Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus

We study an optimal mass threshold for normalizability of the Gibbs measures associated with the focusing mass-critical nonlinear Schr\"odinger equation on the one-dimensional torus. In an influential paper, Lebowitz, Rose, and Speer (1988) proposed a critical mass threshold given by the mass of the ground state on the real line. We provide a proof for the optimality of this critical mass threshold. The proof also applies to the two-dimensional radial problem posed on the unit disc. In this case, we answer a question posed by Bourgain and Bulut (2014) on the optimal mass threshold. Furthermore, in the one-dimensional case, we show that the Gibbs measure is indeed normalizable at the optimal mass threshold, thus answering an open question posed by Lebowitz, Rose, and Speer (1988). This normalizability at the optimal mass threshold is rather striking in view of the minimal mass blowup solution for the focusing quintic nonlinear Schr\"odinger equation on the one-dimensional torus.

and Bourgain [6]. A focusing Gibbs measure ρ is a probability measure on functions / distributions with a formal density: where Z denotes the partition function and the Hamiltonian functional H(u) is given by The NLS equation: i∂ t u + ∆u + |u| p−2 u = 0, (1.2) generated by the Hamiltonian functional H(u), has been studied extensively as models for describing various physical phenomena ranging from Langmuir waves in plasmas to signal propagation in optical fibers [44,21,1]. Furthermore, the study of the equation (1.2) from the point of view of the (non-)equilibrium statistical mechanics has received wide attention; see for example [26,6,7,8,47,48,25,9,12]. See also [4] for a survey on the subject, more from the dynamical point of view. Our main goal in this paper is to study the construction of the focusing Gibbs measures on the one-dimensional torus T = R/Z and the two-dimensional unit disc D ⊂ R 2 (under the radially symmetric assumption with the Dirichlet boundary condition) and determine optimal mass thresholds of their normalizability in the critical case. In particular, we resolve an issue in the Gibbs measure construction on T [26, Theorem 2.2] and also answer a question posed by Bourgain and Bulut [9, Remark 6.2] on the optimal mass threshold for the focusing Gibbs measure on the unit disc D. Furthermore, in the case of the one-dimensional torus, we prove normalizability at the optimal mass threshold in spite of the existence of minimal mass blowup solution to NLS at this mass, thus answering an open question posed by Lebowitz, Rose, and Speer [26]. We first go over the case of the one-dimensional torus. Consider the mean-zero Brownian loop u on T, defined by the Fourier-Wiener series: where {g n } n∈Z\{0} denotes a sequence of independent standard complex-valued 1 Gaussian random variables. Then, the law µ 0 of the mean-zero Brownian loop u in (1.3) has the formal density given by The main difficulty in constructing the focusing Gibbs measures comes from the unboundedness-from-below of the Hamiltonian H(u). This makes the problem very different from the defocusing case, which is a well studied subject in constructive Euclidean 1 Namely, Re gn and Im gn are independent real-valued mean-zero Gaussian random variables with vari- quantum field theory. In [26], Lebowitz, Rose, and Speer proposed to consider the Gibbs measure with an L 2 -cutoff: 2 dρ = Z −1 p,K e 1 p´T |u| p dx 1 { u L 2 (T) ≤K} dµ 0 and claimed the following results.
Theorem 1.1. Given p > 2 and K > 0, define the partition function Z p,K by where E µ 0 denotes an expectation with respect to the law µ 0 of the mean-zero Brownian loop (1.3). Then, the following statements hold: (i) (subcritical case) If 2 < p < 6, then Z p,K < ∞ for any K > 0.
(ii) (critical case) Let p = 6. Then, Z 6,K < ∞ if K < Q L 2 (R) , and Z 6,K = ∞ if K > Q L 2 (R) . Here, Q is the (unique 3 ) optimizer for the Gagliardo-Nirenberg-Sobolev inequality on R such that Q 6 L 6 (R) = 3 Q ′ 2 L 2 (R) . There remains a question of normalizability at the optimal threshold K = Q L 2 (R) in the critical case (p = 6). We address this issue in Subsection 1.2.
Lebowitz, Rose, and Speer proved the non-normalizability result for K > Q L 2 (R) in Theorem 1.1 (ii) by using a Cameron-Martin-type theorem and the following sharp Gagliardo-Nirenberg-Sobolev (GNS) inequality on R d : with d = 1 and p = 6. See Section 3 for a further discussion on the sharp GNS inequality.
The threshold value p = 6 and the relevance of the GNS inequality can be understood at an intuitive level by formally rewriting (1.4) as a functional integral with respect to the (periodic) Gaussian free field (= the mean-zero Brownian loop in (1.3)): 2´T |u ′ (x)| 2 dx+ 1 p´T |u(x)| p dx du. (1.6) Applying the GNS inequality (1.5), this quantity is bounded bŷ Thus, when p < 6 or when p = 6 and K is sufficiently small, we expect the Gaussian part of the measure to dominate, and hence the partition function to be finite. Regarding the construction of the focusing Gibbs measure, a pleasing probabilistic proof of Theorem 1.1 based on this idea was given in [26], using the explicit joint density of the times that the Brownian path hits certain levels on a grid. Unfortunately, as pointed out by Carlen, Fröhlich, and Lebowitz [12, p. 315], there is a gap in the proof of Theorem 2.2 in [26]. More precisely, the proof in [26] seems to apply only to the case, where the expectation in the definition of Z p,K is taken with respect to a standard ("free") Brownian motion started at 0, rather than the random periodic function (1.3).
Subsequently, a more analytic proof due to Bourgain appeared in [6], establishing normalizability of the focusing Gibbs measure (i.e. Z p,K < ∞) for (i) 2 < p < 6 and any K > 0 and for (ii) p = 6 and sufficiently small K > 0. His argument combines basic estimates for Gaussian vectors with the Sobolev embedding to identify the tail behavior of the random variable´T |u| p dx, subject to the condition u L 2 (T) ≤ K. It also applies to the case p = 6, but shows only that Z 6,K < ∞ for sufficiently small K > 0.
As the first main result in this paper, we obtain the optimal threshold when p = 6 claimed in Theorem 1.1 (ii) by proving Z 6,K < ∞ for any K < Q L 2 (R) . In particular, our argument resolves the issue in [26] mentioned above. Our proof is closer in spirit to Bourgain's, since it uses the series representation (1.3) of the Brownian loop, as opposed to the path space approach taken in [26]. In Section 2, we go over Bourgain's argument and point out that, in this approach, closing the gap between small K and the optimal threshold seems difficult. We then present our proof of the direct implication of Theorem 1.1 (ii) in Subsection 4.1. As in [26], the idea is to make rigorous the computation suggested by (1.6) by a finite dimensional approximation. As seen in [6], µ is a more natural base Gaussian measure to consider for the nonlinear Schrödinger equations, due to the lack of the conservation of the spatial mean under the dynamics.
We also point out that Theorem 1.1 also holds in the real-valued setting. The same comment applies to Theorems 1.3 and 1.4. For example, this is relevant to the study of the generalized KdV equation (gKdV) on T: Our method also applies to the focusing Gibbs measures on the two-dimensional unit disc D ⊂ R 2 , under the radially symmetric assumption with the Dirichlet boundary condition. In the subcritical case (p < 4), Tzvetkov [47] constructed the focusing Gibbs measures, along with the associated invariant dynamics. His analysis was complemented in [9] by a study of the critical case p = 4, under a small mass assumption. See also [47,48,9] for results in the defocusing case.
Our approach to Theorem 1.1 allows us to establish the optimal mass threshold in the critical case (p = 4), thus answering the question posed by Bourgain and Bulut in [9,Remark 6.2]. We first introduce some notations. Let D = {(x, y) ∈ R 2 : x 2 + y 2 < 1} be the unit disc. Let J 0 (r) be the Bessel function of order zero, defined by , and z n , n ≥ 1, be its successive, positive zeros. Then, it is known [47] that {e n } n∈N defined by e n (r) = J 0 (z n ·) −1 L 2 (D) J 0 (z n r), 0 ≤ r ≤ 1, forms an orthonormal basis of L 2 rad (D), consisting of the radial eigenfunctions of the Dirichlet self-adjoint realization of −∆ on D. Here, L 2 rad (D) denotes the subspace of L 2 (D), consisting of radial functions. Now, consider the random series: z n e n (r), r 2 = x 2 + y 2 , (1.9) where {g n } n∈N is a sequence of independent standard complex-valued Gaussian random variables.
