On the TAP Equations via the Cavity Approach in the Generic Mixed p-Spin Models

Abstract

In 1977, Thouless, Anderson, and Palmer (TAP) derived a system of consistent equations in terms of the effective magnetization in order to study the free energy in the Sherrington–Kirkpatrick (SK) spin glass model. The solutions to their equations were predicted to contain vital information about the landscapes in the SK Hamiltonian and the TAP free energy and moreover have direct connections to Parisi’s replica ansatz. In this work, we aim to investigate the validity of the TAP equations in the generic mixed p-spin model. By utilizing the ultrametricity of the overlaps, we show that the TAP equations are asymptotically satisfied by the conditional local magnetizations on the asymptotic pure states.

Appendices

Proofs of Propositions 4.2 and 4.3

The proofs of Propositions 4.2 and 4.3 are based on the following lemma:

Lemma A.1

There exists a constant $K= K(\beta , h)>0$ such that for all $N\ge 1$ and small $\epsilon >0,$

$$\begin{aligned} \sum _{p\ge 2}\big (\mathbb E\big \langle |A_{p}^\alpha |^{4} \big \rangle _{N,\beta }\big )^{\frac{1}{4}}\le K\quad \text{ and }\quad \sum _{p\ge 2}\big (\mathbb E\big \langle |B_{p}^{\rho }|^4 \big \rangle _{N,\beta }\big )^{\frac{1}{4}}\le K. \end{aligned}$$

Proof

For notation simplicity, we suppress the superscript $\alpha$ and write $A_p^{\alpha }$ as $A_p$. We handle the series of $A_p$ first. Note that for each p, we only need to consider the case, $N\ge p$, otherwise $A_p=0$ by the definition (41). Write

$$\begin{aligned} \mathbb E\big \langle A_{p}^{4} \big \rangle _{N,\beta }&= \mathbb E\bigg \langle \Big (\frac{\beta _{p}^{2}p!}{N^{p-1}}\Big )^2 \Big (\sum _{1\le i_1<\cdots <i_{p-1}\le N-1} g_{i_1,\ldots , i_{p-1},N}\langle \sigma _{i_1} \rangle _{N,\beta }^{\alpha }\cdots \langle \sigma _{i_{p-1}} \rangle _{N,\beta }^\alpha \Big )^{4} \bigg \rangle _{N,\beta }\nonumber \\&=\Big (\frac{\beta _{p}^{2}p!}{N^{p-1}}\Big )^2\sum _{\alpha \in \Sigma _{N}}\sum _{{{\textbf{i}}}_1,\ldots , {{\textbf{i}}}_{4}}\mathbb E\Big [ G_{N,\beta }(\alpha )\prod _{l=1}^{4}g_{{{\textbf{i}}}_l,N} \prod _{k=1}^{p-1}\langle \sigma _{i_{l,k}} \rangle _{N,\beta }^{\alpha }\Big ], \end{aligned}$$

(62)

where the second sum is over all ${{\textbf{i}}}_l = (i_{l,1},\cdots , i_{l, p-1})$, for $l=1,2,3,4$ that are $(p-1)$-tuples with strictly increasing coordinates from $\{1,2, \ldots , N-1\}^{p-1}$. We note that there are $\left( {\begin{array}{c}N-1\\ p-1\end{array}}\right)$ choices for each ${{\textbf{i}}}_l$. Write $g_{i_{l,1},\ldots , i_{l, p-1}, N}$ as $g_{{{\textbf{i}}}_l, N}$ and let $\Delta _p:=\beta _{p}\sqrt{p!}{N^{-(p-1)/2}}.$ We have

$$\begin{aligned} \Delta _p^2 \left( {\begin{array}{c}N-1\\ p-1\end{array}}\right)&=\frac{\beta _p^2p!}{N^{p-1}}\,\frac{(N-1)(N-2)\cdots (N-(p-1))}{(p-1)!}\\&=\beta _p^2p\frac{N-1}{N}\cdot \frac{N-2}{N}\cdots \frac{N-(p-1)}{N}\le \beta _p^2p. \end{aligned}$$

