We conjecture that in the case when the probability distribution of ξ has compact support, the left tail of the density for behaves like exp [− 1+d/2 ] and the right tail decays like exp [− 1+d ].A related conjecture concerns the moments m(l) := Z l of Z , which we conjecture to grow as exp [l 1+2/d ] in the limit l → ∞.If the disorder ξ is Gaussian, then for any β > 0 only a finite number of moments for Z is finite.Thus, one can expect exponential decay of the left tail for .4. The uniqueness of global solutions to the stochastic heat equation and the random Hamilton-Jacobi equation was also proved in dimension d = 1 [8][9][10].The mechanism leading to uniqueness in this case is completely different from our setting.We should also mention that the case d = 1 corresponds to the famous KPZ universality class.
The rest of this paper is organized as follows: In Sect. 2 we derive an expansion for the partition functions and convergence to limiting partition functions.This allows us to state our main result, the factorization formula for Z y,t x,s .We also derive asymptotics for both spatial and temporal correlations of the field of limiting partition functions.In Sect.3, we collect several estimates on transition probabilities for the simple symmetric random walk on Z d .Sections 4 and 5 are devoted to the proof of the factorization formula.Finally, in the appendix we prove the estimates on transition probabilities from Sect. 3.
Notation: Throughout this article the Euclidean norm and inner product in R d are denoted by • and • , respectively.The 1-norm in R d is denoted by • 1 .We simply write a ≡ b to indicate that a ≡ b (mod 2).For functions A and B, potentially of several variables, we write A B or A is dominated by B to denote that A ≤ cB for some constant c > 0. The constant c may depend on the dimension d, the inverse temperature β and the law of the disorder (e.g., through λ defined in ( 6)), or on scaling parameters such as σ from Theorem 3 or ξ from Sect.4.1.However, c is not allowed to depend on any time or space variables such as t and z.The same remark applies to every constant introduced in this paper.Finally, in order to simplify notation, we will write z to indicate that we are summing over all z = (z 1 , . . ., z r ) ∈ (Z d ) r , where the value of r will be clear from the context.

Setting and Main Result
Let γ = (γ n ) n∈Z be a discrete-time simple symmetric random walk on Z d , d ≥ 3, starting at point x ∈ Z d at time s ∈ Z, with corresponding probability measure P x,s and corresponding expectation E x,s .As d ≥ 3, γ is transient.For integers t > s and y ∈ Z d , we denote the probability measure obtained from P x,s by conditioning on the event {γ t = y} by P y,t x,s .The corresponding expectation is denoted by E y,t x,s .We also set q z t : = P 0,0 (γ t = z).Let (ξ(x, t)) x∈Z d ,t∈Z be a collection of i.i.d.random variables with corresponding probability measure Q and corresponding expectation • .These constitute the random potential in our setting.We assume that c(β) := e βξ(0,0) < ∞ for β > 0 sufficiently small.To a sample path of γ over a time interval [s, t], we assign the random action For integers s < t, x, y ∈ Z d , and inverse temperature β > 0, we define the random partition functions Z y,t x,s : = c(β) As shown in the proof of Theorem 2 in [3], Z y,t x,s can be written as Similarly, one obtains the expansions and

Convergence to Limiting Partition Functions
As in [3], define It is well known that q z t t − d 2 (see, e.g., [11] or Lemma 7).Therefore, as d ≥ 3, there is a constant C > 0 such that We also define The following convergence statement for partition functions corresponds to Theorem 1 in [3].
Theorem 1 For β so small that α d λ < 1, the following holds: As t → ∞, Z t x,s converges in L 2 (Q) to a limiting partition function Z ∞ x,s .
Remark 2 Due to symmetry, we also have that exists in the sense of L 2 (Q) for all y ∈ Z d and t ∈ Z.

Remark 3
As pointed out by Bolthausen [2], (Z t x,s ) t≥s is a martingale with respect to the filtration F t := σ (ξ(y, u) : s ≤ u ≤ t, y ∈ Z d ), so convergence to the limiting partition functions also holds Q-almost surely by the Martingale Convergence Theorem.

