Spectral Gap in Mean-Field O(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}(n)$$\end{document}-Model

We study the dependence of the spectral gap for the generator of the Ginzburg–Landau dynamics for all O(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal O(n)$$\end{document}-models with mean-field interaction and magnetic field, below and at the critical temperature on the number N of particles. For our analysis of the Gibbs measure, we use a one-step renormalization approach and semiclassical methods to study the eigenvalue-spacing of an auxiliary Schrödinger operator.


Introduction and Main Results
1.1. O(n)-model. The model we are concerned with in this article is the generator of the Ginzburg-Landau dynamics, or Langevin dynamics, of the mean-field O(n)-model in the critical and supercritical regime β ≥ n, as defined precisely in Section 2. Our objective is to study the scaling of the spectral gap in terms of the system size N , for all the numbers of components n ≥ 1, and including the cases with or without external magnetic field, in the low temperature and critical regime, extending the study of the subcritical regime β < n in [BB19]. When β < n, the spectral gap of the generator remains open uniformly in N and for any number of components n, in the full temperature range.
The mean-field O(n)-model is defined by the energy function h, σ (x) (1.1) acting on spin configurations σ : {1, .., N } → S n−1 where MF is the mean-field Laplacian and h ∈ R n an external magnetic field. For our study of spectral gaps, we consider the Ginzburg-Landau dynamics associated with the Gibbs measure dρ ∝ e −β H (σ ) with Hamilton function (1.1). The inverse temperature parameter β is such that lower temperatures (higher β) favors alignment of spins. The study of mean-field O(n)-models is motivated by the fact that their behavior approximates that of the full O(n)-model on high-dimensional tori [Ell85,LLP10].

State of the art and motivation.
The study of spectral gaps in O(n) models is a popular problem that has received a lot of attention over the last decades. The study of logarithmic Sobolev (and other functional) inequalities is a classical and very effective tool to study concentration of measures and to quantify the relaxation rates, i.e. the mixing properties, of the dynamics. In particular, the spectral gap (the speed of relaxation) is determined by the constant in the Log-Sobolev inequalities. We define the spectral gap to be the size of the gap between 0 and the rest of the spectrum of the associated generator L, defined in (2.3). The gap then can be also characterized by where Var ρ is the variance relative to the equilibrium measure ρ. All these quantities will be specified for our setting in the following section. For further background on functional inequalities see [Gro93,BE85,Led99,Led01,GZ03,ABC+00] and references therein.
There are only few general approaches for the study of spectral gaps of spin systems, using log-Sobolev inequalities, available and many of them rely on an asymptotic study of log-Sobolev inequalities [LY93,SZ92a,SZ92b,SZ92c] or [MO13] for a more recent result in that direction. In the article [BB19], a simpler proof for a log-Sobolev inequality was provided for bounded and unbounded spin systems and sufficiently high temperatures. The novelty of the approach in [BB19] is the combination of the study of log-Sobolev inequalities with a simple renormalization group approach to decompose the stationary measure in a way that makes it accessible to simple Bakry-Émery techniques.
Inspired by the method in [BB19], we invoke the same one-step renormalization group procedure to reduce the high-dimensional problem to the study of a low-dimensional renormalized measure and a fluctuation measure. In the subcritical regime β < n, which is the regime analyzed in [BB19], the renormalization of the equilibrium measure is particularly efficient, since the renormalized potential is strictly convex such that the Bakry-Émery criterion can be directly applied to this measure and implies that the spectral gap remains open. This renormalization group method has recently also been successfully applied in the study of the spectral gap for hierarchical spin models [BB] and for a lattice discretization of a massive Sine-Gordon model [BB20].
The low temperature regime, which is the regime we are concerned about within this article, has a non-convex renormalized potential. In this regime, after a single renormalization step, the renormalized potential is not convex. This makes the asymptotic analysis much more difficult and requires new methods: While we analyze the Ising model, n = 1, without magnetic field, directly using explicit criteria for spectral gap and log-Sobolev inequalities [BG99,BGL14], we heavily use the equivalence between the generator of the Ginzburg-Landau dynamics and a Schrödinger operator to analyze multi-component,n ≥ 2, O(n)-models. This analysis builds heavily upon ideas by Simon [Sim83,CFKS87] and Helffer-Sjöstrand [HS85,HS87] who developed effective semiclassical methods to study the low-lying spectrum of Schrödinger operators in the semiclassical limit (which in our case corresponds to N → ∞). These results are discussed thoroughly in the final chapters of [NH05]. In this article however, we have to study the spectrum of Schrödinger operators beyond the harmonic approximation. In this case, the limiting operator is not explicitly diagonalizable anymore and the spacing between eigenvalues is no longer linear in the semiclassical parameter N , the number of spins.
The mixing time of the Glauber dynamics of the mean-field Ising model (O(1)) without magnetic field has been carefully analyzed in [DLP09a,DLP09b]. There it is shown-among others-that the mixing time in the subcritical regime β < 1 is N log(N ), the scaling at the critical point N 3/2 for β = 1 and in the supercritical regime β > 1 it is exponential growing in N . This is to be compared to a spectral gap that remains open for β < 1, closes like N −1/2 for β = 1 and closes exponentially fast also for β > 1. Thus, the mixing time for the Glauber dynamics are -up to a factor 1/N -comparable to our findings on the spectral gap, cf. Theorem 1.
Our main result on the mean-field Ising model in the supercritical regime β > n is stated in the following Theorem: Theorem 1 (Spectral gap-Supercritical Mean-field Ising models, β > 1). Let N be the number of spins and n the number of components. For the supercritical mean-field Ising model (n = 1, β > 1), the spectral gap λ N of the generator • for the case of small magnetic fields |h| < h c , closes as N → ∞ exponentially fast, where γ 1 (β) ≤ γ 2 (β) ∈ R are the two smallest numbers satisfying the condition • For critical magnetic fields |h| = h c , the spectral gap does not close faster than (N −1/3 ) anymore. • Finally, for strong magnetic fields |h| > h c , it is bounded away from zero uniformly in N .
In the case of supercritical multi-component systems (n ≥ 2, β > n) without magnetic fields, it is the rotational invariance of the model that leads to a closing spectral gap as N tends to infinity. To capture this property, we call a function f : (S n ) N → R radial, if it is only a function of the norm of the mean spin |σ |. Our main results for all multi-component systems in the supercritical regime β > n are summarized in the following Theorem: Theorem 2 (Spectral gap-Supercritical Mean-field O(n)-models, β > n ≥ 2). Let N be the number of spins and n the number of components. For the supercritical mean-field O(n)-models(n ≥ 2, β > n), the spectral gap λ N of the generator • is bounded away from zero uniformly in the number of spins for all h ∈ R n \{0}.
We also analyze the behavior of the spectral gap at the critical point β = n and h = 0. Using a discrete Fourier analysis approach implemented in Section 6 for the Ising case n = 1 and a direct asymptotic analysis for all higher component systems n ≥ 2, we find a different asymptotic of the spectral gap from both the supercritical β > n (exponentially fast closing) and subcritical β < n (spectral gap remains open) regimes: Theorem 3 (Spectral gap-Critical Mean-field O(n) models, β = n). For all critical, β = n, h = 0 mean-field O(n)-models the spectral gap closes as λ N = (N −1/2 ). In particular, the rate N −1/2 is attained for the magnetization We emphasize that at the critical points (β = n, h = 0), the gap does no longer close once a non-zero magnetic field is present: The proof of Theorem 4 is along the lines of Theorem 1 in the regime h > h c and follows from Proposition 4.2 in the Ising-case, n = 1, and in the multi-component case, n ≥ 2, from Proposition 5.4.