Also, there exists a constant $C>0$ independent of $p,N,{{\textbf{i}}}$ such that for any $0\le k_1,k_2,k_3,k_4\le 4$, if we let $d=k_1+k_2+k_3+k_4,$ then

$$\begin{aligned} \Bigl |\partial ^{k_1}_{g_{{{\textbf{i}}}_1,N}}\partial ^{k_2}_{g_{{\textbf{i}}_2,N}} \partial ^{k_3}_{g_{{{\textbf{i}}}_3,N}}\partial ^{k_4}_{g_{{\textbf{i}}_4,N}} G_{N,\beta }(\alpha )\Bigr |&\le C\Delta _p^{d}G_{N,\beta }(\alpha )\le C\Delta _p^d(p-1)^dG_{N,\beta }(\alpha ) \end{aligned}$$

(63)

and

$$\begin{aligned} \Bigl |\partial ^{k_1}_{g_{{{\textbf{i}}}_1,N}}\partial ^{k_2}_{g_{\textbf{i}_2,N}} \partial ^{k_3}_{g_{{{\textbf{i}}}_3,N}}\partial ^{k_4}_{g_{{\textbf{i}}_4,N}} \prod _{l=1}^4\prod _{k=1}^{p-1}\langle \sigma _{i_{l,k}} \rangle _{N,\beta }^{\alpha }\Bigr |&\le C\Delta _p^{d}(p-1)^d. \end{aligned}$$

(64)

These can be established by an induction argument on d. Now we divide the collection of $({{\textbf{i}}}_1,{{\textbf{i}}}_2,{\textbf{i}}_3,{{\textbf{i}}}_4)$ into three cases and compute, respectively, an upper bound for the summand in (62) under each case. In the following discussion, $C_1,C_1',C_2,C_2',\ldots$ are absolute constants independent of N and p.

• Case I: all 4 tuples are distinct. Applying Gaussian integration by part and the chain rule, we get

  $$\begin{aligned}&\ \ \ \Big |\mathbb EG_{N,\beta }(\alpha )g_{{{\textbf{i}}}_1,N}g_{{\textbf{i}}_2,N}g_{{{\textbf{i}}}_3,N}g_{{{\textbf{i}}}_4,N} \prod _{l=1}^{4}\prod _{k=1}^{p-1}\langle \sigma _{i_{l,k}} \rangle _{N,\beta }^{\alpha }\Big | \le C_1\Delta _p^4(p-1)^4\mathbb EG_{N,\beta }(\alpha ) . \end{aligned}$$

  Since the number of choices for $({{\textbf{i}}}_1,{{\textbf{i}}}_2, {\textbf{i}}_3, {{\textbf{i}}}_4)$ in Case I are no more than $\left( {\begin{array}{c}N-1\\ p-1\end{array}}\right) ^4$, the summation in (62) for Case I is bounded by

  $$\begin{aligned}&\Delta _p^4\sum _{\alpha \in \Sigma _{N}}\sum _{{\tiny \mathrm Case\,\,I}} \Big |\mathbb EG_{N,\beta }(\alpha )g_{{{\textbf{i}}}_1,N}g_{{{\textbf{i}}}_2,N}g_{{{\textbf{i}}}_3,N}g_{{{\textbf{i}}}_4,N} \prod _{l=1}^{4}\prod _{k=1}^{p-1}\langle \sigma _{i_{l,k}} \rangle _{N,\beta }^{\alpha }\Big |\\&\quad \le \Delta _p^4\cdot \left( {\begin{array}{c}N-1\\ p-1\end{array}}\right) ^4\cdot C_1\Delta _p^4(p-1)^4\le C_1\beta _p^8p^8. \end{aligned}$$

• Case II: there are three distinct tuples in $({{\textbf{i}}}_1, {{\textbf{i}}}_2, {{\textbf{i}}}_3, {{\textbf{i}}}_4)$. Without loss of generality, suppose ${{\textbf{i}}}_1 = {{\textbf{i}}}_2$ and they are both different from distinct ${{\textbf{i}}}_3,{{\textbf{i}}}_4$. In this case, again using Gaussian integration by part twice and the chain rule, each summand in (62) is bounded in absolute value by

  $$\begin{aligned} \Bigl |\mathbb EG_{N,\beta }(\alpha )g^2_{{{\textbf{i}}}_1,N}g_{{\textbf{i}}_3,N}g_{{{\textbf{i}}}_4,N}\prod _{l=1}^{4} \prod _{k=1}^{p-1}\langle \sigma _{i_{l,k}} \rangle _{N,\beta }^{\alpha }\Bigr |\le C_2 \Delta _p^2(p-1)^2\mathbb Eg_{{{\textbf{i}}}_1,N}^2 G_{N,\beta }(\alpha ). \end{aligned}$$