Proof of Theorem 1
We follow the approach in [3].The right-hand side of (4) has an orthogonality structure which we will exploit.Since h(z, s) and h(z , s ) are independent if z = z or if s = s , and since h(z, s) = 0, we have with Jensen's Inequality and Fubini's Theorem that (Z t x,s ) 2 is bounded from above by Since The following theorem gives a rate of convergence to the limiting partition function Z ∞ x,s , which is needed to prove the factorization formula in Theorem 3.
Theorem 2 For β so small that α d λ < 1 and for θ ∈ (0, Proof For an integer t ≥ s, let which is monotone increasing in t.Set Then, for t > s, The expression in (11) is dominated by and The expression in ( 10) is dominated by

Factorization Formula
The following factorization formula for the partition function Z y,t x,s with fixed starting and endpoint is the main result of this article.
Theorem 3 Let β be so small that α d λ < 1.For every σ ∈ (0, 1), no matter how close to 1, there exists θ = θ(σ ) > 0 such that for all x, y ∈ Z d and s < t with x − y < (t − s) σ , the partition function Z y,t x,s has the representation where the error term δ y,t x,s defined by the formula above satisfies Theorem 3 is proved in Sect. 4. Notice that the formula is similar to the ones obtained by Sinai in [3, Theorem 2] and Kifer in [6, Theorem 6.1].However, we show that the error term is small not only within the diffusive regime x −y < O(t −s) 1 2 , but also for x −y < (t −s) σ with σ arbitrarily close to 1.This extension beyond the diffusive regime is nontrivial because the error term in ( 12) is multiplied by the random-walk transition probability q y−x t−s , which is itself extremely small for x − y ≥ (t −s) 1 2 .In a forthcoming publication, we rely heavily on a continuous-time version of Theorem 3 to prove a uniqueness statement for global solutions to the semi-discrete stochastic heat equation.

Correlations for the Field of Limiting Partition Functions
As mentioned in Sect. 1, the distribution for the field of limiting partition functions (Z ∞ x,s ) x∈Z d ,s∈Z is an interesting object to study, with several important questions still open.Below, we state asymptotics for the spatial and temporal correlations of this field.
Theorem 4 Let β be so small that α d λ < 1.Then the spatial and temporal correlations for the field of limiting partition functions (Z ∞ x,s ) x∈Z d ,s∈Z have the following asymptotics.

lim
It is necessary to take the limit in part (1) along sequences (y n ) such that y n 1 ≡ 0 for all n, as Z ∞ 0,0 and Z ∞ y,0 are independent if y 1 ≡ 1.A similar observation applies to the limit in part (2).The proof of Theorem 4 relies on the following estimates for simple symmetric random walk on Z d , d ≥ 3.

Lemma 5
The following statements hold: where G denotes the Green's function for simple symmetric random walk on Z d .Theorem 4.3.1 in [11] implies that lim follows.To prove (2), first notice that for every even s ∈ N 0 , It is well known (see, e.g., [12,Chapter 1]) that Thus, for s even and ≥ 2T , Since was arbitrarily chosen, we obtain (2).

Proof of Theorem 4
For y ∈ Z d such that y 1 ≡ 0 and t ∈ N, the expansion in (4) along with the properties of h(z, s) yield Along the lines of the proof of Theorem 2, one can easily show that, as t → ∞, the expression on the right-hand side converges to Therefore, and part (1 ) follows from Lemma 5.The proof of part ( 2) is similar and we omit it.

3 Transition Probabilities for the Simple Symmetric Random Walk
In this section we collect several estimates on transition probabilities for the discrete-time simple symmetric random walk on Z d , some of which are presumably well known to specialists.Appendix A will be devoted to the proofs of the results presented here.Let (γ n ) n∈N 0 be a discrete-time simple symmetric random walk on Z d starting at the origin.
Lemma 6 There exist constants c 1 , c 2 > 0 such that the following holds: For every σ ∈ ( 3 4 , 1) and σ ∈ (σ, 1), there exists T ∈ N such that for every t ≥ T and y ∈ Z d with q y t > 0 and y ≤ t σ , Lemma 7 There exists c 1 > 0 such that for every y ∈ Z d and for every linear functional ϕ on In particular, Fix a linear functional ϕ on R d such that |ϕ(x)| ≤ x for every x ∈ R d .To simplify notation, we set ϕ j : = ϕ(e j ) for 1 ≤ j ≤ d, where {e j } is the standard basis in R d .Define for every θ where i is the imaginary unit.Notice that for every θ ∈ R d , where 0 is the zero vector in R d .Furthermore, Notice also that is 2π-periodic in every argument, so it will be convenient to work with the cube C: = (− π 2 , 3π 2 ] d .It is not hard to see that the inequality ( 16) is strict for all θ ∈ C except for θ 0 : = (0, . . ., 0) and θ 1 : = (π, . .