Theorem 1.3. Given p > 2 and K > 0, define the partition function Z p,K by

10)
where E denotes an expectation with respect to the law of the random series (1.9). Then, the following statements hold: (i) (subcritical case) If p < 4, then Z p,K < ∞ for any K > 0.
(ii) (critical case) Let p = 4. Then, Z 4,K < ∞ if K < Q L 2 (R 2 ) , and Z 4,K = ∞ if K > Q L 2 (R 2 ) , where Q is the optimizer in the Gagliardo-Nirenberg-Sobolev inequality (1.5) on R 2 such that Q 4 L 4 (R 2 ) = 2 ∇Q 2 L 2 (R 2 ) . Part (i) of Theorem 1.3 is due to Tzvetkov [47]. In [9], Bourgain and Bulut considered the critical case (p = 4) and proved Z 4,K < ∞ if K ≪ 1, and Z 4,K = ∞ if K ≫ 1, leaving a gap. Theorem 1.3 (ii) answers the question posed by Bourgain and Bulut in [9]. See Remark 6.2 in [9]. We present the proof of Theorem 1.3 (ii) in Subsection 4.2 and Section 5. In Subsection 4.2, we prove Z 4,K < ∞ for the optimal range K < Q L 2 (R 2 ) by following our argument for Theorem 1.1 on T. We point out that some care is needed here due to the growth of the L 4 -norm of the eigenfunction e n ; see (4.15). In Section 5, we prove Z 4,K = ∞ for K > Q L 2 (R 2 ) . Our argument of the non-normalizability follows closely that on T by Lebowitz, Rose, and Speer [26]. 1.2. Integrability at the optimal mass threshold. We now consider the normalizability issue of the focusing Gibbs measure on T in the critical case (p = 6) at the optimal threshold K = Q L 2 (R) , at which a phase transition takes place. Before doing so, let us first discuss the situation for the associated dynamical problem, namely, the focusing quintic NLS, (1.2) with p = 6. On the real line, the optimizer Q for the sharp Gagliardo-Nirenberg-Sobolev inequality (1.5) is the ground state for the associated elliptic problem (see (3.3) below). Then, by applying the pseudo-conformal transform to the solitary wave solution Q(x)e 2it , we obtain the minimal mass blowup solution to the focusing quintic NLS on R. Here, the minimality refers to the fact that any solution to the focusing quintic NLS on R with u L 2 (R) < Q L 2 (R) exists globally in time; see [49].
In [34], Ogawa and Tsutsumi constructed an analogous minimal mass blowup solution u * with u * L 2 (T) = Q L 2 (R) to the focusing quintic NLS on the one-dimensional torus T. It was also shown that, as time approaches a blowup time, the potential energy 1 6´| u * | 6 dx tends to ∞. In view of the structure of the partition function Z 6,K= Q L 2 (R) in (1.4), this divergence of the potential energy seems to create a potential obstruction to the construction of the focusing Gibbs measure in the current setting. In [8], Bourgain wrote "One remarkable point concerning the normalizability problem for Gibbs-measures of NLS in the focusing case is its close relation to blowup phenomena in the classical theory. Roughly speaking, this may be understood as follows. After normalization, the measure would be forced to live essentially on "blowup data", which however is incompatible with the invariance properties under the flow." In spite of the existence of the minimal mass blowup solution, we prove that the focusing critical Gibbs measure is normalizable at the optimal threshold K = Q L 2 (R) .
In view of the discussion above, Theorem 1.4 was unexpected and is rather surprising. Theorem 1.4 answers an open question posed by Lebowitz, Rose, and Speer in [26]. See Section 5 in [26]. Moreover, together with Theorem 1.1 (ii), Theorem 1.4 shows that the partition function Z 6,K is not analytic in the cutoff parameter K, thus settling another question posed in [26,Remark 5.2]. Compare this with the subcritical case (p < 6), where the analyticity result of the partition function on the parameters (including the inverse temperature, which we do not consider here) was proved by Carlen, Fröhlich, and Lebowitz [12] (for slightly different Gibbs measures).
The proof of Theorem 1.4 is presented in Section 6 and constitutes the major part of this paper, involving ideas and techniques from various branches of mathematics: probability theory, functional inequalities, elliptic PDEs, spectral analysis, etc. We break the proof into several steps: (1) In the first step, we use a profile decomposition and establish a stability result for the GNS inequality (1.5); see Lemma 6.2. When combined with our proof of Theorem 1.1, this stability result shows that if the integration is restricted to the complement of U ε = {u ∈ L 2 (T) : u − Q L 2 (T) < ε} for suitable ε ≪ 1, then the resulting partition function is finite. (In fact, we must exclude a neighborhood of the orbit of the ground state Q under translations, rescalings, and rotations, but we ignore this technicality here.) Thus, the question is reduced to the evaluation of the functional integralˆU (1.11) in the neighborhood U ε of (the orbit of) the ground state Q, where H(u) is as in (1.1) with p = 6. (2) In the second step, we show that when ε > 0 is sufficiently small, i.e. when U ε lies in a sufficiently small neighborhood of the soliton manifold M = e iθ Q δ,x 0 : 0 < δ < δ * , x 0 ∈ T, and θ ∈ R , where Q δ,x 0 denotes the dilated and translated ground state (see (6.6)), we can endow U ε with an orthogonal coordinate system in terms of the (small) dilation parameter 0 < δ < δ * , the translation parameter x 0 ∈ T, the rotation parameter θ ∈ R/(2πZ), and the component v ∈ L 2 (T) orthogonal to the soliton manifold M. See Propositions 6.3 and 6.7. (3) We then introduce a change-of-variable formula and reduce the integral (1.11) to an integral in δ, x 0 , θ, and v. See Lemma 6.8. (4) In Subsection 6.6, we reduce the problem to estimating a certain Gaussian integral with the integrand given by for some small η > 0. Here, A = A(δ, η) denotes the operator on H 1 (T) defined in (6.189): is as in (6.177). See also (6.46). We point out that the operator A is closely related to the second variation δ 2 H of the Hamiltonian. See Lemma 6.14. In view of the compactness of the operator A, the issue is further reduced to estimating the eigenvalues of 1 2 id +(1 − η 2 )A. Subsection 6.7 is devoted to the spectral analysis of the operator A.
For readers' convenience, we present the summary of the proof of Theorem 1.4 in Subsection 6.8.
We now state a dynamical consequence of Theorems 1.1 and 1.4.
Corollary 1.5. Let p = 6. Consider the Gibbs measure ρ with the formal density where µ is the law of the Ornstein-Uhlenbeck loop in (1.7). If K ≤ Q L 2 (R) , then the focusing quintic NLS, (1.2) with p = 6, on T is almost surely globally well-posed with respect to the Gibbs measure ρ. Moreover, the Gibbs measure ρ is invariant under the NLS dynamics. By imposing that the Ornstein-Uhlenbeck loop in (1.7) is real-valued (i.e. g −n = g n , n ∈ Z) or by replacing µ in (1.12) with the law µ 0 of the real-valued mean-zero Brownian loop in (1.3) (with g −n = g n , n ∈ Z \ {0}), a similar result holds for the focusing quintic generalized KdV, (1.8) with p = 6.
Strictly speaking, in the case of gKdV, the invariance is known only under the gauged dynamics (which is needed to prove local well-posedness in [13]) at this point. We, however, expect that the invariance of the Gibbs measure also holds for the (ungauged) gKdV; see [39].
When K < Q L 2 (R) , Corollary 1.5 follows from the deterministic local well-posedness results for the quintic NLS [5] and the quintic gKdV [13] in the spaces containing the support of the Gibbs measure, combined with Bourgain's invariant measure argument [6]. When K = Q L 2 (R) , the density e 1 6´T |u| 6 dx 1 { u L 2 (T) ≤K} is only in L 1 (µ) and thus Bourgain's invariant measure argument is not directly applicable. In this case, however, the desired claim follows from the corresponding result for K = Q L 2 (R) − ε, ε > 0, and the dominated convergence theorem by taking ε → 0. Remark 1.6. (i) A result analogous to Theorem 1.4 presumably holds for the twodimensional radial Gibbs measure on D studied in Theorem 1.3. Moreover, we expect the analysis on D to be slightly simpler since the problem on D has fewer symmetries (in particular, no translation invariance). In order to limit the length of this paper, however, we do not pursue this issue here.
