Scaling Limits and Critical Behaviour of the $4$-Dimensional $n$-Component $|\varphi |^4$ Spin Model

Abstract

We consider the $n$-component $|\varphi |^4$ spin model on ${\mathbb {Z}}^4$, for all $n \ge 1$, with small coupling constant. We prove that the susceptibility has a logarithmic correction to mean field scaling, with exponent $\frac{n+2}{n+8}$ for the logarithm. We also analyse the asymptotic behaviour of the pressure as the critical point is approached, and prove that the specific heat has fractional logarithmic scaling for $n =1,2,3$; double logarithmic scaling for $n=4$; and is bounded when $n>4$. In addition, for the model defined on the $4$-dimensional discrete torus, we prove that the scaling limit as the critical point is approached is a multiple of a Gaussian free field on the continuum torus, whereas, in the subcritical regime, the scaling limit is Gaussian white noise with intensity given by the susceptibility. The proofs are based on a rigorous renormalisation group method in the spirit of Wilson, developed in a companion series of papers to study the 4-dimensional weakly self-avoiding walk, and adapted here to the $|\varphi |^4$ model.

Appendix: Bounds on Renormalisation Group Map

We now prove Lemma 4.1, the second bound of (4.49), and Lemma 4.3. These are restated here as Lemmas 5.4, 5.5, and 5.6, respectively. (Lemma 5.5 does more, in preparation for the proof of Lemma 5.6.) This involves a detailed analysis of the sequence $u_j$, as well as estimates on second derivatives of the renormalisation group flow with respect to the initial condition $\nu _0$.

From (3.60), we recall that the sequence $u_j$ is defined by

$$\begin{aligned} u_j = \sum _{i=0}^{j-1} \delta u_{i+1}. \end{aligned}$$

(5.1)

The coupling constants $\delta u_{j+1}$ are given by

$$\begin{aligned} \delta u_{j+1} = \delta u_{{\mathrm{pt}}}(V_j) + R^{\delta u}_{j+1}(V_j,K_j), \end{aligned}$$

(5.2)

with $(V_j,K_j)$ the renormalisation group flow of Theorem 3.6, $\delta u_{\mathrm{pt}}$ defined in (3.27), and $R^{\delta u}_+$ the $\delta u$ component of (3.52).

1.1 The Coupling Constant $u$: Proof of Lemma 4.1

We begin with the following lemma concerning $\delta u_{\mathrm{pt}}$ of (3.27).

Lemma 5.1

The coefficients in (3.27) are continuous in $m^2 \in [0,\delta )$ and are uniformly bounded by $O(L^{-dj}{\vartheta }_j)$.

Proof

Except for the coefficient of $g^2$, the claim follows from (3.29)–(3.31) and the facts that $C_{0,0} = O(L^{-2j})$, $\Delta C_{0,0} = O(L^{-4j})$ by (2.11) (the $\delta [(\Delta w)^{(2)}]$ term can be handled similarly to $\zeta$).

We fix any $k>0$ and set $M_j = (1+m^2L^{2j})^{-k}$. With a $k$-dependent constant, $M_j=O({\vartheta }_j)$. The remaining bound to be established is

$$\begin{aligned} \delta [w^{(4)}] - 4 C_{0,0}w^{(3)} + 2 \Delta C_{0,0} (w^3)^{(**)} - 6 C_{0,0}^2 w^{(2)} = O(M_jL^{-4j}). \end{aligned}$$

(5.3)

The left-hand side of (5.3) is equal to

$$\begin{aligned} 4 \sum _x w_x^3 (C_x-C_0-\frac{1}{2} x_1^2 \Delta C_0) + 6 \sum _x w_x^2 (C_x^2-C_0^2) + 4\sum _x w_x C_x^3 + \sum _x C_x^4. \end{aligned}$$

(5.4)