. , π).
Lemma 8 There exist ρ 1 , ρ 2 > 0 such that the following holds: For every t ∈ N and for every z ∈ Z d such that z ≤ ρ 1 t and q z t > 0, there exists a linear functional 5 ) q z t e ϕ(z) and q z t e ϕ(z) t − d 2 y∈Z d q y t e ϕ(y) .
Lemma 9 There exist constants ρ, c > 0 such that for every t, t ∈ N and z, z ∈ Z d with z ≤ ρt and q z t > 0, one has

Proof of Theorem 3
The main idea behind the factorization formula, which goes back at least to [3], is that there is strong averaging for times neither too close to s nor too close to t.
Consider the representation of Z y,t x,s in (2).For fixed r , i 1 , . . ., i r , and z 1 , . . ., z r , the random walk is pinned to the points z 1 , . . ., z r at the corresponding times i 1 , . . ., i r .The proof of Theorem 1 suggests that the contribution to Z y,t x,s from r on the order of (t − s) is negligible.If r is not on the order of (t − s), at least one of the gaps i j − i j−1 must be in some sense large (see Sect. 4.1).In Sect.4.2.2, we show that the contribution to Z y,t x,s coming from two or more large gaps is negligible as well.Thus, the main contribution comes from having exactly one large gap i j − i j−1 , which is then on the order of (t − s).In order for q z j −z j−1 i j −i j−1 to be positive, z j−1 must be close to x and z j must be close to y.The transition probability q z j −z j−1 i j −i j−1 is then close to q y−x t−s .Notice that to prove Theorem 3, it is enough to show that if α d λ < 1, then for every σ ∈ (0, 1) there exists θ > 0 such that lim t→∞ t θ sup This is because for a fixed realization ω of the disorder, δ y,t x,s (ω) can be written as , where ω is obtained by shifting ω in space and time.The distribution of the disorder is invariant under such shifts.
For t ∈ N 0 and r ∈ {1, . . ., t + 1}, let With this notation, the expansion in (2) becomes Z y,t where one should recall from Sect. 1 the notational shorthand z for summation over all z = (z 1 , . . ., z r ) ∈ (Z d ) r .The first step is to split the double sum into terms according to the size of the largest gap between indices, as discussed in Sect.4.1.

Large and Huge Gaps
If there exist σ ∈ (0, 1) and θ > 0 such that (13) (the convergence of the error term in the factorization formula) holds, then (13) also holds for σ and θ , where σ can be any value in (0, σ ).There is then no loss of generality in assuming that σ > 3/4, and one may even think of σ as being very close to 1.For a collection of indices 0 the gaps are the differences between consecutive indices, i.e., i To quantify what it means to have many gaps, we fix positive constants κ 1 , κ 2 ∈ ( 1 2 (3σ − 1), σ ) such that κ 1 < κ 2 .Let T κ 2 ∈ N be so large that 2(t − t κ 2 ) > t for all t ≥ T κ 2 .Then define Note that k(t) grows with t like t κ 1 .We say that a collection of indices 0 To classify the size of a gap between indices, fix another constant ξ such that 0 < ξ < min 1 − σ, κ 2 − κ 1 }.One should think of ξ as being very close to 0. Note that ξ + κ 1 < 1 and that ξ < κ 1 , the latter because of ξ < We say that the gap between two consecutive indices i j−1 and i j is Observe that the size of the largest gap is necessarily greater than t/(r + 1) ≥ t 1−κ 1 ≥ t ξ , so there is at least one large gap.A huge gap is necessarily large.If there is only one large gap, then all other gaps are of size less than t ξ , so this large gap is even huge.Thus, if there is no huge gap, there are at least two large ones.Since t must be greater than T κ 2 in order for k(t) ≥ 1 to hold, we have 2(t − rt ξ ) > 2(t − t κ 1 +ξ ) > 2(t − t κ 2 ) > t, so there can be at most one huge gap.Note, however, that a huge gap is not necessarily the only large one.
Let us introduce some more notation.Fix r ∈ N and t ∈ N 0 .For m ∈ N such that 1 ≤ m ≤ r + 1, define the following set of r -tuples: For t so large that k(t) ≥ 1, we decompose the expansion of Z y,t 0,0 in (18) as follows: where, B y,t With this decomposition in hand, Theorem 3 follows immediately from the following lemma.
Sections 4.2 and 4.3 are devoted to the proof of this lemma.