(ii) In [37], Quastel and the first author constructed the focusing Gibbs measure conditioned at a specified mass, provided that the mass is sufficiently small in the critical case (p = 6). This answered another question posed in [26]. See also [12,10]. However, the argument in [37], based on Bourgain's approach, is not quantitative. Thus, it would be of interest to further investigate this problem to see if the focusing Gibbs measure conditioned at a specified mass in the critical case (p = 6) can be indeed constructed up to the optimal mass threshold as in Theorem 1.4. Remark 1.7. In view of the minimal mass blowup solution to NLS, the normalizability of the focusing critical Gibbs measure at the optimal threshold K = Q L 2 (R) in Theorem 1.4 was somehow unexpected. As an afterthought, we may give some reasoning for this phenomenon, referring to certain properties of the minimal mass blowup solution on the real line (which are not known in the periodic setting).
The first result is the uniqueness / rigidity of the minimal mass blowup solution on R due to Merle [30], which states that if an H 1 -solution u with u L 2 (R) = Q L 2 (R) to the focusing quintic NLS on R blows up in a finite time, it must be the minimal mass blowup solution up to the symmetries of the equation. See also an extension [27] of this result for rougher H s -solution, s > 0, which holds only on R d for d ≥ 4 under the radial assumption. While an analogous result is not known in the periodic setting (and in low dimensions), these results may indicate non-existence of rough blowup solutions at the critical threshold K = Q L 2 (R) . Theorem 1.4 shows that this non-existence claim holds true probabilistically.
Another point is instability of the minimal mass blowup solution. The minimal mass blowup solution is intrinsically unstable because a mass subcritical perturbation leads to a globally defined solution. See also [32]. Such instability may be related to the fact that we do not see (rough perturbations of) the minimal mass blowup solution probabilistically Lastly, we mention the situation for the focusing quintic gKdV on the real line. While finite time blowup solutions to gKdV "near" the ground state are known to exist [31,28], it is also known that there is no minimal mass blowup solution to the focusing quintic gKdV on the real line [29]. Thus, from the gKdV point of view, the normalizability in Theorem 1.4 is perhaps naturally expected (but we point out that analogues of the results in [31,28,29] are not known on T). Remark 1.8. While the construction of the defocusing Gibbs measures has been extensively studied and well understood due to the strong interest in constructive Euclidean quantum field theory, the (non-)normalizability issue of the focusing Gibbs measures, going back to the work of Lebowitz, Rose, and Speer [26] and Brydges and Slade [11], is not fully explored. See related works [43,9,12,35,40,36,46] on the non-normalizability (and other issues) for focusing Gibbs measures. In particular, recent works such as [35,36] employ the variational approach due to Barashkov and Gubinelli [2] and establish certain phase transition phenomena. Let us conclude this introduction by quoting Bourgain [8]: "It seems worthwhile to investigate this aspect [the (non-)normalizability issue of the focusing Gibbs measures] more as a continuation of [26] and [11] ." 1.3. Notations. We write A B to denote an estimate of the form A ≤ CB for some C > 0, Similarly, we write A ∼ B to denote A B and B A and use A ≪ B when we have A ≤ cB for some small c > 0. We may use subscripts to denote dependence on external parameters; for example, A p B means A ≤ C(p)B, where the constant C(p) depends on a parameter p.
In the following, we deal with complex-valued functions viewed as elements in real Hilbert and Banach spaces. In particular, with M = T, D, or R, the inner product on H s (M ) is given by (1.13) Note that with the inner product (1.13), the family {e 2πinx } n∈Z does not form an orthonormal basis of L 2 (T). Instead, we need to use {e 2πinx , ie 2πinx } n∈Z as an orthonormal basis of L 2 (T). A similar comment applies to the case of the unit disc D. We point out that the series representations such as (1.3) are not affected by whether we use the inner product (1.13) with the real part or that without the real part. For example, in (1.3), we have g n e 2πinx = (Re g n )e 2πinx + (Im g n ) ie 2πinx .
Here, the right-hand side is more directly associated with the the inner product (1.13), while the left-hand side is associated with the inner product without the real part. Given N ∈ N, we denote by π N the Dirichlet projection (for functions on T) onto frequencies {|n| ≤ N }: (1.14) and we set We also define π =0 to be the orthogonal projection onto the mean-zero part of a function: (1.16) and set π 0 = id −π =0 . Given k ∈ Z ≥0 := N ∪ {0}, let P k be the Littlewood-Paley projection onto frequencies of order 2 k defined by (1.17) Similarly, set Given measurable sets A 1 , . . . , A k , we use the following notation: where E denotes an expectation with respect to a probability distribution for u under discussion. This paper is organized as follows. In Section 2, we review Bourgain's argument from [6], which will be used in our proof of the direct implication of Theorem 1.1 (ii) in Subsection 4.1.
In Section 3, we go over the Gagliardo-Nirenberg-Sobolev inequality (1.5) on R d and discuss its variants on T and D. In Section 4, we then establish the direct implications of Theorem 1.1 (ii) on the one-dimensional torus T (Subsection 4.1) and Theorem 1.3 (ii) on the unit disc D (Subsection 4.2). In Section 5, we prove the non-normalizability claim in Theorem 1.3 (ii). Finally, we prove normalizability of the focusing critical Gibbs measure at the optimal mass threshold (Theorem 1.4) in Section 6.

Review of Bourgain's argument
In this section, we reproduce Bourgain's argument in [6] for the proof of Theorem 1.1 (i). Part of the argument presented below will be used in Subsection 4.1. Let 2 < p ≤ 6 and u denote the mean-zero Brownian loop u in (1.3). Rewriting (1.4) as we see that it suffices to show that there exist C > 0 and sufficiently large c > 0 such that for all sufficiently large λ ≫ 1. Given k ∈ Z ≥0 , we set u k = P k u, u ≤k = P ≤k u, and u ≥k = P ≥k u, where P k , P ≤k , and P ≥k are as in (1.17), (1.18), and (1.19). By subadditivity, we have for any k: where {λ j } ∞ j=k is a sequence of positive numbers such that Then, by using Sobolev's inequality in the form of Bernstein's inequality, we have Thus, with (1.3), the probability on the right-hand side of (2.2) is bounded by where C 0 = (2π) −1 C. The next lemma follows from a simple calculation involving moment generating functions of Gaussian random variables. See for example [38] for a proof.
Lemma 2.1. Let {X n } n∈N be independent standard real-valued Gaussian random variables. Then, we have By applying Lemma 2.1, we can bound the probability (2.5) by By choosing λ j = λ(1 − 2 −r )2 kr 2 −jr for 0 < r < 1 p , both conditions (2.7) and (2.3) are satisfied for all large k (and j ≥ k). For such k, the probability in (2.6) is then bounded by Summing over j ≥ k in (2.2), we find that By applying Bernstein's inequality again with the restriction u L 2 (T) ≤ K, we have Hence, by setting it follows from (2.9) that (2.10) Therefore, from (2.8) and (2.10), we obtain for all sufficiently large λ ≫ 1. Note that the exponent 4p p−2 beats the exponent p in (2.1) if (i) p < 6 or (ii) p = 6 and K is sufficiently small. Determining the optimal threshold for K would presumably require a delicate optimization of λ j in (2.2), an exact Gaussian tail bound to replace the appraisal (2.6), and an optimal inequality to replace the applications of Bernstein's inequality in (2.4) and (2.9) to determine the precise tail behavior of u L p (T) given u L 2 (T) ≤ K. We did not attempt this calculation. Even if it is possible to carry out, such an approach would likely lead to a less transparent argument than the one we propose in Section 4. Moreover, our argument is easily adapted to the case of the two-dimensional unit disc D.