Since $\nabla ^e\nabla ^{-e}C_0 = \nabla ^e\nabla ^{e} C_{0} + O(\Vert \nabla ^3 C\Vert _\infty )$, and by invariance under lattice rotations, $x_1^2\Delta C_0$ can be replaced by $\sum _{i,j=1}^d x_i x_j \nabla ^{e_i}\nabla ^{e_j}C_0 + O(\Vert \nabla ^3 C\Vert _\infty )$. By a discrete Taylor approximation (e.g., as in the proof of [26, Lemma 3.5]),

$$\begin{aligned} \sum _x w_x^3 (C_x-C_0-\frac{1}{2} x_1^2 \Delta C_0)&= \sum _x w_x^3 O(|x|^3 \Vert \nabla ^3 C\Vert _\infty ). \end{aligned}$$

(5.5)

Therefore, using (2.11) to estimate the $C_n$ (e.g., $\Vert \nabla ^3 C\Vert _\infty = M_jL^{-5j}$), and since $C_n$ is supported in a cube with $O(L^{4n})$ points, we obtain

$$\begin{aligned} \sum _x w_x^3 (C_x-C_0-\frac{1}{2} x_1^2 \Delta C_0)&= O(M_jL^{-5j}) \sum _{j \ge i\ge l\ge m} \sum _{x} C_{i;x}C_{l;x}C_{m;x} |x|^3\nonumber \\&= O(M_jL^{-5j}) \sum _{j \ge i \ge l} L^{-2i} L^{-2l} L^{5l}\nonumber \\&= O(M_jL^{-5j}) \sum _{j \ge i} L^{i} = O(M_jL^{-4j}). \end{aligned}$$

(5.6)

Similarly,

$$\begin{aligned} \sum _x w_x^2 (C_x^2-C_0^2) = O(M_j L^{-6j}) \sum _x w_x^2 |x|^2 = O(M_jL^{-6j}) \sum _{k\le j} L^{2k} = O(M_jL^{-4j}). \end{aligned}$$

(5.7)

The last two terms in (5.4) are $O(M_jL^{2j} L^{-6j}) = O(M_jL^{-4j})$ and $O(M_jL^{4j}L^{-8j}) = O(M_jL^{-4j})$ as claimed. This completes the proof. $\square$

The proof of Lemma 4.1 uses the following definition and proposition from [14]. The proof of the proposition is given in [14, Proposition 8.3].

Definition 5.2

• (i) A map $(V,K,m^2) \mapsto F(V,K,m^2)$ acting on a subset of $\mathcal {V}\times \mathcal {K}_j \times [0,\delta )$ with values in a Banach space $E$ is a continuous function of the renormalisation group coordinates at scale-j, if its domain includes $\mathbb {D}_j(\tilde{s}_j) \times \tilde{\mathbb {I}}_{j+1}(\tilde{m}^2)$ for all $\tilde{s}_0 \in [0,\delta ) \times (0,\delta )$, and if its restriction to the domain $\mathbb {D}_j(\tilde{s}_j) \times \tilde{\mathbb {I}}_{j+1}(\tilde{m}^2)$ is continuous as a map $F: \mathbb {D}_j(\tilde{s}_j) \times \tilde{\mathbb {I}}_{j+1}(\tilde{m}^2) \rightarrow E$, for all $\tilde{s}_0 \in [0,\delta ) \times (0,\delta )$. We also say that $F$ is a $C^0$ map of the renormalisation group coordinates.

• (ii) For $k \in {\mathbb {N}}$, a map $F$ is a $C^k$ map of the renormalisation group coordinates at scale-j, if it is a $C^0$ map of the renormalisation group coordinates, its restrictions to the domains $\mathbb {D}_j(\tilde{s}_j) \times \tilde{\mathbb {I}}_{j+1}(\tilde{m}^2)$ are $k$-times continuously Fréchet differentiable in $(V,K)$, and every Fréchet derivative in $(V,K)$, when applied as a multilinear map to directions $\dot{V}\in \mathcal {V}^{p}$ and $\dot{K}\in \mathcal {W}^{q}$, is jointly continuous in all arguments, $m^{2}, V,K, \dot{V}, \dot{K}$.