Proof of Part 2: No Huge Gaps
Let t ∈ N be so large that k(t) ≥ 1.Then where Now we estimate M t,r (y).Let r ∈ N such that 1 ≤ r ≤ k(t), and y ∈ Z d such that y ≤ t σ and q y t > 0. Given i = (i 1 , . . ., i r ) ∈ I 2 (t, r ) such that i 1 = 0 and i r = t, set t 1 : = i 1 , t 2 : = i 2 −i 1 , . . ., t r : = i r −i r −1 , t r +1 : = t−i r .And given z = (z 1 , . . ., z r ) ∈ (Z d ) r , set x 1 : = z 1 , x 2 : = z 2 − z 1 , . . ., x r : = z r − z r −1 , x r +1 : = y − z r .This change of variables yields For i ∈ I 2 (t, r ), there is no huge gap and hence there are at least two large ones.Let i.e., (l + 1) gives the number of large gaps in i.There are (r + 1) possible slots for the largest gap (which is then also a large gap), and r l possible slots for the other l large gaps once the largest gap has been fixed.Together with (24), this yields the estimate where M t,r ,l (y) := The sum on the right-hand side is taken over all t 1 , . . ., t r +1 ∈ N and x 1 , . . ., x r +1 ∈ Z d that satisfy the four conditions under the summation sign.In the special case l = r , the fourth condition is void.
For given positive integers t l+1 , . . ., t r strictly less than t ξ , set and for x l+1 , . . ., x r ∈ Z d , set If l < r , this lets us write where M t t,r ,l (x ): We now search for a bound for M t t,r ,l (x ) when t is sufficiently large.
Claim 4.1 There exist constants C, C , T > 0 such that for every integer t ≥ T and for all r , l, t , x as above, and, in the special case l = r, M t,r ,r (y) t −ξ/4 (q We use Claim 4.1 to estimate M t,r ,l (y) from (26) as follows: Then we combine the above estimate with (25) to obtain Finally, combining this estimate with ( 22) and (23), we obtain Fix t ∈ N so large that k(t) ≥ 1 as well as t σ ≤ ρ 1 t and ρ 2 t σ −1 ≤ 1.Let y ∈ Z d such that y ≤ t σ and q y t > 0. The conditions ρ 2 t σ −1 ≤ 1 and y ≤ t σ imply in particular that ϕ ≤ 1 for the linear function ϕ corresponding to t and y.Let t 1 , . . ., t l , t r +1 ∈ N and x 1 , . . ., x l , x r +1 ∈ Z d such that the conditions under the summation sign in (27) hold.In the special case l = r , replace t and x with t and y, respectively, here and in the remainder of the proof.By Lemma 7, there exists a constant c 1 > 0 such that , where in the third line we used the fact that ϕ is a linear functional, and in the fourth line we used (17), where was defined in (15).Since t < t and (0) ≥ 1, it follows from (28) that (0) t ≤ (0) t t d/2 q y t e ϕ(y) .As a result, for all positive integers t 1 , . . ., t l , t r +1 such that t 1 + • • • + t l + t r +1 = t and t r +1 ≥ t 1 , . . ., t l ≥ t ξ , one has max Furthermore, the sum t r+1 over all tuples (x 1 , . . ., x l , x r +1 ) such that t , and by Lemma 9 there exist constants c, ρ > 0 such that for t so large that t σ ≤ ρt.Therefore, , where P(t): In the second line of the estimate above, we also used that ϕ ≤ ρ 2 y /t.
Together with Lemma 14 from the appendix, we obtain the following estimate on M t t,r ,l (x ): where C > 0 is a constant.It remains to bound (t/t ) d/2 P(t).We estimate the following expressions involved in (t/t ) d/2 P(t) like so: where r j=l+1 x j ≤ r j=l+1 t j is valid under the assumption that q x 1 t 1 . . .q x r+1 t r+1 > 0.Then, using y ≤ t σ , we obtain for some constant C > 0. This completes the proof of Claim 4.1.