Organization of the article.
The article is organized as follows: • In Section 2 we introduce the mean-field O(n)-model. • In Section 3 we introduce the renormalized methods.
• In Section 4 we analyze the mean-field Ising model in the supercritical regime β > 1 and prove Theorem 1.
• In Section 5 we analyze the higher-component mean-field O(n)-models in the supercritical regime β > n and prove Theorem 2. • In Section 6 we study the critical regime and prove both Theorems 3 and 4. • Our article contains an appendix that contains technical details and further details on numerical methods.
We say that f (z) = (g(z)) if there are k 1 , k 2 > 0 and z 0 such that for all z ≥ z 0 we have k 1 g(z) ≤ f (z) ≤ k 2 g(z). The expectation with respect to a measure μ is written denoted by E μ (X ). The normalized surface measure on the n sphere is denoted as d S S n . We write 1l to denote a vector or matrix whose entries are all equal to one and id for the identity map. Finally, we introduce the notation [N ] := {1, ..., N } . The eigenvalues of a self-adjoint matrix A shall be denoted by

The Mean-Field O(n)-Model
We study the mean-field O(n)-model with spin configuration σ : [N ] → S n−1 and introduce the mean-field Laplacian The mean spin is defined as σ := 1 N x∈ [N ] σ (x). The energy of a spin configuration σ is given by the Curie-Weiss Hamiltonian (2.1) where the constant vector h ∈ R n represents an external magnetic field and β is the inverse temperature of the system. The critical temperature for the O(n)-models is β = n and we study both regimes: the supercritical regime β > n and the critical regime β = n. The dynamics we consider is the continuous-time Ginzburg-Landau dynamics to the invariant distribution of the mean-field O(n)-model which is the Gibbs measure S n−1 are the Laplace-Beltrami and gradient operator on S n−1 acting on spin i, respectively. We recall that for the Ising model n = 0 and a function F : S 0 → R, the gradient is given by Studying the operator L on the weighted space L 2 (S n−1 ) N , dρ makes this generator self-adjoint. The quadratic form of the generator (2.3) is just a rescaled Dirichlet form .