Sharp Gagliardo-Nirenberg-Sobolev inequality
The optimizers for the Gagliardo-Nirenberg-Sobolev interpolation inequality with the optimal constant: play an important role in the study of the focusing Gibbs measures. The following result on the optimal constant C GNS (d, p) and optimizers is due to Nagy [33] for d = 1 and Weinstein [49] for d ≥ 2. See also Appendix B in [45].
on H 1 (R d ). Then, the minimum is attained at a function Q ∈ H 1 (R d ) which is a positive, radial, and exponentially decaying solution to the following semilinear elliptic equation on R d : with the minimal L 2 -norm (namely, the ground state). Moreover, we have See [19] for a pleasant exposition, including a proof of the uniqueness of positive solutions to (3.3), following [24].
The scale invariance of the minimization problem implies that these inequalities also hold on the finite domains T and D (essentially) with the same optimal constants.
for any u ∈ H 1 (T).
(ii) Let p > 2. Then, we have Proof. For the first claim, see Lemma 4.1 in [26]. As for the second part, extend u on D tō u ∈ H 1 (R 2 ) by settingū ≡ 0 on R 2 \ D and apply (3.1) on R 2 . (ii) (Non-existence of optimizers on T and D). On T and D (with the Dirichlet boundary condition), there is no optimizer for the Gagliardo-Nirenberg-Sobolev inequality (3.1). If there were an optimizer u on D with the Dirichlet boundary condition, we can extend u on D to a function in H 1 (R 2 ) by setting u ≡ 0 on R 2 \ D, which would be a (non-zero) non-negative optimizer for (3.1) on R 2 with compact support. which is a contradiction (see Theorem 1.1 in [19]). As for the torus case, we first note that (3.1) does not hold for constant functions. Moreover, given a function u ∈ H 1 (T), by simply considering u + C for large C ≫ 1 and observing that the left-hand side of (3.1) (on T) grows faster than the right-hand side as C → ∞, we see that the inequality (3.1) on T does not hold for (non-constant) functions in general, unless they have mean zero. Hence, we restrict our attention to mean-zero functions.
For mean-zero functions on T, Sobolev's inequality on T (see [3]) and an interpolation yield the GNS inequality for some constant C > 0: for any u ∈ H 1 0 (T) := u ∈ H 1 (T) :´T u dx = 0 . In fact, the GNS inequality (3.5) on T for mean-zero functions holds with C = C GNS (1, p) coming from the GNS inequality (3.1) on R. Suppose that u ∈ H 1 0 (T) is a real-valued mean-zero function on T. Then, by the continuity of u, there exists a point x 0 ∈ T such that u(x 0 ) = 0. By setting for |x| > 1 2 , we can apply the GNS inequality (3.1) on R to v and conclude that the GNS inequality (3.5) on T holds for u with C = C GNS (1, p). Now, given complex-valued u ∈ H 1 0 (T), write u = u 1 + iu 2 , where u 1 = Re u and u 2 = Im u. Since u has mean zero on T, its real and imaginary parts also have mean zero. In particular, there exists x j ∈ T such that u j (x j ) = 0, j = 1, 2. Hence, by the argument above, we see that for u 1 and u 2 , the GNS inequality (3.5) holds with C = C GNS (1, p). We now proceed as in Step 2 of the proof of Theorem A.1 in [19]. By Hölder's inequality (in j), we have Then, by the triangle inequality, the GNS inequality (3.5) with C = C GNS (1, p) for u j , j = 1, 2, and (3.6), we obtain This proves the GNS inequality (3.5) for u ∈ H 1 0 (T) with C = C GNS (1, p). Note that the argument above shows that the GNS inequality (3.5) on T with C = C GNS (1, p) indeed holds for any u belonging to a larger class: as in (3.2) and consider the minimization problem over H 1 00 (T): inf It follows from (3.7) that this infimum is bounded below by C GNS (1, p) −1 > 0. Suppose that there exists a mean-zero optimizer u * ∈ H 1 0 (T) ⊂ H 1 00 (T) for (3.8). Then, by adapting the argument in Step 2 of the proof of Theorem A.1 in [19] (see (3.6) and (3.7) above) to the case of the one-dimensional torus T, we see that either (i) one of Re u * or Im u * is identically equal to 0, or (ii) both Re u * and Im u * are optimizers for (3.8) and | Re u * | = λ| Im u * | for some λ > 0. In either case, one of Re u * or Im u * is an optimizer for (3.8). Without loss of generality, suppose that Re u * is a (non-zero) optimizer for (3.8). Note that Re u * ∈ H 1 00 (T) and that its positive and negative parts also belong to H 1 00 (T) (but not to H 1 0 (T)). Then, by the argument in Step 2 of the proof of Theorem A.1 in [19] once again, we conclude that Re u * is either non-negative or non-positive. This, however, is a contradiction to the fact that u (and hence Re u * ) has mean zero on T. This argument shows that there is no mean-zero optimizer for the GNS inequality (3.5) on T.

4.
Integrability below the threshold 4.1. On the one-dimensional torus. We first present the proof of the direct implication in Theorem 1.1 (ii). Namely, we show that the partition function for the converse. All the norms are taken over the one-dimensional torus T, unless otherwise stated.
In the following, E denotes an expectation with respect to the mean-zero Brownian loop u in (1.3) (in particular u has mean-zero on T). Given λ > 0, we use the notation (1.20) and write Here, we used the fact that P ≥0 = Id on mean-zero functions. Fir Note that the sets E k 's are disjoint and that N k=1 1 E k increases to 1 { u L p >λ} almost surely as N → ∞ since u in (1.3) belongs almost surely to L p (T) for any finite p. Hence, by the monotone convergence theorem, we obtain The first term on the right-hand side of (4.2) is clearly finite for any finite λ, K > 0. Hence, in view of (4.1), it suffices to show that Given an integer p, we have Integrating and applying Hölder's inequality followed by Young's inequality, we have, for any u satisfying u ≥k L p ≤ λ, for some small ε > 0 (to be chosen later), where in the last step we used Hence, by letting Now, let λ = 1 and p = 6. By Lemma 3.2 (i) for some small m > 0 (to be chosen later) with u L 2 ≤ K, there is a constant C(m) > 0 such that (4.6) is now bounded by By Hölder's inequality, (4.7) By (2.8), we have Recall that for X ∼ N R (0, 1) and t < 1 2 . Then, using (4.9) and (3.4) in Proposition 3.1, the expectation in (4.7) is bounded by (4.10) Note that, under K < Q L 2 (R) and (4.5), we can choose m, ε, η > 0 sufficiently small such that guaranteeing the application of (4.9) in the computation above. Finally, summing (4.3) over k ∈ N with (4.6), (4.7), (4.8), and (4.10), we have Therefore, we conclude that the partition function Z 6,K is finite for any K < Q L 2 (R) . This completes the proof of Theorem 1.1. Remark 4.1. As mentioned in Remark 1.2, Theorem 1.1 (ii) holds for the Ornstein-Uhlenbeck loop in (1.7). We first note that (2.8) also holds for the Ornstein-Uhlenbeck loop u since (2.5) holds even if we replace n 2 by (2π) −2 + n 2 . Hence, from (1.7) and (4.8), we have where E µ denotes an expectation with respect to the law of the Ornstein-Uhlenbeck loop u and ∂ x = 1 − ∂ 2 x . The rest of the argument follows as above, thus establishing Theorem 1.1 (ii) for the Ornstein-Uhlenbeck loop u in (1.7).

4.2.
On the two-dimensional disc. Next, we prove the normalizability of the Gibbs measure on D stated in Theorem 1.3 (ii). Namely, we show that the partition function Z 4,K in (1.10) is finite, provided that K < Q L 2 (R 2 ) . The proof is based on a computation analogous to that in Subsection 4.1. As we see below, however, we need to proceed with more care, partially due to the eigenfunction estimate, which makes the computation barely work on D; compare (4.11) and (4.23).
We first recall the following simple corollary of Fernique's theorem [18]. See also Theorem 2.7 in [15]. 4 In particular, we have for any t > 1.