Proposition 5.3 Let $j < N(\mathbb {V})$, $k \in {\mathbb {N}}_0$, and let $F$ be a $C^k$ map of the renormalisation group coordinates at scale-$j$. Then, for every $p \le k$, all $s_0 \in [0,\delta ) \times (0,\delta )$, the derivative $D_{V_0}^p F(s_0)$ exists, and

$$\begin{aligned} s_0 \mapsto D_{V_0}^p F(s_0)\,\text {is a continuous map}\,\,[0,\delta ) \times (0,\delta ) \rightarrow L^p(\mathcal {V}, E), \end{aligned}$$

(5.8)

where $L^p(\mathcal {V},E)$ is the space of $p$-linear maps from $\mathcal {V}$ to $E$ with the operator norm.

The following lemma is a restatement of Lemma 4.1.

Lemma 5.4

For $(m^2,g_0) \in [0,\delta )^2$, the limit $u_\infty = \lim _{j\rightarrow \infty }u_j$ exists, is continuous in $(m^2,g_0) \in [0,\delta )^2$, and obeys

$$\begin{aligned} u_\infty&= \lim _{j\rightarrow \infty }u_j = u_j + O(L^{-4j}{\vartheta }_j\bar{g}_j). \end{aligned}$$

(5.9)

In particular, since $u_0=0$, $u_\infty = O(g_0)$.

Proof

The first term on the right-hand side of (5.2) is $O(L^{-dj}{\vartheta }_j\bar{g}_j)$, by Lemma 5.1 and Theorem 3.6. The second term is $O(L^{-dj}{\vartheta }_j\bar{g}_j^3)$, by Theorems 3.5–3.6. Thus $\delta u_{j+1}= O(L^{-4j}{\vartheta }_j \bar{g}_j)$.

By Theorems 3.5–3.6, $U_+=(\delta u_+,V_+)$ is a continuous function of the renormalisation group coordinates at scale $j$. Thus, by Proposition 5.3, $(m^2,g_0) \mapsto \delta u_{j+1}$ is a continuous function on $[0,\delta ) \times (0,\delta )$. Since $\delta u_{j+1} = O(L^{-4j}{\vartheta }_j \bar{g}_j) = O(g_0) \rightarrow 0$ as $g_0\downarrow 0$, it follows that $\delta u_{j+1}$ is continuous on $[0,\delta )^2$.

The existence of the limit $u_\infty$ and (5.9) follow immediately from the estimate $\delta u_{j+1}= O(L^{-4j}{\vartheta }_j \bar{g}_j)$. Since $\delta u_{j+1} = O(L^{-4j})$, the sum (5.1) converges uniformly on $(m^2,g_0) \in [0,\delta )^2$ as $j\rightarrow \infty$, so $u_\infty$ is also continuous on $[0,\delta )^2$. This completes the proof. $\square$

1.2 Derivatives of Flow

For a function $f = f(m^2,g_0,\nu _0,z_0)$, we recall the notation

$$\begin{aligned} f'&= \frac{\partial }{\partial \nu _0} f(m^2,g_0, \nu _0^c(m^2,g_0), z_0^c(m^2,g_0)),\end{aligned}$$

(5.10)

$$\begin{aligned} f''&= \frac{\partial ^2}{\partial \nu _0^2} f(m^2,g_0, \nu _0^c(m^2,g_0), z_0^c(m^2,g_0)), \end{aligned}$$

(5.11)

with $(z_0^c,\nu _0^c)$ as in Theorem 3.6. As in [14, Lemma 8.6],

$$\begin{aligned} \check{\mu }_j' = L^{2j}\left( \frac{\check{g}_j}{g_0}\right) ^{\gamma } (c(m^2, g_0) + O(\vartheta _j \check{g}_j)), \end{aligned}$$

(5.12)

where $c(m^2,g_0) = 1+O(g_0)$, and

$$\begin{aligned} \check{g}_j', \check{z}_j' = O({\vartheta }_j \check{\mu }_j'\check{g}_j^2), \quad \Vert K_j'\Vert _{\mathcal {W}_j} = O({\vartheta }_j \check{\mu }_j'\check{g}_j^2). \end{aligned}$$

(5.13)

The following lemma gives similar bounds for second derivatives, via an extension of the proof of [14, Lemma 8.6].