  It follows that

  $$\begin{aligned}&\Delta _p^4\sum _{\alpha \in \Sigma _{N}}\sum _{{\tiny \mathrm Case\,\,II}} \Big |\mathbb EG_{N,\beta }(\alpha )g_{{{\textbf{i}}}_1,N}g_{{{\textbf{i}}}_2,N}g_{{{\textbf{i}}}_3,N}g_{{{\textbf{i}}}_4,N} \prod _{l=1}^{4}\prod _{k=1}^{p-1}\langle \sigma _{i_{l,k}} \rangle _{N,\beta }^{\alpha }\Big |\\&\quad \le C_2'\Delta _p^4\cdot \left( {\begin{array}{c}N-1\\ p-1\end{array}}\right) ^3\cdot \Delta _p^2(p-1)^2 \cdot \mathbb Eg_{{{\textbf{i}}}_1,N}^2\\&\quad \le C_2'\Delta _p^6 \left( {\begin{array}{c}N-1\\ p-1\end{array}}\right) ^3(p-1)^2\le C_2'\beta _p^6p^5 \end{aligned}$$

• Case III: there are no more than two distinct tuples. In this case, we have three possibilities, each bounded in absolute value respectively as follows:

  $$\begin{aligned} \Bigl |\mathbb EG_{N,\beta }(\alpha )g^2_{{{\textbf{i}}}_1,N}g^2_{{{\textbf{i}}}_2,N}\prod _{l=1}^{4} \prod _{k=1}^{p-1}\langle \sigma _{i_{l,k}} \rangle _{N,\beta }^{\alpha }\Bigr |&\le \mathbb EG_{N,\beta }(\alpha )g^2_{{{\textbf{i}}}_1,N}g^2_{{{\textbf{i}}}_2,N} ,\\ \Bigl |\mathbb EG_{N,\beta }(\alpha )g^1_{{{\textbf{i}}}_1,N}g^3_{{{\textbf{i}}}_2,N}\prod _{l=1}^{4} \prod _{k=1}^{p-1}\langle \sigma _{i_{l,k}} \rangle _{N,\beta }^{\alpha }\Bigr |&\le \mathbb EG_{N,\beta }(\alpha ) \left| g^1_{{{\textbf{i}}}_1,N}g^3_{{{\textbf{i}}}_2,N}\right| ,\\ \Bigl |\mathbb EG_{N,\beta }(\alpha )g^4_{{{\textbf{i}}}_1,N}\prod _{l=1}^{4} \prod _{k=1}^{p-1}\langle \sigma _{i_{l,k}} \rangle _{N,\beta }^{\alpha }\Bigr |&\le \mathbb EG_{N,\beta }(\alpha )g^4_{{{\textbf{i}}}_1,N}. \end{aligned}$$

  Consequently,

  $$\begin{aligned}&\Delta _p^4\sum _{\alpha \in \Sigma _{N}}\sum _{{\tiny \mathrm Case\,\,III}} \Big |\mathbb EG_{N,\beta }(\alpha )g_{{{\textbf{i}}}_1,N}g_{{{\textbf{i}}}_2,N}g_{{{\textbf{i}}}_3,N}g_{{{\textbf{i}}}_4,N} \prod _{l=1}^{4}\prod _{k=1}^{p-1}\langle \sigma _{i_{l,k}} \rangle _{N,\beta }^{\alpha }\Big |\\&\quad \le C_3\Delta _p^4\cdot \left( {\begin{array}{c}N-1\\ p-1\end{array}}\right) ^2\cdot \left[ \mathbb Eg^2_{{{\textbf{i}}}_1,N}g^2_{{{\textbf{i}}}_2,N} + \mathbb E\left| g^1_{{{\textbf{i}}}_1,N}g^3_{{{\textbf{i}}}_2,N}\right| + \mathbb Eg^4_{{{\textbf{i}}}_1,N}\right] \\&\quad \le C_3'\Delta _p^4\cdot \left( {\begin{array}{c}N-1\\ p-1\end{array}}\right) ^2 \le C_3'\beta _p^4p^2. \end{aligned}$$