Recall from [47, Lemmas 2.1 and 2.2] the asymptotic formula for the eigenvalue z n : and the eigenfunction estimate: As in Section 2, we define the spectral projection of v by We also define v ≤k and v ≥k in an analogous manner. In the following, E denotes an expectation with respect to the random Fourier series v in (1.9) and all the norms are taken over the unit disc D ⊂ R 2 unless otherwise stated. By Minkowski's integral inequality with (4.14) and (4.15), we have Then, by Markov's inequality and choosing r ≫ 1, we have without using any fine property of X. Then, (4.12) and (4.13) follow in view of Remark 2.8 in [15].
for any j ∈ Z ≥0 . Then, applying Lemma 4.2 and (4.16) with suitable ε j ∼ j −2 such that for some constant C > 0, uniformly in k ∈ Z ≥0 and λ ≥ 1. Then, it follows from the Borel-Cantelli lemma that By definition, F k 's are disjoint and, from (4.18), we have This implies that N k=1 1 F k increases to 1 { v L 4 >1} almost surely as N → ∞. Starting from Z 4,K in (1.10), we reproduce the computations in (4.2), (4.4), and (4.6) with p = 4 and λ = 1 by replacing u in (1.3), the sets E k , and the integrals over [0, 1] with v in (1.9), the sets F k , and integrals over D, respectively. We find where δ = δ(4) = 48ε is as in (4.5) with p = 4. As before, it remains to show that the series in (4.19) is convergent.
where Q is the optimizer for the Gagliardo-Nirenberg-Sobolev inequality (3.1) on R 2 . Then, by recallingˆD and applying Hölder's inequality, the expectation in the summands on the right-hand side of (4.19) is bounded by (4.20) The first factor on the right-hand side of (4.20) can be computed exactly as in (4.10), using (4.9). Namely, provided that it is finite and equals (4.22) Given K < Q L 2 (R 2 ) , we choose δ and η sufficiently small such that (4.21) holds. Then, from (4.20) with (4.17) and (4.22), we conclude that Z 4,K in (4.19) is bounded by This proves the normalizability of the Gibbs measure on D claimed in Theorem 1.3 (ii).

Non-integrability above the threshold on the disc
In this section, we discuss the non-normalizability of the Gibbs measure on D stated in Theorem 1.3 (ii). Namely, when K > Q L 2 (R 2 ) , we prove Here, Q is the ground state on R 2 as in Proposition 3.1. Since the proof of (5.1) is essentially identical to that for the non-normalizability of the Gibbs measure on the torus T (see [26, Theorem 2.2 (b)]), we keep our presentation brief.
Then, by setting it follows from Proposition 3.1 (in particular (3.4)) that H R 2 (Q) = 0. As a result, we have Then, we have where we assume f to be radial for the last identity and ∂ r denotes the directional derivative in the radial direction.
Then, thanks to the exponential decay of the ground state Q on R 2 , it follows from (5.2), (5.3), and (5.4) that for some A 1 , A 2 , A 3 > 0 and sufficiently small η 0 > 0, uniformly in large ρ ≫ 1. We need the following simple calculus lemma which follows from a straightforward computation. See Lemma 4.2 in [26].
for p = 2, 4. Here, C 0 (D) denotes the collection of continuous complex-valued functions on D, vanishing on the boundary ∂D.
Given ε > 0, let B ε (Q α,ρ ) ⊂ C 0 (D) be the ball of radius ε centered at Q α,ρ : Then, from Lemma 5.1 and (5.5), there exists small ε > 0 such that : v, radial and P denotes the Gaussian probability measure with respect to the random series (1.9).

Integrability at the optimal mass threshold
In this section, we present the proof of Theorem 1.4. Namely, we show that the Z 6,K < ∞ when K = Q L 2 (R) .
We fix d = 1, p = 6, and C GNS = C GNS (1,6). In this section, we prove the normalizability when u is the Ornstein-Uhlenbeck loop in (1.7): where n = (1 + 4π 2 |n| 2 ) 1 2 . Namely, expectations are taken with respect to the law µ of the Ornstein-Uhlenbeck loop in (6.1). In this non-homogeneous setting, the problem is not scaling invariant and some extra care is needed. See for example the proofs of Lemma 6.2 and Proposition 6.3. In Remark 6.17, we indicate the necessary modifications for handling the case of the mean-zero Brownian loop in (1.3).
6.1. Rescaled and translated ground state. In the following, we compare a function u on the circle T to translations and rescalings of the ground state Q. For this purpose, we introduce the L 2 -invariant scaling operator D λ on R by Then, given δ > 0, we set While the ground state Q is defined on R, we now interpret it as a function of Given x 0 ∈ T, we also introduce the translation operator τ x 0 defined by . The rescaled and translated version of the ground state: plays an important role in our analysis. By this definition, we have Q δ = Q δ,0 . In the following, we use ∂ δ and ∂ x 0 to denote differentiations with respect to the scaling and translation parameters, respectively. When there is no confusion, we also denote by D λ and τ x 0 the scaling and translation operators for functions on the real line. Note that since Q decays exponentially as |x| → ∞ on R, we have as δ → 0.
Remark 6.1. From (6.6), we have which is an even function. On the other hand, we have which is an odd function. By writing Similarly, for given x 0 ∈ T, we have (6.10) By parity considerations of these functions centered at x = x 0 , we also conclude that they 6.2. Stability of the optimizers of the GNS inequality. We begin by establishing stability of the optimizers of the Gagliardo-Nirenberg-Sobolev inequality (3.1); if u is "far" from all rescalings and translations of the ground state Q in the L 2 -sense, then the GNS inequality (3.1) is "far" from being sharp in the sense of (6.11) below.
Given γ > 0, define the set S γ by setting Here, P ≤k u denotes the Dirichlet projector onto the frequencies {|n| ≤ 2 k } defined in (1.18) and π =0 denotes the projection onto the mean-zero part defined in (1.16). Fix γ > 0. Young's inequality and (6.11) yield for any u ∈ S γ , where π 0 = id −π =0 . Then, by repeating the argument in Subsection 4.1 with (6.12), we can show that The main goal of this subsection is to prove the following "stability" result. In view of (6.13), this lemma allows us to restrict our attention to a small neighborhood of the orbit of the ground state Q in the subsequent subsections. Lemma 6.2. Given any ε > 0, there exists γ(ε) > 0 such that the following holds; suppose that, given any small for all 0 < δ < δ * , x 0 ∈ T, and θ ∈ R. Then, we have u ∈ S γ(ε) .
By definition (6.11) of S γ , for each n ∈ N, there exists k n ∈ N such that since u n / ∈ S γn . Then, from (3.1) and (6.18) with (6.16) and γ n → 0, we see that as n → ∞. In view of the upper bound (6.16), we also have u n L 2 (T) → Q L 2 (R) . Hence, by the Pythagorean theorem, we obtain Furthermore, we claim that Otherwise, we would have P ≤kn u n Ḣ1 (T) ≤ C < ∞ for all n ∈ N and thus there exists a subsequence, still denoted by {P ≤kn u n } n∈N , converging weakly to some u inḢ 1 (T). Then, from the compact embedding 5 ofḢ 1 (T) into L 2 (T) ∩ L 6 (T), we see that P ≤kn u n converges strongly to u in L 2 (T) ∩ L 6 (T). Hence, from (6.16) and then (6.18), we obtain . This would imply that u is a mean-zero optimizer of the Gagliardo-Nirenberg-Sobolev inequality (3.1) on the torus T, which is a contradiction. See Remark 3.3. Therefore, (6.21) must hold.
By continuity, there exists a point x n ∈ T such that and by linear interpolation for 1 2 < |x| ≤ 1 2 + β n , where the addition here is understood mod 1. Then, from (6.22), we have |v n (± 1 2 )| ≤ P ≤kn u n L 2 (T) . Moreover, from (6.23) with (6.16) and (6.21), we have where With the scaling operator D λ as in (6.2), let Then, from (6.19), (6.24), and (6.27), we have Recall that we work with mean-zero functions on T.