Lemma 5.5

Let $(m^2,g_0)\in [0,\delta ) \times (0,\delta )$, let $(z_0,\nu _0)=(z_0^c,\mu _0^c)$. Then

$$\begin{aligned} \check{\mu }_j'', \check{g}_j'', \check{z}_j'' = O({\vartheta }_j(\check{\mu }_j')^2 \check{g}_j), \quad \Vert K_j''\Vert _{\mathcal {W}_j} = O({\vartheta }_j(\check{\mu }')^2\check{g}_j). \end{aligned}$$

(5.14)

Proof

The proof is by induction, with the induction hypothesis that there exist constants $M_1,M_2 > 0$ such that

$$\begin{aligned} |\check{\mu }_j''|, |\check{g}_j''|, |\check{z}_j''| \le M_1 {\vartheta }_j(\check{\mu }_j')^2 \check{g}_j, \quad \Vert K_j''\Vert _{\mathcal {W}_j} \le M_2 {\vartheta }_j(\check{\mu }_j')^2 \check{g}_j. \end{aligned}$$

(5.15)

This case $j=0$ is trivial since the left-hand sides are $0$. The advancement of the induction uses the fact that, by (5.12),

$$\begin{aligned} \frac{\check{g}_{j}}{\check{g}_{j+1}} = 1+O(\check{g}_j), \quad \frac{\check{\mu }_j'}{\check{\mu }_{j+1}'} = L^{-2}(1+O(\check{g}_j)). \end{aligned}$$

(5.16)

Also, assuming $\Omega \le L$, we have ${\vartheta }_j/{\vartheta }_{j+1} \le L$. Assuming also that $L \ge 4$, we therefore have

$$\begin{aligned} {\vartheta }_j \check{g}_j (\check{\mu }'_j)^2 \le \frac{2L}{L^4} {\vartheta }_{j+1} (\check{\mu }_{j+1}')^2 \check{g}_{j+1}&\le \frac{1}{2L^2}{\vartheta }_{j+1} \check{g}_{j+1} (\check{\mu }_{j+1}')^2 \le \frac{1}{2} {\vartheta }_{j+1} (\check{\mu }_{j+1}')^2 \check{g}_{j+1} . \end{aligned}$$

(5.17)

As in (3.66), (3.67), we write the recursion relation for $(\check{V}_j,K_j)$ as

$$\begin{aligned} \check{V}_{j+1} = \bar{\phi }_j(\check{V}_j) + \check{R}_{j+1}^{(0)}(\check{V}_j,K_j), \quad K_{j+1}=\check{K}_{j+1}(V_j,K_j). \end{aligned}$$

(5.18)

With $F$ equal to either $\check{R}_{j+1}^{(0)}$ or $\check{K}_{j+1}$, the chain rule gives

$$\begin{aligned} F''(\check{V}_j,K_j)&= D_{\check{V}}F(\check{V}_j,K_j)\check{V}_j'' + D_KF(\check{V}_j,K_j)K_j'' + D_{\check{V}}^2F(\check{V}_j,K_j)\check{V}_j'\check{V}_j'\nonumber \\&\quad + D_K^2F(\check{V}_j,K_j)K_j'K_j' + 2D_{\check{V}}D_KF(\check{V}_j,K_j)\check{V}_j'K_j \end{aligned}$$

(5.19)

(here $D_{\check{V}}D_KF(\check{V},K)AB$ denotes the second derivative of $F$ with derivative in the variable $\check{V}$ taken in direction $A$ and derivative in $K$ taken in direction $B$). We use $\Vert \cdot \Vert$ to denote either the norm $\Vert \cdot \Vert _{\mathcal {V}}$ on $\mathcal {V}\cong {\mathbb {R}}^3$ or the norm $\Vert \cdot \Vert _{\mathcal {W}_j}$. By the versions of (3.55) and (3.56) for $\check{R}_+,\check{K}_+$ discussed below (3.65), and by (5.15),

$$\begin{aligned} \Vert D_VF(\check{V}_j,K_j)\check{V}_j''\Vert&\le O({\vartheta }_j \check{g}_j^2)M_1(\check{\mu }_j')^2\check{g}_j\end{aligned}$$