Combining all three cases, we have

$$\begin{aligned} \mathbb E\big \langle A_{p}^{4} \big \rangle _{N,\beta }&\le C_4\bigl (\beta _{p}^{4}p^2 + \beta _{p}^{6}p^{5}+\beta _p^{8}p^8\bigr ). \end{aligned}$$

Since $\sum _{p\ge 2} 2^{p}\beta _{p}^{2} < \infty$, we have $\beta _{p}^{2}= o(2^{-p})$ as $p\rightarrow \infty$. Choosing $p_0$ large enough such that $\beta _{p} \le 2^{-p/2}$ and $p^2 < 2^{p/4}$ for all $p>p_0$, it follows that

$$\begin{aligned} \sum _{p>p_{0}}\bigl (\mathbb E\big \langle A_{p}^{4} \big \rangle _{N,\beta }\bigr )^{1/4}&\le C_4 \sum _{p> p_{0}}\bigl (\beta _{p}^{4}p^2 + \beta _{p}^{6}p^{5}+\beta _p^{8}p^8\bigr )^{1/4}\le 3C_4 \sum _{p> p_{0}}\beta _{p}p^{2}\\&\le 3C_4 \sum _{p>p_0} 2^{-p/2}p^2 \le 3C_4 \sum _{p>p_0} 2^{-p/4} <\infty . \end{aligned}$$

For the summability for the series of $B_p^\rho$, the proof is essentially the same; the only change is that in (62), $\langle \sigma _{j} \rangle _{N,\beta }^{\alpha }$ will be replaced by $s_{j}^\rho$. Notice that $|s_j^\rho | \le 1$ and any partial derivatives of $s_{j}^\rho$ of degree $d\le 4$ with respect to the variables $(g_{{{\textbf{i}}},N})_{{{\textbf{i}}}}$ are bounded by $\Delta _p^d$ up to an absolute constant independent of p, N and ${{\textbf{i}}}$. For example,

$$\begin{aligned} \bigg |\frac{\partial s_{j}^{\rho }}{\partial g_{{{\textbf{i}}}, N}} \bigg |&=\bigg | \Delta _p \frac{\big \langle \tau _j\tau _{i_1}\cdots \tau _{i_{p-1}}\tau _N\sinh (X_{N,\beta }(\tau )+h) \big \rangle _{N-1,\beta '}^{\rho }}{\big \langle \cosh (X_{N,\beta }(\tau )+h) \big \rangle _{N-1,\beta '}^{\rho }}\Big .\\&\qquad \Big .- \Delta _p \frac{s_j^{\rho }\big \langle \tau _{i_1}\cdots \tau _{i_{p-1}}\tau _N\sinh (X_{N,\beta }(\tau )+h) \big \rangle _{N-1,\beta '}^{\rho }}{\big \langle \cosh (X_{N,\beta }(\tau )+h) \big \rangle _{N-1,\beta '}^{\rho }}\bigg |\le 2\Delta _p. \end{aligned}$$

More general partial derivatives can be controlled by an induction argument on the number of differentiations. This implies that (64) with $\langle \sigma _{j} \rangle _{N,\beta }^{\alpha }$ replaced by $s_j^\rho$ is also valid. We omit the rest of the details. $\square$

Proof of Proposition 4.2

Similar to (65), we have

$$\begin{aligned} \mathbb E\bigg \langle \Big [\sum _{p>p_0}A_p^\alpha \Big ]^{4} \bigg \rangle _{N, \beta } \le \Big (\sum _{p>p_0}\bigl (\mathbb E\big \langle A_{p}^{4} \big \rangle _{N,\beta }\bigr )^{\frac{1}{4}}\Big )^{4}. \end{aligned}$$

Since $\bigl (\mathbb E\big \langle A_{p}^{4} \big \rangle _{N,\beta }\bigr )^{1/4}$ is summable, as proved in Lemma A.1, the right hand side can be made arbitrarily small by choosing $p_0$ sufficiently large. The other assertion can be treated similarly. $\square$