Since {w n } n∈N is a bounded sequence in H 1 (R), we can invoke the profile decomposition [20, Proposition 3.1] for the (subcritical) Sobolev embedding: H 1 (R) ֒→ L 6 (R). See also Theorem 4.6 in [23]. There exist J * ∈ Z ≥0 ∪ {∞}, a sequence {φ j } J * j=1 of non-trivial H 1 (R)functions, and a sequence {x j n } J * j=1 for each n ∈ N such that up to a subsequence, still denoted by {w n } n∈N , we have Here, an equality holds if and only if J * = 0 or 1. If J * = 0, then it follows from (6.36) that w n tends to 0 in L 6 (R) as n → ∞. Then, from (6.31), we see that w n tends to 0 in L 2 (R). This is a contradiction to (6.30). Hence, we must have J * = 1. In this case, (6.38) (with J * = 1) holds with equalities and thus we see that φ 1 is an optimizer for the Gagliardo-Nirenberg-Sobolev inequality on R.
Hence, we conclude from Remark 3.3 that there exist σ = 0, λ > 0, and x 0 ∈ R such that From the last step in (6.38) (with J * = 1 and an equality) with (6.30), we have which implies that σ = e iθ for some θ ∈ R. From (6.32) and (6.33) with J = J * = 1, we see that τ −x 1 n w n converges weakly to φ 1 in L 2 (R) (which follows from the (weak) convergence of τ −x 1 n w n to φ 1 in L 6 (R)), while (6.30) implies convergence of the L 2 -norms. Hence, we obtain strong convergence in L 2 (R): (which tends to 0 as n → ∞). Thus, it follows from (6.39) that as n → ∞. In the following, we denote by Q λn,yn the dilated and translated ground state Q, viewed as a periodic function on T, and by τ yn D λn Q the dilated and translated ground state viewed as a function on the real line. By possibly choosing a subsequence, we assume that y n ≥ 0 without loss of generality. Write , − 1 2 + y n ) =: I 1,n ∪ I 2,n . Note that while Q λn,yn and τ yn D λn Q coincide on I 1,n , they do not coincide on I 2,n . Thanks to the exponential decay of the ground state Q, we have On the other hand, on I 2,n , we have Q λn,yn (x) = λ as n → ∞. Finally, by combining (6.20), (6.23), and (6.43), we obtain a contradiction to (6.17). This completes the proof of Lemma 6.2.
6.3. Orthogonal coordinate system in a neighborhood of the soliton manifold. In view of Lemma 6.2 and (6.13), in order to prove Z 6,K= Q L 2 (R) < ∞, it suffices to show that for some 0 < δ < δ * , x 0 ∈ T, and θ ∈ R . Our main goal in this subsection is to endow U ε (δ * ) with an "orthogonal" coordinate system.
Proof. We first show that the claimed result holds true in the case of the real line. Given where the inner product on L 2 (R) is real-valued as defined in (1.13).
By another translation and rotation, we may assume that x 1 = 0 and θ 1 = 0 in (6.61). Namely, we have Hence, to finish the proof, we show that (6.66) guarantees that u lies in the image of F = F δ 1 ,0,0 . For this purpose, we recall the following version of the inverse function theorem; see [16,Theorem 26.29]. See also [16,Lemma 26.28]. In the following, · denotes the operator norm.
Our goal is thus to estimate the quantity κ in (6.67), with f replaced by F = F δ 1 ,0,0 , to conclude from (6.66) that u is in the image of F . We begin by computing dF (δ 1 , 0, 0, v 1 ) −1 and its norm. Lemma 6.5. There is a constant C > 0 such that for all sufficiently small δ 1 > 0 and v 1 L 2 (T) ≪ 1.
We assume Lemma 6.5 for now and proceed with the proof of Proposition 6.3. The proofs of this lemma and Lemma 6.6 below will be presented at the end of this subsection.
We conclude this subsection by presenting the proofs of Lemmas 6.5 and 6.6. In the following, ·, · denotes the inner product in L 2 (T).
Proof of Lemma 6.5. Let (α, β, where the fourth term on the right-hand side is as in (6.69).
This completes the proof of Lemma 6.5.
Next, we present the proof of Lemma 6.6.
Before proceeding to the proof of Proposition 6.7, let us first discuss properties of truncated solitons. Note that the frequency truncation operator π N is parity-preserving; π N Q δ is an even function for any δ > 0. It also follows from (6.8) and (6.9) that π N ∂ δ Q δ is an even function, while π N ∂ x 0 Q δ is an odd function. Hence, they are orthogonal in H k (T), k ∈ Z ≥0 . Moreover, the operator π N also commutes with the pointwise conjugation, so π N Q δ , π N ∂ δ Q δ , and π N ∂ x 0 Q δ are all real functions. Therefore, π N iQ δ is orthogonal 6 to both π N ∂ δ Q δ and π N ∂ x 0 Q δ in H k (T), k ∈ Z ≥0 . By a similar consideration centered at x = x 0 , we also conclude that π N e iθ ∂ δ Q δ,x 0 , π N e iθ ∂ x 0 Q δ,x 0 , and π N ie iθ Q δ,x 0 are pairwise orthogonal in H k (T), k ∈ Z ≥0 .
In the following, we use F R to denote the Fourier transform of a function on the real line. By a change of variables and the exponential decay of Q, we have for 0 < δ ≪ 1, provided that δ|n| 1. With A 1 as in (6.83), an analogous computation yields for 0 < δ ≪ 1, provided that δ|n| 1. Fix small γ > 0 and set δ * = δ * (N ) > 0 such that Then, by a Riemann sum approximation with (6.140), we then have for δ * < δ ≪ 1, where π R N denotes the Dirichlet projection onto frequencies {|ξ| ≤ N } for functions on the real line. Since the estimate above holds independently of the base point x 0 ∈ T, we have uniformly for δ * < δ ≪ 1 and x 0 ∈ T. On the other hand, by integration by parts 2K times together with the exponential decay of the ground state and (6.141), we have (6.143) Then, with π ⊥ N = id −π N , it follows from (6.141) and (6.143) that for any K ∈ N, uniformly in δ * < δ ≪ 1 and x 0 ∈ T. Similarly, we have for any k, K ∈ N, uniformly in δ * < δ ≪ 1 and x 0 ∈ T.

6.5.
A change-of-variable formula. Our main goal is to prove the bound (6.44). In the remaining part of the paper, we fix small δ * > 0 and set where U ε (δ * ) is defined in (6.45) for given small ε > 0. Recalling the low regularity of the Ornstein-Uhlenbeck loop in (1.7), we only work with functions u ∈ H s (T) \ H 1 2 (T), s < 1 2 , for the µ-integration. Thus, with a slight abuse of notations, we redefine U ε = U ε (δ * ) in (6.45) to mean U ε \ H 1 2 (T). Namely, when we write U ε = U ε (δ * ) in the following, it is understood that we take an intersection with H Given small δ > 0 and x 0 ∈ T, let V δ,x 0 ,0 = V δ,x 0 ,0 (T) be as in (6.46) with θ = 0. In view of Proposition 6.3, the convention above: U ε = U ε ∩ H 1 2 (T) c , and the fact that Q δ,x 0 ∈ H 1 (T), we redefine V δ,x 0 ,0 to mean V δ,x 0 ,0 ∩ H 1 (T). Namely, when we write V δ,x 0 ,0 in the following, it is understood that we take an intersection with H 1 (T). Now, let µ ⊥ δ,x 0 denote the Gaussian measure with V δ,x 0 ,0 ⊂ H 1 (T) as its Cameron-Martin space. Then, we have the following lemma on a change of variables for the µ-integration over U ε .
Lemma 6.8. Fix K > 0 and sufficiently small δ * > 0. Let F (u) ≥ 0 be a functional of u on T which is continuous in some topology H s (T), s < 1 2 , such that F ≤ C for some C and F (u) = 0 if u L 2 (T) > K. Then, there is a locally finite measure dσ(δ) on (0, δ * ) such that where the domain of the integration on the right-hand side is to be interpreted as Moreover, σ is absolutely continuous with respect to the Lebesgue measure dδ on (0, δ * ), satisfying | dσ dδ | δ −20 . The proof of Lemma 6.8 is based on a finite dimensional approximation and an application of the following lemma. Lemma 6.9. Let M ⊂ R n be a closed submanifold of dimension d and N be its normal bundle. Then, there is a neighborhood U of M such that U is diffeomorphic to N via the map φ : N → U given by for x ∈ M and v ∈ T x M ⊥ . Furthermore, the following estimate holds for any non-negative measurable function f : R n → R and any open set V ⊂ {(x, v) : |v| ≤ 1} ⊂ N : Here, the measure dσ is defined by where dω(x) is the surface measure on M and {t k (x)} d k=1 = {(t 1 k (x), . . . , t n k (x))} d k=1 is an orthonormal basis for the tangent space T x M with the expression ∇t k (x) defined by for any coordinate chart ϕ on a neighborhood of x ∈ M such that y = (y 1 , . . . , y d ) = ϕ(x) and d(ϕ −1 )(y) is an isometry with its image. Note that the constant in (6.158) is independent of the dimension n of the ambient space R n .