(5.20)

$$\begin{aligned} \Vert D_V^2F(\check{V}_j,K_j)\check{V}_j'\check{V}_j'\Vert&\le O({\vartheta }_j \check{g}_j) (\check{\mu }_j')^2 \end{aligned}$$

(5.21)

$$\begin{aligned} \Vert D_VD_KF(\check{V}_j,K_j)\check{V}_j'K_j'\Vert&\le O(\check{g}_j^{-1}) \check{\mu }_j' ({\vartheta }_j \check{\mu }_j' \check{g}_j^2 )\end{aligned}$$

(5.22)

$$\begin{aligned} \Vert D_K^2F(\check{V}_j,K_j)K_j'K_j'\Vert&\le O({\vartheta }_j^{-1} \check{g}_j^{-10/4}) ({\vartheta }_j \check{\mu }_j' \check{g}_j^2 )^2 \le O({\vartheta }_j (\check{\mu }_j')^2 \check{g}_j^{3/2} )\end{aligned}$$

(5.23)

$$\begin{aligned} \Vert D_K\check{R}_{j+1}^{(0)}(\check{V}_j,K_j)K_j''\Vert&\le O(M_2){\vartheta }_j(\check{\mu }_j')^2 \check{g}_j\end{aligned}$$

(5.24)

$$\begin{aligned} \Vert D_K\check{K}_{j+1}(\check{V}_j,K_j)K_j''\Vert&\le M_2{\vartheta }_j (\check{\mu }_j')^2 \check{g}_j . \end{aligned}$$

(5.25)

This implies, for $M_2 \gg 1$,

$$\begin{aligned} \Vert (\check{R}_{j+1}^{(0)})''(\check{V}_j,K_j)\Vert \le O(M_2) {\vartheta }_j (\check{\mu }_j')^2 \check{g}_j, \quad \Vert \check{K}_{j+1}''(\check{V}_j,K_j)\Vert \le 2M_2 {\vartheta }_j (\check{\mu }_j')^2 \check{g}_j. \end{aligned}$$

(5.26)

The second bound and (5.17) immediately advance the induction for $K_j''$. For $\check{g}_{j+1}''$, we use (5.18). The second derivative of the first term of (5.18) can be bounded using the recursion (3.43) for $\bar{g}$, Proposition 3.2 to estimate the coefficients, and (5.13) and the induction hypothesis (5.15) to estimate the first and second derivatives. With (5.26), this gives

$$\begin{aligned} |\check{g}_{j+1}''|&\le ((1+O(g_j))M_1+O(M_2)){\vartheta }_j (\check{\mu }_j')^2 \check{g}_j . \end{aligned}$$

(5.27)

Therefore,

$$\begin{aligned} |\check{g}_{j+1}''|&\le \frac{1}{2}(M_1+O(M_2)) {\vartheta }_{j+1} (\check{\mu }_{j+1}')^2 \check{g}_{j+1} \le M_1 {\vartheta }_{j+1} (\check{\mu }_{j+1}')^2 \check{g}_{j+1}, \end{aligned}$$

(5.28)

by (5.17) for the second inequality, and using $M_1 \gg M_2$ in the last inequality. The estimates for $\check{z}_j'', \check{\mu }_j''$ are analogous, with the difference that for $\check{\mu }_j''$ there is an additional factor $L^{2}$ (which is bounded analogously, using the second rather than the third inequality in (5.17)). This completes the proof. $\square$

1.3 Derivatives of $u$: Proof of Lemma 4.3

The following lemma is a restatement of Lemma 4.3.