Proof of the Central Lemma, Part 3: The Main Contribution
Let t be so large that k(t) ≥ 1 and let y ∈ Z d such that y ≤ t σ and q y t > 0. For i ∈ I 1 (t, r , m) and z ∈ (Z d ) r , define where the factor with the hat is absent; in other words, we set the transition probability corresponding to the huge gap equal to 1.

Now decompose B y,t
3 further, depending on the position of the huge gap 1) at the begining, 2) in the middle, or 3) at the end, as follows: (30) and the error terms are given by L y,t Notice that F y,t are well-defined even if q y t = 0. We first show that the contribution from each error term is negligible.

Lemma 11
There exists θ > 0 such that Proof It is enough to show that there exists θ > 0 such that for i ∈ {1, 2, 3}, lim t→∞ t θ sup y: y ≤t σ ,q For t so large that k(t) ≥ 1 and for y ∈ Z d such that y ≤ t σ and q y t > 0, one has λ r a i (r ) , where a i (r ): and where the sum x is taken over all being close to q y t in the following sense: Let ρ > 0 be the constant from Lemma 9, and assume that t is so large that t Using this claim we can bound sup y: y ≤t σ ,q y t >0 (L Let θ ∈ (0, min{2/5, 1 − σ − ξ }).The definition of the Landau symbol O(t − 2 5 ) implies that there exist constants C, T > 0 such that for every t > T , where, in the third line, we used that (e x − 1)/x ≤ e x for every x > 0. Hence, For ϕ ∈ (α d λ, 1) and t so large that α d λ exp(c 3 t σ +ξ −1 ) ≤ ϕ, the series on the right is dominated by the convergent series ϕ r (c 3 r 2 + Cr).Dominated convergence and (32) then imply (31) for θ ∈ (0, max{2/5, 1 − σ − ξ }).
To complete the proof of Lemma 11, it remains to prove Claim 4.2.

Proof of Claim 4.2 Let
which is of smaller order than t − t .Finally, t − t and y − x 1 have the same parity because q y t > 0 and q x 1 t 1 . . .q x r t r > 0. Now, we derive an upper bound on |q for some constant c 1 > 0. If q y t > q y−x t−t , we argue as follows: Let t ∈ N be so large that Using that (a − 1) 2 ≤ a 2 − 1 for every a ≥ 1, in either case (33) or (34), we have the following bound: where c 3 > 0 is a constant.
In order to deal with the F i 's defined in (30), the strategy is to first define suitable truncations of the partition functions.Fix ξ 1 , ξ 2 satisfying 0 < ξ 1 < ξ 2 < ξ, and notice that since ξ + σ < 1, we have ξ 1 + σ < 1.Now set and T y,t where q y t, r +1 (i, z) and q y t, 1(i, z) are defined according to (29), with q y t, r +1 (i, z) not depending on y.Notice that T t 0,0 and T y,t 0 are truncations of the partition functions Z t 0,0 and Z y,t 0 , respectively (see ( 4) and ( 3)).The convergence statement in (21) will follow from the lemmas below.

Main Contribution
Define the set Recall the notation q y t, m (i, z) from (29).Making the change of summation indices r : = r + s and m: = r + 1 in (39), one has The In order to prove (35), it is then enough to show existence of θ > 0 such that for i = 1, 2, 3, For i = 1, 3, one has ( f y,t where the expression in (42) is dominated by This implies (41) for θ < (ξ 2 − ξ 1 )( d 2 − 1).Case i = 2.The expression in (42) is dominated by From this estimate we deduce (41) for every θ > 0. Case i = 3.The expression in (42) is dominated by which converges to 0 as t → ∞ faster than any polynomial by the same argument as in the case i = 2.