Proof of Proposition 4.3

Note that for any $p_1, p_2,p_3, p_{4}\ge 2$, using Hölder’s inequality yields

$$\begin{aligned} \mathbb E\big \langle A_{p_{1}}^\alpha A_{p_2}^\alpha A_{p_3}^\alpha A_{p_{4}}^\alpha \big \rangle _{N,\beta } \le \Big (\mathbb E\big \langle |A_{p_1}^\alpha |^{4} \big \rangle _{N,\beta }\mathbb E\big \langle |A_{p_2}^\alpha |^{4} \big \rangle _{N,\beta }\mathbb E\big \langle |A_{p_3}^\alpha |^{4} \big \rangle _{N,\beta }\mathbb E\big \langle |A_{p_{4}}^\alpha |^{4} \big \rangle _{N,\beta }\Big )^{\frac{1}{4}}, \end{aligned}$$

which implies that

$$\begin{aligned} \mathbb E\big \langle \big [X_{N,\beta }\bigl (\langle \sigma \rangle _{N,\beta }^{\alpha }\bigr )\big ]^{4} \big \rangle _{N, \beta } = \mathbb E\bigg \langle \Big (\sum _{p\ge 2}A_{p}^\alpha \Big )^{4} \bigg \rangle _{N,\beta }\le \Big (\sum _{p\ge 2}\big (\mathbb E\big \langle |A_{p}^\alpha |^{4} \big \rangle _{N,\beta }\big )^{\frac{1}{4}}\Big )^{4}. \end{aligned}$$

(65)

By the Cauchy–Schwarz inequality and Lemma A.1,

$$\begin{aligned} \mathbb E\big \langle \big [X_{N,\beta }(\langle \sigma \rangle _{N,\beta }^{\alpha }){\mathbb {1}}_{\{G_{N-1,\beta '}(A_{\ominus }^{\rho })<\delta \}}\big ]^2 \big \rangle _{N, \beta }&\le \Big (\mathbb E\big \langle X^4_{N,\beta }(\langle \sigma \rangle _{N,\beta }^{\alpha }) \big \rangle _{N,\beta }\mathbb E\big \langle {\mathbb {1}}_{\{G_{N-1,\beta '}(A_{\ominus }^{\rho })<\delta \}} \big \rangle _{N, \beta } \Big )^{1/2}\\&\le \sqrt{K}\Big (\mathbb E\big \langle {\mathbb {1}}_{\{G_{N-1,\beta '}(A_{\ominus }^{\rho })<\delta \}} \big \rangle _{N, \beta } \Big )^{1/2}. \end{aligned}$$

Thus, (43) follows from (22). The proof of (44) is exactly the same. $\square$

Proof of Lemma 4.4

Proof of Lemma 4.4

Fist of all, for any $\tau \in \Sigma _{N-1}, \varvec{\tau }= (\tau ^1,\ldots ,\tau ^{p-1}) \in \Sigma _{N-1}^{p-1}$, $X_{N,\beta }$ and $Z_{N,p}$ are centered Gaussian random variables with variances bounded by $C_\beta$ and p respectively, which result in

$$\begin{aligned} \mathbb E_{g_{\cdot N}}\bigl \langle |Z_{N,p}(\varvec{\tau })|^{k}\bigr \rangle _{N-1,\beta '}&\le p^{k/2}(k-1)!!,\\ \mathbb E_{g_{\cdot N}}\bigl \langle \cosh ^{k}(X_{N,\beta }(\tau )+h)\bigr \rangle _{N-1,\beta '}&\le e^{C_\beta k^2/2}\cosh ^k(h),\\ \mathbb E_{g_{\cdot N}}\cosh ^k(X_{N,\beta }(\tau )+h)&\le e^{C_\beta k^2/2}\cosh ^k(h). \end{aligned}$$

Using the nested structure (20) and the Hölder inequality with p conjugate exponents $2r(p-1),2r(p-1),\ldots ,2r(p-1)$, and $2r/(2r-1)$, we have

$$\begin{aligned} |D_p^\alpha -D_p^\rho |^{2r}&\le \bigg \langle \prod _{l=1}^{p-1}\cosh (X_{N,\beta }(\tau ^l)+h)\bigg (\prod _{l=1}^{p-1}{\mathbb {1}}_{A_{\oplus }^{\rho }}(\tau ^l) - \prod _{l=1}^{p-1}{\mathbb {1}}_{A_{\ominus }^{\rho }}(\tau ^l)\bigg ) \bigg \rangle _{N-1,\beta '}^{2r}\\&\le \big \langle \cosh ^{2r(p-1)} (X_{N,\beta }(\tau )+h) \big \rangle _{N-1,\beta '}\bigg \langle \bigg (\prod _{l=1}^{p-1}{\mathbb {1}}_{A_{\oplus }^{\rho }}(\tau ^l) - \prod _{l=1}^{p-1}{\mathbb {1}}_{A_{\ominus }^{\rho }}(\tau ^l)\bigg )^{\frac{2r}{2r-1}} \bigg \rangle _{N-1,\beta '}^{2r-1}\\&\le \big \langle \cosh ^{2r(p-1)} (X_{N,\beta }(\tau )+h) \big \rangle _{N-1,\beta '}\bigg \langle \prod _{l=1}^{p-1}{\mathbb {1}}_{A_{\oplus }^{\rho }}(\tau ^l) - \prod _{l=1}^{p-1}{\mathbb {1}}_{A_{\ominus }^{\rho }}(\tau ^l) \bigg \rangle _{N-1,\beta '}, \end{aligned}$$