Lemma 5.6

Let $(m^2,g_0)\in (0,\delta )^2$, and let $(z_0,\nu _0)=(z_0^c,\nu _0^c)$. There exist $u_\infty '$ continuous in $(m^2,g_0) \in [0,\delta )^2$ and $u_\infty ''$ continuous in $(m^2,g_0) \in (0,\delta )^2$ such that

$$\begin{aligned} u'_\infty&= \lim _{N\rightarrow \infty } u_N' = \frac{n}{2 } \sum _{j=1}^\infty \check{\nu }_j' C_{j;0,0} + O(g),\end{aligned}$$

(5.29)

$$\begin{aligned} u''_\infty&= \lim _{N\rightarrow \infty } u_N'' = -\frac{n}{2(8+n)} \sum _{j=0}^\infty \beta _j (\check{\nu }_j')^2 + O(1). \end{aligned}$$

(5.30)

The convergence $u_N'\rightarrow u_\infty '$ is uniform in $(m^2,g_0) \in [0,\delta )^2$, and the convergence $u_N'' \rightarrow u_\infty ''$ is uniform on compact subsets of $(m^2,g_0) \in (0,\delta )^2$.

In the proof of Lemma 5.6, we use the transformed variables $(\check{V},K) = (T(V),K)$. As in (3.65), the corresponding version of (5.2) is

$$\begin{aligned} \delta u_{j+1} = \bar{\phi }^{\delta u}_{j}(\check{V}_j) + \check{R}^{\delta u}_{j+1}(\check{V}_j,K_j), \end{aligned}$$

(5.31)

where the map $\bar{\phi }^{\delta u}$ is given by (3.46), and where $\check{R}^{\delta u}_+$ is the transformed version of $R^{\delta u}_+$, defined by $\check{R}^{\delta u}_+(\check{V},K) = \delta u_+(T^{-1}(\check{V}),K) - \bar{\phi }^{\delta u}(\check{V})$. As noted around (3.65), the estimates stated for $R_+$ in Theorem 3.5 hold mutatis mutandis for $\check{R}_+$. In particular,

$$\begin{aligned} \Vert D_V^p D_K^q \check{R}_+^{\delta u}\Vert _{L^{p,q}} \le {\left\{ \begin{array}{ll} O(\tilde{\vartheta }\tilde{g}^{3-p}) &{} (p\ge 0,\, q=0)\\ O(\tilde{g}^{1-p-q}) &{} (p\ge 0,\, q = 1,2). \end{array}\right. } \end{aligned}$$

(5.32)

We recall that the norm (2.32) appearing on the left-hand side of (5.32) scales the $\delta u$ component by a factor $L^{4j}$. Thus, when estimating absolute values of derivatives of $\check{R}^{\delta u}$, we obtain an additional factor $O(L^{-4j})$.

Proof

We first note that, by Lemma 5.1, Theorems 3.5–3.6, and (5.2), $\delta u_+$ is a $C^2$ function of the renormalisation group coordinates at scale-$j$. By Proposition 5.3, each of $\delta u_+,\delta u_+', \delta u_+''$ is therefore continuous on $[0,\delta ) \times (0,\delta )$ (in particular, the derivatives exist).

We now prove the convergence and bounds for $u_j'$. By (3.46) and Lemma 5.1, and by (5.12)–(5.13),

$$\begin{aligned} (\bar{\phi }^{\delta u}_{j+1})' = O(L^{-4j} {\vartheta }_j \check{\mu }_j'). \end{aligned}$$

(5.33)

Similarly, by (5.12)–(5.13), (5.32), and the chain rule,

$$\begin{aligned} (\check{R}_{j+1}^{\delta u})'= O(L^{-4j} {\vartheta }_j \check{\mu }_j' \check{g}_j^2). \end{aligned}$$

(5.34)