Renormalized Measure and Mathematical Preliminaries
We start with the definition of entropy with respect to probability measures: Definition 3.1 (Entropy). For a probability measure μ on some Borel set the entropy Ent μ (F) of a positive measurable function F : Instead of studying the generator of the dynamics directly, we apply a one step renormalization first [BBS19, Sec. 1.4]: Definition 3.2 (Renormalized quantities). The renormalized single spin potential V n associated with the mean-field O(n)-model for ϕ ∈ R n is defined as where I is the modified Bessel function of the first kind. The N -particle renormalized measure is defined for a normalizing constant ν Definition 3.3 (Fluctuation measure). For any ϕ ∈ R n , there is a probability measure μ ϕ , the fluctuation measure, on S n−1 N defined as A straightforward calculation shows that the stationary measure dρ can be decomposed into the fluctuation and renormalized measure such that E ρ (F) = E ν N (E μ ϕ (F)).
Example 1. In the case of the Ising model (n = 1) the renormalized potential is For the XY model (n = 2) the renormalized potential reads where I is the modified Bessel function of the first kind. For the Heisenberg model (n = 3) one finds For the N -asymptotic study of eigenvalues we observe that the renormalized potential grows quadratically at infinity such that V n ∈ L ∞ (R n ).
The Ginzburg-Landau dynamics dynamics for the renormalized measure is then given by the self-adjoint operator L ren : D(L ren ) ⊂ L 2 (R n , dν N ) → L 2 (R n , dν N ), satisfying (3.7) The renormalized generator L ren satisfies The renormalized Schrödinger operator with null space spanned by e −N V n is the operator defined by conjugation − ren = e −N V n /2 L ren e N V n /2 Definition 3.4 (LSI and SGI). Let μ be a Borel probability measure on R n . We say that μ satisfies a logarithmic Sobolev inequality LSI(k) iff for all smooth functions f . The LSI(k) implies [Led99, Prop. 2.1] that μ satisfies a spectral gap inequality SGI(k) Thus, in light of the characterisation (1.2), the spectral gap of L ren is by (3.8) precisely the constant in the SGI of the renormalized measure.
Remark 1. If f vanishes outside a set of measure μ( ) < 1 and if μ satisfies a SGI(k) then (3.10) For Borel probability measures μ on R there is an explicit characterization of the measures satisfying a LSI [BG99, Theorem 5.3]: Any such measure μ satisfies a LSI(k) iff there exist absolute constants K 0 = 1/150 and K 1 = 468 such that the optimal value k in the LSI(k) satisfies K 0 (D 0 + D 1 ) ≤ 1/k ≤ K 1 (D 0 + D 1 ) for finite D 0 and D 1 . Let m be the median of μ and p(t) dt the absolutely continuous part of μ with respect to Lebesgue measure. The constants D 0 and D 1 are given by   We continue by observing that the fluctuation measures satisfy a LSI( 2 γ n ) independent of h or ϕ. This follows for n = 1 with γ n = 4 from a simple application of the tensorization principle to the classical bound on the Bernoulli distribution [ABC+00, Led01,SC97]. For number of components n ≥ 2 one can use the results from [ZQM11].
Proposition 3.5. Let the renormalized measure ν N satisfy a LSI(λ), then the full equilibrium measure ρ satisfies a LSI and if the renormalized measure ν N satisfies a SGI(λ), then the equilibrium measure ρ satisfies a SGI Proof. The proof of the SGI is as follows: For the SGI we obtain the decomposition (3.14) To bound the second term in the above estimate, we compute using the Cauchy-Schwarz inequality and the spectral gap inequality for fluctuation measures μ ϕ on the sphere, defined by (3.4) such that, see [BB19, Theorem 1, (11)-(15)], We then use that by the explicit expression (3.2) Inserting this into (3.15) we find that Thus, we have using Cauchy-Schwarz that We can then use that by Cauchy-Schwarz again (3.18) Finally, inserting this into (3.17) and using the LSI for the fluctuation measure, we find which after inserting this bound into (3.14) implies the claim. To prove the LSI we follow [BB19] and write For the second term we have from applying the Cauchy-Schwarz inequality: By doubling the variables σ x , σ x , we write where in the last two lines we applied CS inequality and used that |σ This gives Overall we have .

The Mean-Field Ising Model
Without loss of generality, we assume h ≥ 0 when studying the Ising model. We define the critical magnetic field strength in the Ising model The critical magnetic field strength is chosen in such a way that for fields h < h c (β) there are two potential wells in the renormalized potential landscape, see Figure 1, whereas for h ≥ h c (β) there is only one, see Figure 2 in subsection 4.3 where this case is discussed.