Proof. Given x ∈ M , let w 1 (x), . . . , w n−d (x) be an orthonormal basis of T x M ⊥ . Consider the map where α = (α 1 , . . . , α n−d ). Recalling that {w j (x)} n−d j=1 is an orthonormal basis of T x M ⊥ , it follows from the area formula (see [17, Theorem 2 on p. 99]) with (6.157), (6.159), and V ⊂ {(x, v) : |v| ≤ 1} and then applying a change of variables that where J ψ is the determinant of the differential of the map ψ. Hence, the bound (6.158) follows once we prove Recall that the tangent space of M ×R n−d at the point (x, α) is isomorphic to T x M ×R n−d . Then, denoting by {e j } n−d j=1 the standard basis of R n−d , it follows from (6.159) that 7 dψ[e j ] = w j , j = 1, . . . , n − d.
Hence, by taking (t 1 , . . . , t d , e 1 , . . . , e d ) (and (t 1 , . . . , t d , w 1 , . . . , w d ), respectively) as the (orthonormal) basis of the domain T x M × R n−d (and the codomain R n , respectively), the matrix representation of dψ is given by Thus, we have for any α ∈ R n−d with |α| ≤ 1. Therefore, the bound (6.160) (and hence (6.158)) follows once we prove sup 1≤h,k≤d uniformly in α ∈ R n−d with |α| ≤ 1. By differentiating the orthogonality relation w j (x), t k (x) R n = 0, we obtain for any τ ∈ T x M . Thus, from (6.161), (6.163), Cauchy-Schwarz inequality with |α| ≤ 1, and the orthonormality of {w j } n−d j=1 , we have where we used Cauchy-Schwarz inequality once again in the last step. This proves (6.162) and hence concludes the proof of Lemma 6.9.
We now present the proof of Lemma 6.8.
6.6. Further reductions. The forthcoming calculations aim to prove (6.44). We first apply the change of variables (Lemma 6.8) to the integral in (6.44). In the remaining part of this section, we set K = Q L 2 (R) and use ·, · to denote the inner product in L 2 (T), unless otherwise specified. Note that the the integrand is neither bounded nor continuous. The lack of continuity is due to the sharp cutoff 1 { u L 2 (T) ≤K} . Since the set of discontinuity has µ-measure zero, we may start with a smooth cutoff and then pass to the limit. We can also replace this integrand by a bounded one, as long as the bounds we obtain are uniform, at the cost of a simple approximation argument which we omit. By applying Lemma 6.8 to the integral in (6.44) and using also the translation invariance of the surface measure (namely, the independence of all the quantities from x 0 ), we havê where µ ⊥ δ = µ ⊥ δ,0 and (6.169) From (6.168) and (6.169) with (6.4), we have where we used Re(v 2 ) = 2(Re v) 2 − |v| 2 to get the fourth term on the right-hand side. By the sharp Gagliardo-Nirenberg-Sobolev inequality (Proposition 3.1), we have Q δ 6 L 6 (R) = 3 Q ′ δ 2 L 2 (R) and thus we have 1 6ˆT uniformly in 0 < δ ≤ 1, thanks to the exponential decay of Q (as in (6.7)). By Young's inequality, the sum of the last four terms in (6.170) is bounded by ηˆT Q 4 δ 2(Re v) 2 + 1 2 |v| 2 dx + C ηˆT |v| 6 dx (6.172) for any 0 < η ≪ 1 and a (large) constant C η > 0. Hence, from (6.167), (6.170), (6.171), and (6.172) together with Proposition 6.3, we havê As a consequence of the argument presented in Section 4.1, we have the following integrability result. Lemma 6.10. Given any C ′ η > 0, there exists small ε 1 > 0 such that uniformly in 0 < δ ≪ 1.
Thus, by Rellich's lemma, we see that A is a compact operator on V ′ ⊂ H 1 (T) and thus the spectrum of A consists of eigenvalues. Recalling that V ′ ⊂ H 1 (T) in (6.177) is the Cameron-Martin space for µ ⊥⊥ δ , we see that evaluating the integral in (6.191) is equivalent to estimating the product of the eigenvalues of 1 2 id +(1−η 2 )A. Thus, the rest of this subsection is devoted to studying the eigenvalues of A. We present the proof of Proposition 6.12 at the end of this subsection.
Since A preserves the subspace of Re H 1 (T) consisting of real-valued functions and also the subspace Im H 1 (T) consisting of purely-imaginary-valued functions, in studying the spectrum of A, we split the analysis onto these two subspaces. On the real subspace Re H 1 (T), A coincides with the operator On the other hand, identifying Im H 1 (T) with Re H 1 (T) by the multiplication with i, we see that, on the imaginary subspace, A coincides with the operator In the following, we proceed to analyze the eigenvalues of A 1 and A 2 . In Proposition 6.13, we provide a lower bound for the eigenvalues of 1 2 id +(1 − η 2 )A. Then, by comparing A j with simple operators whose spectrum can be determined explicitly (Lemma 6.16), we establish asymptotic bounds on the eigenvalues in Proposition 6.15. Proposition 6.13. There exists a constant ε 0 > 0 such that for any δ, η > 0 sufficiently small, the smallest eigenvalue of A = A(δ, η) is greater than − 1 2 + ε 0 .
Proof. Fix ε 0 > 0. Suppose that there exists a sequence {δ n } n∈N of positive numbers tending to 0 and w j n ∈ V j (δ n ), j = 1, 2, such that for j = 1, 2, where c 1 = 5 2 and c 2 = 1 2 . Let A j n denote the Schrödinger operator given by where P V j (δn) denotes the projection onto V j (δ n ) in Re L 2 (T). Then, by the min-max principle (see [42, Theorem XIII.1]), the minimum eigenvalue λ j n is non-positive. We denote Noting that v, w Ḣ1 (R) = v, −∂ 2 x w L 2 (R) and that Q is a smooth function, we can view V j (x 0 ) as a subspace of Re L 2 (R). We now define the operator T j = T j (x ′ ), j = 1, 2, on Re L 2 (R) by , if x ′ ∈ R, (6.205) and Here, P denotes the projection onto V j (x ′ ) in Re L 2 (R). Then, from (6.197), (6.201), (6.202), (6.205), and (6.206) along with the smoothness and exponential decay of Q and its derivatives, we have as n → ∞, where λ n = δ 2 n λ n ≤ 0. Note that from (6.198), we have −C ≤ λ n ≤ 0 for any n ∈ N. Thus, passing to a subsequence, we may assume that λ n → λ for some λ ≤ 0 and thus from (6.207), we obtain Noting that T j is semi-bounded, let λ 0 be the minimum of the spectrum of T j . By the min-max principle (see [42, Theorem XIII.1]), we have for every v ∈ H 1 (R). Therefore, from (6.208), we obtain that λ 0 ≤ λ ≤ 0.
Since Q is a Schwartz function, the essential spectrum of the Schrödinger operator S j in (6.205) is equal to [1, ∞); see [22,. Moreover, T j in (6.205) differs from S j by a finite rank perturbation and thus its essential spectrum is [1, ∞); see [22,]. In particular, λ 0 ≤ 0 does not lie in the essential spectrum of T j . Namely, λ 0 belongs to the discrete spectrum of T j . Hence, there exists v ∈ H 2 (R) such that When |x ′ | = ∞, it is easy to see from (6.206) that (6.209) can not hold for λ 0 ≤ 0, since T j is a strictly positive operator on Re H 1 (R). When x ′ ∈ R, we invoke the following lemma.
Similarly, the operator viewed as an operator on L 2 (R), is strictly positive on V 2 := V 2 (0) defined in (6.204).