The latter bound is obtained when the derivative acts in the $\mu _j$ or $K_j$ direction, with the derivatives in the $g_j,z_j$ directions smaller by a factor $O(\check{g}_j^2)$. By (4.27), it follows in particular that $\delta u_{j+1}' = O(L^{-4j}\check{\mu }_j') = O(L^{-2j})$. Thus $\delta u_{j+1}'$ is summable, uniformly in $(m^2,g_0)\in [0,\delta )^2$. Since $\delta u_{j+1}'$ is continuous in $(m^2,g_0) \in [0,\delta )^2$, as noted in the first paragraph of the proof, this implies that $u'_\infty$ is also continuous on $[0,\delta )^2$, as claimed. By (5.12)–(5.13), and since the coefficients of $\bar{\phi }^{\delta u}$ are uniformly bounded, the dominant contribution in (3.46) is given by $n\check{\nu }_j' C_{j;0,0}$, and its sum over $j$ yields the main term of (5.29). The other terms in (3.46) as well as $(R_{j+1}^{\delta u})'$ are bounded by $O({\vartheta }_j L^{-4j} \check{g}_j \check{\mu }_j') = O(L^{-2j}\check{g}_j)$, whose sum is $O(g)$ as claimed.

We now consider $u_N''$. By (5.12)–(5.14), the dominant contribution in (3.46) for $(\bar{\phi }^{\delta u})''$ is given by the term proportional to

$$\begin{aligned} (\check{\nu }_j^2)'' = 2(\check{\nu }_j')^2 + 2 \check{\nu }_j''\check{\nu }_j = 2(\check{\nu }')^2 (1+O({\vartheta }_j\check{g}_j^2)), \end{aligned}$$

(5.35)

with the other terms bounded by $O(L^{-4j} {\vartheta }_j (\check{\mu }_j')^2 \check{g}_j^2)$. To see the latter, observe that differentiating every monomial in (3.46) gives either one factor from $\check{g}_j,\check{z}_j,\check{\mu }_j$ multiplied with one of $\check{g}_j'',\check{z}_j'',\check{\mu }_j''$, which is $O(L^{-4j} {\vartheta }_j (\check{\mu }_j')^2 \check{g}_j^2)$, or two factors of $\check{g}_j',\check{z}_j',\check{\mu }_j'$ of which the largest is $(\check{\mu }')^2$, i.e., (5.35), with all other combinations bounded by $O(L^{-4j} {\vartheta }_j (\check{\mu }_j')^2 \check{g}_j^2)$. From (5.12)–(5.14) and (5.32), it similarly follows that

$$\begin{aligned} (\check{R}^{\delta u}_{j+1})'' = O(L^{-4j} {\vartheta }_j (\check{\mu }_j')^2 \check{g}_j^2), \end{aligned}$$

(5.36)

which is obtained when both derivatives act in the $\check{\mu }_j$ direction, or if one acts in $\check{\mu }_j$ direction and one in the $K_j$ direction. With (3.46)–(3.47) for the coefficient of the $\check{\nu }^2$ term, since $\delta _j[w^{(2)}] = \beta _j/(8+n)$, it follows that

$$\begin{aligned} u_N'' = - \frac{n}{2(8+n)} \sum _{j=0}^{N-1} \beta _j (\check{\nu }_j')^2 + \sum _{j=0}^{N-1} O({\vartheta }_j \check{g}_j (\check{\nu }_j')^2). \end{aligned}$$

(5.37)

The second term on right-hand side is bounded by

$$\begin{aligned} \sum _{j=0}^\infty \vartheta _j \check{g}_j (\check{\nu }_j')^2 = O(1) \sum _{j=0}^\infty \vartheta _j \frac{\bar{g}_j^{1+2\gamma }}{g_0^{2\gamma }} = O(1), \end{aligned}$$

(5.38)

by (4.55).

We finally show that $u_N \rightarrow u_\infty$ uniformly in $m^2 \in [\varepsilon , \delta )$, for any $\varepsilon \in (0,\delta )$. It suffices to show that this holds for the restriction of both sums in (5.37) to $j \ge j_{\varepsilon } = \lfloor \log _L \varepsilon \rfloor$. Then the summands are uniformly bounded by $O({\vartheta }_j) = O(2^{-(j-j_{\varepsilon })})$, from which the claim is immediate. Thus $u_N'' \rightarrow u_\infty ''$ compactly on $(m^2,g_0) \in (0,\delta )^2$, and since $u_j''$ is continuous, it follows that $u_\infty ''$ is also continuous on $(0,\delta )^2$, as claimed. This completes the proof. $\square$