Lower bound on spectral gap in weak field h < h c (β) regime.
We start by showing that the inverse spectral gap in the Ising model in the case of subcritical magnetic fields, i.e. h < h c (β), converges at most exponentially fast to zero as the number of spins, N , increases.
We start by showing a LSI with exponential constant for the renormalized measure. This implies by Prop. 3.5 that such an LSI must also hold for the full many-particle measure dρ.
Let β > 1 and h < h c (β) such that V 1 is a double well potential where the depth of the smaller well is denoted by small (V ), cf. Fig. 1. The mean-field Ising model satisfies Proof. The renormalized potential V 1 has on [0, ∞) a global minimum with positive second derivative at some ϕ min satisfying ϕ min = tanh(βϕ min + h). This follows since the renormalized potential (3.2) reduces to and the critical points of this potential are easily found to satisfy ϕ = tanh(βϕ + h), see also [BBS19, Lemma 1.4.6]. For small temperatures, i.e. β → ∞, one has ϕ min = 1 + O(1). We first consider h = 0: In this case, the median of the renormalized measure is located precisely at ϕ = 0 and ϕ min > 0 is one of the two non-degenerate global minima of the renormalized potential (the other minimum is located at −ϕ min by axisymmetry).
An application of Laplace's principle, see [Won01, Ch. II,Theorem 1], shows that for all x > 0 : (4.1) The supremum of (4.1) is attained at x = ϕ min such that Here, we used that for x > ϕ min we get by Laplace's principle and thus lim N →∞ The case x = ϕ min can be treated analogously. Hence, we obtain for the constant D 1 as in (3.11) The symmetry of the distribution for h = 0 implies then that D 0 = D 1 .
We now consider h > 0: The renormalized potential possesses a unique global minimum at some ϕ min and the median of the renormalized measure converges to this point ϕ min , see Fig. 1 Hence, it suffices to verify the LSI bounds (3.11) for m = ϕ min as argued in Remark 2. Arguing as in (4.2) yields for h > 0 and x < ϕ min : by taking x to be the minimum of the smaller well of the renormalized potential. For the constant D 1 we get on the other hand for x > ϕ min , since the renormalized potential is monotonically increasing on [ϕ min , ∞), such that D 1 is negligible compared with D 0 .

Upper bound on spectral gap in weak field
The upper bound on the spectral gap is obtained by finding an explicit trial function saturating the SGI. For this construction, we use the notation and results of Lemma B.1.
In order to fix ideas first, we assume h = 0. We start by observing that the mean spin σ can only take values in the set M := {−1, −1 + 2/N , ..., 1} . The weights of the stationary measure dρ are given by functions where we used (2.1).
We also introduce trial functions f N : {±1} N → R for the spectral gap inequality given by with indicator function 1l and γ 3 (β) is the largest solution to ϕ = tanh(βϕ + h). Since f N depends only on the mean spin, we can identify them with functions g N : For the L 2 norm of the f N we find where Z is the normalization constant of the full measure dρ. For the gradient of f N we find . (4.8) Using (3.10) with μ( ) = 1 2 implies by comparing (4.7) with (4.8) that the constant γ in the SGI is bounded from below by We recall from the discussion in Lemma B.1 that the continuous approximation η N (i) attains its maximum in the limit at i = γ 3 (β) and the summand 1 η N (i) in the second sum attains its maximum in the limit at i = 0.
Thus it suffices to study the asymptotic of the logarithm of the leading order summands in (4.9) using the asymptotic behaviour of ζ N := ∂ s log(η N (s)) given in (B.1) Here, we used integration of the inverse function to obtain the last line and (3.5) in the last one. In the case of a positive weak magnetic field h (4.10) Proceeding as above in (4.7) we obtain for the L 2 norm the lower bound For the Dirichlet form we find, as for (4.8), for some C > 0 . (4.12) We can apply (3.10) with μ( ) = 1 1−ε for some ε > 0 since the trial function (4.10) vanishes to the right of the global maximum such that by comparing (4.11) with (4.12) the constant γ in the SGI is bounded from below by (4.13) The weight η N ,h (i) in the first sum attain their maximum (in the limit) at i = γ 1 (β) and the summands 1 η N ,h (i) in the second sum attain their maximum at i = γ 2 (β). To explicitly state an upper bound on the spectral gap it suffices to study the asymptotic of the logarithm of the leading order summands

Spectral gap in strong magnetic field regime h > h c (β).
Next, we study the case of strong magnetic fields for the Ising model, that is V 1 has at most one root, for β > 1.
We also include the case β = 1 and h = 0. Unlike in the case of weak magnetic fields, in which case the constant in the LSI for the renormalized measure is exponentially increasing in the number of spins, the spectral gap of the renormalized measure is now linearly increasing in the number of spins. Responsible for this uniform gap is the local uniform convexity at the minimum of the renormalized potential. More precisely, we have V 1 (ϕ) = β(ϕ − tanh(βϕ + h)) and V 1 (ϕ) = β(1 − β sech(βϕ + h) 2 ). (4.14) Thus, V 1 (ϕ) = 0 yields ϕ ± = −h±arccosh( Proof. Since the renormalized Schrödinger operator and renormalized generator are unitarily equivalent up to a factor, see (3.9), the semiclassical eigenvalue distribution stated in [Sim83, Theo. 1.1] implies the statement of the Proposition: It follows immediately from the renormalized Schrödinger operator (3.9) that the low-lying eigenfunction of ren accumulate at the unique non-degenerate (the second derivative is non-zero) potential well and the spectral gap of the renormalized measure grows linearly in N . The result then follows from Prop. 3.5.