This lemma shows that (6.209) can not hold for λ ≤ 0 when x ′ ∈ R. Therefore, we arrive at a contradiction to (6.195), whether x ′ ∈ R or |x ′ | = ∞. This concludes the proof of Proposition 6.13 (modulo the proof of Lemma 6.14 which we present below).
We now present the proof of Lemma 6.14.
Consider the Hamiltonian H(u) = H R (u): By the sharp Gagliardo-Nirenberg-Sobolev inequality (Proposition 3.1), we know that H : H 1 (R) → R has a global minimum at Q, when restricted to the manifold u L 2 (R) = Q L 2 (R) . In view of (6.3), we see that u = Q and λ = −1 satisfy the following Lagrange multiplier problem: . By a direct computation, the second variation of H(u) + G(u) at Q in the direction v is given by 2 B 1 Re v, Re v L 2 (R) + 2 B 2 Im v, Im v L 2 (R) , while the constraint G(u) = 0 gives v, Q L 2 (R) = 0. Then, from the second derivative test for constrained minima, we obtain that B 1 w, w L 2 (R) ≥ 0 on w ∈ Re H 1 (R) : w, Q L 2 (R) = 0 , (6.212) From (3.3), we have B 2 Q = 0. Also by differentiating (6.3), we obtain B 1 Q ′ = 0. Moreover, by the elementary theory of Schrödinger operators, 12 the kernel of the operator B j , j = 1, 2, has dimension 1. In particular, when j = 2, B 2 Q = 0 implies (6.210) for any v ∈ V 2 . Hereafter, we use the fact that the essential spectrum of B j is [1, ∞) and that the spectrum of B j on (−∞, 1) consists only of isolated eigenvalues of finite multiplicities. 13 In the following, we focus on B 1 . With Q ⊥ = (span(Q)) ⊥ , let P Q ⊥ denote the L 2 (R)projection onto Q ⊥ given by 12 If there is a linearly independent solution v to Bj v = 0, then by considering the Wronskian, we obtain that v ′ (x) → 0 as |x| → ∞. This in particular implies v / ∈ L 2 (R) ∪Ḣ 1 (R). 13 This claim follows from Weyl's criterion ([41, Theorem VII.12]).
Then, from (6.212), we have Obviously, the quadratic form (6.213) vanishes for v = Q. On Q ⊥ , we have P Q ⊥ B 1 v = 0 if and only if Recalling that the restriction of B 1 to (Q ′ ) ⊥ is invertible, we see that the condition (6.214) holds at most for a two-dimensional space. Recall from Remark 6.1 that Q ′ (= −∂ x 0 Q) is orthogonal to Q in L 2 (R). Moreover, by differentiating (6.3) in δ, we have As a result, it follows from (6.203) that there exists θ > 0 such that for any v ∈ V 1 .
Next, we establish asymptotic bounds on the eigenvalues of A j , j = 1, 2. We achieve this goal by comparing A j to a simpler operator whose spectrum is studied in Lemma 6.16 below. Proof. The first bound (6.215) is just a restatement of Proposition 6.13. In the following, we establish the asymptotic behavior (6.216) of λ j n and µ j n . For j = 1, 2, define an operator T j by where c 1 = 5 2 and c 2 = 1 2 . Since T j is the composition of a bounded multiplication operator and (1−∂ 2 x ) −1 , which is a compact operator on H 1 (T), it is compact and thus has a countable sequence of eigenvalues accumulating only at zero. We label the negative eigenvalues as −λ j 1 ≤ −λ j 2 ≤ · · · ≤ 0, while the positive eigenvalues are labeled asμ j 1 ≥μ j 2 ≥ · · · ≥ 0. Since A j is the composition of T j with a projection, A j is also compact. Then, by the min-max principle, we have −λ j n ≤ −λ j n ≤ −λ j n+3 andμ j n ≤ µ j n ≤μ j n+3 . (6.218) Hence, it suffices to prove the asymptotic bounds (6.216) for the eigenvalues −λ j n andμ j n of T j .
We estimate the eigenvaluesλ j n andμ j n for large n by comparing T j with an operator with piecewise constant coefficients, whose spectrum can be computed explicitly. Fix a > 0 such that 10Q 4 (a) ≤ 1. Given small δ > 0, define the function S = S δ on [− 1 2 , 1 2 ) by setting .
We first complete the proof of Proposition 6.15, assuming Lemma 6.16. It follows from the asymptotics (6.220) and the min-max principle that λ n 1 n 2 andμ n δ −2 n 2 (6.221) for the eigenvalues −λ j n andμ j n of T j . Then, the asymptotic bounds (6.216) for the eigenvalues of −λ j n and µ j n of A j follows from (6.218) and (6.221).
We now present the proof of Lemma 6.16.
Proof of Lemma 6.16. Noting that S defined in (6.219) is a bounded operator, we see that R = (1 − ∂ 2 x ) −1 S is a compact operator on H 1 (T) and thus has a countable sequence of eigenvalues {λ n } n∈N accumulating only at 0 with the associated eigenfunctions {f n } n∈N forming a complete orthonormal system in H 1 (T). Moreover, since the operator R has even and odd functions as invariant subspaces, we assume that any such eigenfunction f n is even or odd. The eigenvalue equation: shows that the second derivative of any eigenfunction is piecewise continuous. With C 0 = 5 Q 4 L ∞ (R) − 2 = 5 · 6 − 2 = 28, define Then, by dropping the subscript n, we can rewrite the eigenvalue equation (6.222) as 1 2 ]. (6.224) Let A = |A 0 | and B = |B 0 |. In the following, we carry out case-by-case analysis, depending on the signs of A 0 and B 0 . It follows from (6.223) that λ > 0 implies A 0 > B 0 , while λ < 0 implies B 0 < 0. This in particular implies that the case A 0 < 0 < B 0 can not happen for any parameter choice. On the other hand, the general solution to (6.224) on (aδ, 1 2 ] is given by f (x) = α cosh(B(x − aδ)) + β sinh(B(x − aδ)), x ∈ (aδ, 1 2 ]. Thus, with the notations: To enforce the periodicity condition f ′ ( 1 2 ) = 0, we need 0 = f ′ ( 1 2 ) = B cos(Aaδ) sinh(B(2 −1 − aδ)) − A sin(Aaδ) cosh(B(2 −1 − aδ)).
In order to show uniqueness, it suffices to show that for every A with F 1 (A) = 1, we have F ′ 1 (A) > 0. We first note that, for 0 ≤ A ≪ δ −1 , we have F 1 (A) A 2 δ.
However, since A, B, a > 0, we see that this condition can never be satisfied for 0 < δ ≪ 1.
This completes the proof of Lemma 6.16.
Finally, we conclude this subsection by presenting the proof of Proposition 6.12.
For readers' convenience, we go over how we choose the parameters. We first choose η > 0 in (6.172) sufficiently small such that Proposition 6.12 holds. Next, we fix small ε 1 = ε 1 (η) > 0 such that Lemma 6.10 holds. Then, Proposition 6.3 determines small ε = ε(ε 1 ) > 0 and δ * = δ * (ε 1 ) > 0. See also Lemma 6.11, where we need smallness of δ = δ(η, ε 1 ) > 0. Finally, we fix γ = γ(ε) > 0 by applying Lemma 6.2. This completes the proof of Theorem 1.4. Remark 6.17. In this section, we presented the proof of Theorem 1.4, where the base Gaussian process is given by the Ornstein-Uhlenbeck loop in (6.1). When the base Gaussian process is given by the mean-zero Brownian loop in (1.3), the same but simpler argument gives Theorem 1.4. For example, in the case of the mean-zero Brownian loop, we can omit (6.12) and the reduction to the mean-zero case at the beginning of the proof of Lemma 6.2. In the proof of Proposition 6.3, we introduced V γ 0 δ,x 0 ,θ in (6.47) in order to use a scaling argument in the non-homogeneous setting. In the case of the mean-zero Brownian loop, we can simply use V 0 δ,x 0 ,θ , i.e. γ 0 = 0. The rest of the proof remains essentially the same.
(DFG, German Research Foundation) under Germany's Excellence Strategy-EXC-2047/1-390685813, through the Collaborative Research Centre (CRC) 1060. L.T. was also supported by the Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University, and the University of Edinburgh.