Proof of Lemma 12, (36) and (37): Convergence for One Huge Gap at the Start or the End
We only show the convergence statement in (37) as the proof of (36) is analogous.Write F y,t where for i = 1, 2, and , H 2 r : = I 1 (t, r , r + 1).For i = 1, 2, one has Convergence in the cases i = 1 and i = 2 works then as in the proof of (35).

Proof of Lemma 13: Convergence to Limiting Partition Functions
Let us first show that the truncated partition function T t 0,0 converges to the limiting partition function Z ∞ 0,0 in the L 2 sense and obtain a rate of convergence.We will prove that there exists θ > 0 such that One has , where It is then enough to show existence of θ > 0 such that We have ) holds for i = 2 and for every θ > 0.Moreover, 2 , so (44) holds for i = 1 and θ ∈ (0, (ξ 2 − ξ 1 )( d 2 − 1)).This implies (43).Combining (43) with Theorem 2, one obtains in particular that there exists θ > 0 such that To complete the proof of Lemma 13, notice that Therefore, we obtain the desired result by applying the Cauchy-Schwarz Inequality to the two summands on the right-hand side, and using (45) together with the conditions of Theorem 2.3.11 in [11].Thus, there exists ρ > 0 such that for every i ∈ N and z ∈ Z d satisfying z < ρi, we have Now, fix σ ∈ ( 3 4 , 1), σ ∈ (σ, 1), and let T ∈ N be so large that 1 + t σ < (t − 1) σ and t σ < ρ 2|||L||| t for every t ≥ T , where |||L||| is the operator norm of L. We distinguish between two cases: t is either even or odd.Even case.If t = 2m for some m ∈ N, then we can prove a slightly stronger statement: Claim A.1 There exist constants c 1 , c 2 > 0, independent of σ and σ , such that (14) holds for every even t ≥ T and y ∈ Z d with q y t > 0 and y ≤ t σ .
The difference to the conclusion of Lemma 6 is that the estimate holds for y ≤ t σ and not just for y ≤ t σ .To prove this claim, fix t = 2m ≥ T and y ∈ Z d such that q y 2m > 0 and y ≤ (2m) σ .Then q y 2m = q Ly m .Since Ly ≤ |||L||| y ≤ |||L|||t σ < ρm, one has Odd case.Now, suppose t = 2m +1 ≥ T for some m ∈ N. Fix y ∈ Z d such that q y 2m+1 > 0 and y ≤ (2m + 1) σ .Let E be the set of standard unit vectors in R d and their additive inverses.Then Since y − z < 1 + t σ < (t − 1) σ = (2m) σ and q y−z 2m > 0 for every z ∈ E, then using Claim A.1, we can bound q y 2m+1 from below as follows: There exist c 1 , c 2 > 0 such that In addition to t ≥ T , assume that t is so large that exp − d 2 Plugging this into the right-hand side of (A1), we obtain the desired estimate.
Then we find 1 (2π) d C (θ) t dθ − q z t e ϕ(z) ≤ j∈{0,1} We now estimate the expression on the right-hand side.First, we show that the linear functional ϕ can be chosen in such a way that for j ∈ {0, 1}, The idea is to choose ϕ as a function of z and t in such a way that the linear term in the Taylor expansion of (θ)e −iz•θ/t around θ j vanishes, i.e., (θ j )∇ e −iz•θ j /t + e −iz•θ j /t ∇ (θ j ) = 0.

: Proofs of Lemmas 12 and 13 5 . 1
Proof of Lemma 12, (35): Convergence for One Huge Gap in the Middle One has

d l=1 cosh(ϕ l ) , 1 ≤
k ≤ d, by virtue of (A2).Let F : R d → R d be given by F(x 1 , . . ., x d ): = d k=1 sinh(x k ) d l=1 cosh(x l ) e k .For r > 0 and x ∈ R d , let B r (x) denote the open Euclidean ball of radius r centered at x. Since F(0) = 0 and det D F(0) = 1 d d = 0, the Inverse Function Theorem yields existence of ρ 1 > 0 and an open neighborhood U of 0 such that F : U → B ρ 1 (0) is a diffeomorphism.Therefore, for every t ∈ N and z ∈ Z d