Critical magnetic fields in n = 1.
Proposition 4.3. Let h = h ± and β > 1. The spectral gap of the radial renormalized Schrödinger operator grows as (N 2/3 ) and in particular, the spectral gap of the full measure does not close faster than (N −1/3 ).
Proof. Let λ := N /2 and consider the Schrödinger operator, defined in (3.9), for the renormalized potential and auxiliary Schrödinger operators, which are obtained as the Taylor expansion of (4.16) on L 2 (R) localized to the two critical points, the inflection point ϕ ± and the global minimum ϕ * . We then define j ∈ C ∞ c ((−2, 2); [0, 1]) such that j (x) = 1 for |x| ≤ 1 and from this functions (4.18) Invoking then unitary maps U ϕ * , U ϕ ± ∈ L(L 2 (R)) defined as shows that the two Schrödinger operators in (4.17) are in fact unitarily equivalent, up to multiplication by powers of λ, to the λ-independent Schrödinger operators (4.20) respectively. Both operators have discrete spectrum and that inf(Spec(S ϕ ± )) > 0 is shown in Section C. We illustrate the behaviour of the smallest eigenvalues of S ϕ ± in Figure 4. More precisely, we have that Taylor expansion of the potential at the respective critical point and the estimate on the gradient (4.18) imply that Let 0 = e 1 < e 2 ≤ .. be the eigenvalues (counting multiplicities) of S ϕ * and 0 < f 1 ≤ f 2 ≤ ... the ones of S ϕ ± and choose τ such that λe n+1 > τ > λe n and λ 2/3 f m+1 > τ > λ 2/3 f m with P i being the projection onto the eigenspace to all eigenvalues of S i below τ. The IMS formula, see [CFKS87,(11.37)] for a version on manifolds, implies that (4.23) On the other hand, it follows that In particular, we find From (4.23) we then conclude that for some C > 0 This implies the claim of the Proposition, since rank (J 0 H ± P J 0 ) ≤ n.
In particular, the lowest possible eigenvalue e 1 = 0 of the renormalized Schrödinger operator is of course attained as the nullspace of the renormalized Schrödinger operator H ϕ * is non-trivial. This shows that the spectral gap of the renormalized Schrödinger operator grows at least proportional to λ 2/3 .

Multi-component O(n)-Models
5.1. n ≥ 2: Zero magnetic field, h = 0. Let h = 0 then the renormalized potential for n ≥ 2 is radially symmetric and possesses a critical point at ϕ = 0. In the supercritical case, i.e. β > n, the renormalized potential possesses another critical radius r = ϕ ∈ (0, 1), see Fig. 6. To see this, we differentiate the renormalized potential  It is now obvious that r = 0 is a critical point of the renormalized potential at which lim r ↓0 I n/2 (βr ) where we used that β > n is supercritical. To conclude the existence of precisely one other critical radius r min at which the renormalized potential attains its global minimum it suffices therefore to show that I n/2 (βr ) r I n/2−1 (βr ) decays monotonically to zero. We prove this in Lemma B.2 in the appendix. This implies that also the factor 1 − I n/2 (βr ) r I n/2−1 (βr ) has precisely one root, i.e. the second critical radius.
In the next proposition we show that the radial part of the measure dν N (ϕ) which we denote by dr N := ν (n) N r n−1 e −N V n (r ) dr in the sequel satisfies a SGI with a constant that is uniformly bounded in the number of spins.