where the last inequality holds since $\prod _{l=1}^{p-1}{\mathbb {1}}_{A_{\oplus }^{\rho }}(\tau ^l) - \prod _{l=1}^{p-1}{\mathbb {1}}_{A_{\ominus }^{\rho }}(\tau ^l)\in \{0,1\}$. Since

$$\begin{aligned} \prod _{l=1}^{p-1}{\mathbb {1}}_{A_{\oplus }^{\rho }}(\tau ^l) - \prod _{l=1}^{p-1}{\mathbb {1}}_{A_{\ominus }^{\rho }}(\tau ^l)&\le \sum _{l=1}^{p-1}\mathbb {1}_{A_\oplus ^\rho \setminus A_\ominus ^\rho }(\tau ^l), \end{aligned}$$

we have

$$\begin{aligned} |D_p^\alpha -D_p^\rho |^{2r}&\le (p-1) \big \langle \cosh ^{2r(p-1)} (X_{N,\beta }(\tau )+h) \big \rangle _{N-1,\beta '}\big \langle {\mathbb {1}}_{A_{\oplus }^{\rho }\setminus A_{\ominus }^\rho }(\tau ) \big \rangle _{N-1,\beta '}. \end{aligned}$$

From this, by a change of measure for $\alpha =(\rho , \alpha _N)\sim G_{N,\beta }$ as in Lemma 3.2 and the Cauchy–Schwarz inequality, we obtain the second assertion,

$$\begin{aligned} \mathbb E\big \langle (D_p^\alpha -D_p^{\rho })^{2r} \big \rangle _{N,\beta }&\le \mathbb E\Bigl [\sum _{\rho }G_{N-1,\beta '}(\rho )\big \langle {\mathbb {1}}_{A_{\oplus }^{\rho }\setminus A_{\ominus }^\rho }(\tau ) \big \rangle _{N-1,\beta '}\\&\qquad \cdot \mathbb E_{g_{\cdot N}}\bigl [\cosh (X_{N,\beta }(\rho )+h)\big \langle \cosh ^{2r(p-1)} (X_{N,\beta }(\tau )+h) \big \rangle _{N-1,\beta '}\bigr ]\Bigr ]\\&\le \eta _N p e^{(4r^2 p^2 +1)C_\beta }\cosh ^{2rp}(h). \end{aligned}$$

For the first assertion, we similarly have

$$\begin{aligned} |C_p^\alpha -C_p^{\rho }|^{2r}&\le \left\langle Z_{N,p}^2(\varvec{\tau }) \right\rangle ^r_{N-1,\beta '}\bigg \langle \prod _{l=1}^{p-1}\cosh ^2(X_{N,\beta }(\tau ^l)+h)\bigg (\prod _{l=1}^{p-1}{\mathbb {1}}_{A_{\oplus }^{\rho }}(\tau ^l) - \prod _{l=1}^{p-1}{\mathbb {1}}_{A_{\ominus }^{\rho }}(\tau ^l)\bigg )^2 \bigg \rangle _{N-1,\beta '}^{r}\\&\le (p-1)\big \langle Z_{N,p}^{2r}(\varvec{\tau }) \big \rangle ^r_{N-1,\beta '}\big \langle \cosh ^{2r(p-1)}(X_{N,\beta }(\tau )+h) \big \rangle _{N-1,\beta '}\big \langle {\mathbb {1}}_{A_{\oplus }^{\rho }\setminus A_\ominus ^\rho }(\tau ) \big \rangle _{N-1,\beta '}, \end{aligned}$$