Zero magnetic field-a lower bound on the spectral gap.
When h = 0 and n ≥ 2, then the renormalized Schrödinger operator (3.9) for λ := N /2 is the self-adjoint operator This operator is also rotationally symmetric such that by separating (spherical coordinates) the angular part from the radial part, the remaining radial component rad, ren of the renormalized Schrödinger operator on L 2 ((0, ∞), r n−1 dr) for ∈ N 0 reads rad, Here, the term ( + n − 2) accounts for the eigenvalues of the angular part of the Laplacian. The renormalized potential possesses, when h = 0 and n ≥ 2, exactly two critical radii at which |∂ r V n (r )| 2 = 0. The radii are r = 0 and r = r min , see the beginning of this Section 5.1. However, V n (r ) is strictly concave at 0, i.e. ∂ 2 r V n (0) < 0, and by Lemma B.2 strictly convex at r min such that ∂ 2 r V n (r min ) > 0. This follows from ∂ 2 r V n (r min ) = βr min ∂ r | r =r min 1 − I n/2 (βr ) r I n/2−1 (βr ) > 0, see the beginning of Section 5. By the tensorization principle we already know that the rotational invariance of the renormalized measure implies that the spectral gap inequality for the renormalized measure is at least uniform in N . In our next Proposition we therefore study the low-lying spectrum of the radial component, rad,0 ren , as λ → ∞.
Proposition 5.1. Let h = 0, n ≥ 2 and β > n. The spectral gap of the radial renormalized Schrödinger operator rad,0 ren grows linearly in N . Proof. To study the low-lying spectrum of the radial component of the renormalized Schrödinger operator, let λ := N /2 and consider Schrödinger operators where we use the variable x rather than r to emphasize that the last operator is defined on L 2 (R), unlike the first one which is an operator on L 2 ((0, ∞), r n−1 dr). Observe that in (5.3) we replaced the gradient term of the Schrödinger operator by its Taylor approximation at the critical point. This explains the occurrence of the second derivative at the critical point in (5.3). Invoking the unitary maps U 0 ∈ L(L 2 ((0, ∞), r n−1 dr)) and U r min ∈ L(L 2 (R)) defined as shows that the two Schrödinger operators in (5.3) are in fact unitarily equivalent, up to multiplication by λ, to the λ-independent Schrödinger operators respectively. More precisely, we have that U −1 0 λ S 0 osc U 0 = H 0 osc (λ) and U −1 r min λ S r min osc U r min = H r min osc (λ). (5.6) Since the bottom of the spectrum of the operator S 0 osc is strictly positive S 0 osc ≥ −∂ 2 r V n (0) > 0, we conclude from (5.6) that the bottom of the spectrum of H 0 osc (λ) increases linearly to infinity as λ → ∞.
To connect the low-energy spectrum of the renormalized Schrödinger operator with the above auxiliary operators, take j ∈ C ∞ c (−∞, 2) such that j (x) = 1 for |x| ≤ 1. Then, we define for i ∈ {0, r min } and J := 1 − J 2 r min − J 2 0 . Without loss of generality we can assume that λ is large enough such that J 0 and J r min are disjoint.
Taylor expansion of the potential at 0 and r min respectively and the estimate on the gradient (5.7) imply that Let 0 = e 1 < e 2 ≤ .. be the eigenvalues (counting multiplicities) of S 0 osc ⊕ S r osc and choose τ such that e n+1 > τ > e n with P i being the projection onto the eigenspace to all eigenvalues of H i osc below τ λ. The IMS (Ismagilov, Morgan, and Simon/Sigal) formula, see [CFKS87,(11.37)] for a version on manifolds, implies that rad,0 ren = J rad,0 ren J − |∂ r J | 2 + i∈{0,r min } J i rad,0 (5.8) On the other hand, it follows that By construction, since ∇V n vanishes linearly on the support of J i , we have Since V n is globally bounded anyway, this implies for large λ that J rad,0 ren J ≥ J 2 (cλ 6/5 − λ) ≥ λe n J 2 . (5.9) From (5.8) we then conclude that for some C > 0 rad,0 ren ≥ λe n − Cλ 4/5 + i∈{0,r min } This implies the claim of the Proposition, since More precisely, for the eigenvalues E 1 (λ) ≤ E 2 (λ) ≤ .. of rad,0 ren we have shown that In particular, the lowest possible eigenvalue e 1 = 0 of the renormalized Schrödinger operator is of course attained as the nullspace of the renormalized Schrödinger operator is non-trivial. This shows that the spectral gap of the renormalized Schrödinger operator grows at least linearly in λ in the angular sector = 0. Proof. Since the spectral gap of the radial component of the renormalized measure grows linearly in N and the spectral gap of the angular component is uniform in N , the tensorization principle implies that the full renormalized measure satisfies a SGI that is uniform in N . Due to Proposition 3.5, the spectral gap of the full measure does therefore not close faster than of order 1/N .
For radial functions f , the R n ϕ → E μ ϕ ( f ) maps also into radial functions and therefore the spectral gap of the renormalized measure is only determined by the radial renormalized Schrödinger operator in Prop. 5.1. Using Proposition 3.5 and (3.14), this implies that for radial functions, the gap remains open.
In the next Proposition we show that the rate N −1 in this case is in fact optimal: Proposition 5.3. Let h = 0, n ≥ 2 and β > n. The spectral gap of the full measure ρ of the dynamics decays at least as fast as N −1 .