where the first inequality used the Cauchy–Schwarz inequality and the second inequality was obtained by an analogous argument for $|D_p^\alpha -D_p^\rho |^{2r}$. Via a change of measure for $G_{N,\beta }$ as above, we can then apply the Hölder inequality in the expectation $\mathbb E_{g_{\cdot N}}$ with thee conjugates exponents 3, 3, 3 to get the desired bound,

$$\begin{aligned} \begin{aligned}&\mathbb E\big \langle |C_p^\alpha -C_p^{\rho }|^{2r} \big \rangle _{N,\beta }\\&\quad \le \mathbb E\Bigl [\sum _{\rho }G_{N-1,\beta '}(\rho )\big \langle {\mathbb {1}}_{A_{\oplus }^{\rho }\setminus A_\ominus ^\rho }(\tau ^l) \big \rangle _{N-1,\beta '}\\&\qquad \cdot \mathbb E_{g_{\cdot N}}\bigl [\cosh (X_{N,\beta }(\rho )+h) \big \langle Z^{2r}_{N,p}(\varvec{\tau }) \big \rangle _{N-1,\beta '}\big \langle \cosh ^{2r(p-1)}(X_{N,\beta }(\tau )+h) \big \rangle _{N-1,\beta '}\bigr ]\Bigr ]\\&\quad \le \eta _N(p-1)p^{r}\bigl [(6r-1)!!\bigr ]^{1/3}e^{(3/2+6r^2 p^2)C_\beta }\cosh ^{2rp}(h). \end{aligned} \end{aligned}$$

(66)

$\square$

Proof of Lemma 5.3

Proof of Lemma 5.3

The proof is essentially the same as that for Proposition 4.2. First of all, we claim that there exists a constant $K= K(\beta , h)>0$ such that for all $N\ge 1$ and any small $\epsilon >0$,

$$\begin{aligned} \sum _{p=2}^\infty \bigl (\mathbb E\langle |E_p^\rho |^4\rangle _{N,\beta }\bigr )^{1/4}&\le K. \end{aligned}$$

(67)

This part of the argument is analogous to the proof of Lemma A.1, but in a slightly simpler manner. We begin by rewriting

$$\begin{aligned} \mathbb E\big \langle |E^{\rho }_{p}|^{4} \big \rangle _{N,\beta }&= \mathbb E\bigg \langle \Big (\frac{\beta _{p}^{2}p!}{N^{p-1}}\Big )^2 \bigg (\sum _{1\le i_1<\cdots <i_{p-1}\le N-1}^{N-1} g_{i_1,\ldots , i_{p-1},N}\langle \sigma _{i_1} \rangle _{N,\beta '}^{\rho }\cdots \langle \sigma _{i_{p-1}} \rangle _{N-1,\beta '}^{\rho }\bigg )^{4} \bigg \rangle _{N,\beta }\\&=\Big (\frac{\beta _{p}^{2}p!}{N^{p-1}}\Big )^2\sum _{\alpha \in \Sigma _{N}}\sum _{{{\textbf{i}}}_1,\ldots , {{\textbf{i}}}_{4}}\mathbb E\Big [ G_{N,\beta }(\alpha )\prod _{l=1}^{4}g_{{{\textbf{i}}}_l,N} \prod _{k=1}^{p-1}\langle \sigma _{i_{l,k}} \rangle _{N-1,\beta '}^{\rho }\Big ]. \end{aligned}$$

When applying Gaussian integration by parts to control the last equation, we only need to differentiate $G_{N,\beta }(\alpha )$ with respect to $g_{{{\textbf{i}}}_l, N}$ and the bounds of the partial derivatives of $G_{N,\beta }(\alpha )$ given by (63). An identical argument as in the proof of Lemma 4.2 implies our claim (67), the summability of $\bigl (\mathbb E\langle |E_p^\rho |^4\rangle _{N,\beta }\bigr )^{1/4}$. With this claim, our assertion follows immediately since, similar to (65),

$$\begin{aligned} \mathbb E\bigg \langle \bigg [\sum _{p> p_0}E_p^\rho \bigg ]^4 \bigg \rangle _{N,\beta }\le \bigg (\sum _{p> p_0}\bigl (\mathbb E\langle |E_p^\rho |^4\rangle _{N,\beta }\bigr )^{1/4}\bigg )^4, \end{aligned}$$

and the right hand side can be made arbitrarily small by choosing $p_0$ sufficiently large. $\quad \square$