Proof. We consider the mean-spinσ : In analogy to the spherical harmonics which in cartesian coordinates reads x 1 / x , we consider the function: where η ∈ C ∞ (R; [0, 1]) is a cut-off function such that for fixed 1 > δ > 0: η(t) := 1, when t > δ and 0, when t ≤ δ/2. (5.12) As we want to compute the covariant derivative ∇ σ 1 t (σ ), we consider the parametrisation γ 1 (t) so that γ 1 (0) = σ 1 . Then we define γ (t) := (γ 1 (t), σ 2 , . . . , σ N ) and s(t) := σ (γ (t)). It is then clear that for v := γ 1 (0) we have s (0) = v/N , the first coordinate of which is s 1 (0) = e 1 , v /N . We define then Thus, since f (γ (t)) = s 1 (t) s(t) , we find for the derivative (5.14) Therefore, we see that in terms of the derivative is just The cut-off function η ensures that |σ (σ )| is not small. Therefore we can bound By rotational symmetry we also know that E ρ ( f ) = 0. For the second moment E ρ ( f (σ ) 2 ), we have by rotational invariance again where R N is the error Our aim is now to argue that R N is small as N is large. For β > n we know that the renormalized potential attains its minimum at hyperspheres ∂ B R n (0, r min ). This implies that the renormalized measure concentrates at such ϕ ∈ ∂ B(0, r min ) with exponential tail bounds, i.e. the probability of ϕ away from ∂ B(0, r min ) is exponentially small in N . The fluctuation measure then enforces that also the mean spinσ has to be outside of a ball of radius δ > 0 with high probability. To see this recall that the fluctuation measure can be rewritten as Hence, we can pick δ such that e β ϕ,σ < (1 + N (ϕ))/2 for all ϕ ∈ ∂ B(0, r min ) and |σ | ≤ δ. Hence, we see that for such ϕ This shows that E ρ (1l |σ |≤δ ) = O 1 2N (ϕ) + 1 2 N and hence that R N tends exponentially fast to zero as well, as N tends to infinity, under the condition that β > n.
5.3. Nonzero magnetic fields for n ≥ 2. The situation h = 0 and n ≥ 2 cannot be reduced to a one-dimensional model due to lack of symmetries. Yet, the renormalized Schrödinger operator provides a very elegant tool to show that the spectral gap of the full generator of the Ginzburg-Landau dynamics dynamics remains open as N → ∞.
In fact, whereas the global minimum for h = 0 of the renormalized potential is attained on a hypersphere, the global minimum for h = 0 is attained at a single point, only. This allows us to identify the asymptotic of the low-energy spectrum of the renormalized Schrödinger operator directly with the spectrum of a quantum harmonic oscillator.
Let ϕ c ∈ R n be a critical point of the renormalized potential (3.2). We define the set where λ 1 , ..., λ n comprise the entire spectrum of D 2 V n (ϕ c ). Let e k be the k-th smallest element counting multiplicity in we then have the following Proposition: Proposition 5.4. Let h = 0, β ≥ n, and n ≥ 2. Let E k (λ) denote the k-th lowest eigenvalue of the renormalized generator then this eigenvalue satisfies the asymptotic law lim λ→∞ In particular, the ground state of the renormalized generator in the limit as λ → ∞ is unique and the spectral gap of the renormalized Schrödinger operator remains open and linearly in λ.
Proof. When h = 0 then the renormalized potential has a unique non-degenerate minimum. To see this recall that the renormalized potential reads Introducing the new variable ζ := βϕ + h implies that Computing the gradient of that expression yields where we introduced the auxiliary function g β (r ) := r β − I n/2 (r ) I n/2−1 (r ) . Thus for the gradient to vanish the vectors h and ζ have to be linearly dependent.
Assuming thus that e h = ± e ζ we obtain from setting the gradient to zero the following equation β I n/2 ( ζ ) Thus, when h and ζ are aligned, there is precisely one solution, the global minimum of the renormalized potential, satisfying β I n/2 ( ζ ) with g β ( ζ ) = β −1 h > 0. That the aligned scenario corresponds to the global minimum is evident from the expression of the renormalized potential (5.17). The simplicity of the solution follows since the left hand side β I n/2 ( ζ ) I n/2−1 ( ζ ) is a concave, monotonically increasing function from 0 to β as ζ → ∞.
When h and ζ point in opposite directions, there can, by concavity of the left-hand side, be between zero and two solutions to the equation β I n/2 ( ζ ) with g β ( ζ ) = −β −1 h < 0. In particular, for sufficiently low temperatures there exists a local maximum and a saddle point of the renormalized potential as shown in Figure 7. From differentiating (5.18), the Hessian is given by We note that the Hessian has full rank unless at critical points unless g β ( ζ ) + g β ( ζ ) ( ζ − ζ −1 ) = 0, since g β ( ζ ) = 0 by (5.18) for non-zero magnetic fields.
In addition, there can be only a saddle point which can only happen at one fixed temperature depending on n.
Finally, if the temperature is sufficiently high, yet still such that β > n, there may be no critical point if h and ζ point in opposite directions. This is in particular the case when β = n and h = 0: Taylor expansion at zero yields β I n/2 ( ζ ) where for β = n we find n (n/2) 2 (1+n/2) = 1 and concavity of the function ζ → β I n/2 ( ζ ) show. Thus ∇V n 2 vanishes at not more than three critical points ϕ c on the span of h. In particular, all eigenvalues of D 2 V n are non-negative only at the global minimum of V n by (5.19), since we already established that g ( ζ ) < 0 at the other two. To see that they are strictly positive there, it suffices to analyze for r = ζ (5.20) Hence, we find that In particular, this expression is strictly positive at the global minimum, since g β ( ζ ) > 0 and I ( ζ ) > 0 by general principles, see Lemma B.2. The asymptotic behaviour of the spectrum of the renormalized Schrödinger operator has been computed in [Sim83] and our above representation of follows by noticing that 1 2 D 2 |∇V n | 2 (ϕ c ) = D 2 V n (ϕ c ) 2 > 0.
Since the renormalized Schrödinger operator and renormalized generator are unitarily equivalent up to a factor, the semiclassical eigenvalue distribution stated in [Sim83, Theorem 1.1] implies the statement of the Proposition.

The Critical Regime, Proof of Theo. 3
We conclude our analysis by investigating the critical case β = n and prove Theorem 3. As before, we distinguish between n = 1 and multi-component systems n ≥ 2: 6.1. Critical Ising model. It follows from (4.1), which always vanishes for all x > 0, that the spectral gap, at the critical point β = n = 1, does not close exponentially fast in the number of spins. We want to show in this subsection that it closes at least polynomially, though. For a refined analysis in dimension n = 1, we recall some basic ideas from discrete Fourier analysis: Let f : {±1} N → C be an arbitrary function on the hypercube. The L 2 ({±1} N , 2 −N dμ count ) inner product on the hypercube is defined as (x 1 , .., x N )g(x 1 , .., x N ).
Since the first term on the right-hand side of this equation is always uniformly bounded by the Dirichlet form, as shown in the proof of Proposition 3.5, it suffices to study the behaviour of the second term. Thus, applying the expectation with respect to the fluctuation measure yields by the Fourier decomposition (6.1) In particular, using the explicit form of V 1 (φ), a direct computation yields for all x ∈ [N ] Using that μ ϕ is a product measure, this implies that the full expression for (6.3) is given by (6.4) Hence, we find for the variance (6.5) For the Dirichlet form, we find, with S 1 S 2 denoting the symmetric difference of sets S 1 and S 2 , (6.6) .
In particular, for the magnetization σ (x) (6.7) we obtain an inequality .
While Proposition 6.1 shows that the magnetization leads for critical β = 1 to a spectral gap that closes at least like ∼ N −1/2 , when h = 0, the next Proposition shows that the magnetization does not imply a vanishing spectral gap when h > 0.
On the other hand, we can explicitly calculate using the derivative of the modified Bessel function of the first kind, ∂ z I ν (z) = ν z I ν (z) + I ν+1 (z), and (3.2), the expectation value E μ ϕ (σ (x)) that is independent of x ∈ [N ] for ϕ = 0 For the renormalized potential we find by Taylor expansion, which we shall already specialize to critical temperatures β = n, at zero V n (ϕ) = n 2 + n 3 8 + 4n For the magnetization M (6.7), we can write We then have by radial symmetry of both the renormalized potential and the integrand that at critical temperatures β = n Combining this asymptotic behavior with (6.16) and (6.18) then yields the multicomponent claim of Theorem 3, i.e. the rate N 1/2 is caught for the trial (mean spin) function M and thus the spectral gap is decaying at least with speed N −1/2 .
The following Proposition shows that the upper bound N −1/2 on the spectral gap in the critical regime β = n for all dimensions n ≥ 1, is in fact sharp: Proposition 6.3. Let h = 0 and β = n ≥ 1. The spectral gap of the radial renormalized Schrödinger operator grows as (N 1/2 ) and in particular, the spectral gap of the full measure does not close faster than (N −1/2 ).
Since V n vanishes to second order V n R n ≥ cλ −4/9 on J for some c > 0.
This implies for large λ that J H J ≥ √ λe n J 2 . (6.28) From (6.27) we then conclude that for some C > 0 This implies the claim of the Proposition, since rank (J 0 H n P J 0 ) ≤ n.
In particular, the lowest possible eigenvalue e 1 = 0 of the renormalized Schrödinger operator is of course attained as the nullspace of the renormalized Schrödinger operator is non-trivial. This shows that the spectral gap of the renormalized Schrödinger operator grows at least proportional to √ λ.

Appendix C. SUSY Quantum Mechanics
We use ideas from supersymmetric quantum mechanics, to show positivity and analyze the ground state of several Schrödinger operators appearing in this article: In one dimension, we recall that using operators with real-valued and smooth superpotential W , we can write In particular, W (x) := √ β(β − 1)βx 2 yields operator S ϕ ± defined in (4.20).
Choosing then W (r ) := n 3 (2+n) r 3 , such that A * A = S n with S n in (6.23), we find by solving and for the variance