On the Universality of the Superconcentration in Mixed p-Spin Models

Abstract

Consider the mixed p-spin models with general environments such that the covariance of Hamiltonian process is non-negative. In this paper, we prove the universality of the superconcentration phenomenon. Precisely, we show that the variance of the free energy grows sublinearly in the size of its expectation when the disordered random variable satisfies some moment matching conditions. Additionally, we also study the universality of first and second moments of the free energy of a spin glass model on general hypergraphs.

Appendix: Proof of (10) and (12)

For the completeness of the paper, we present the proof of (10) and (12), though it is similar to that of [1, Lemma 2.2].

Using Taylor’s theorem as in [1, proof of Lemma 2.2], we obtain that

$$\begin{aligned} J&= (y_e \hat{F}(y_e) -\hat{F}'(y_e))-\hat{F}(0)y_e - \sum _{n=1}^{k-1} \hat{F}^{(n)}(0) \Big (\dfrac{y_e^{n+1}}{n!}- \dfrac{y_e^{n}}{(n-1)!} \Big ) \nonumber \\&= -\hat{F}^{(k-1)}(0) \dfrac{y_e^k}{(k-1)!} + \hat{F}^{(k-1)}(x_{e,1}) \dfrac{y_e^k}{(k-1)!} - \hat{F}^{(k)}(x_{e,3}) \dfrac{y_e^{k-1}}{(k-1)!} \end{aligned}$$

(41)

$$\begin{aligned}&= \hat{F}^{(k)}(x_{e,2}) \dfrac{y_e^{k+1}}{k!} - \hat{F}^{(k)}(x_{e,3}) \dfrac{y_e^{k-1}}{(k-1)!} , \end{aligned}$$

(42)

where $x_{e,1},x_{e,2},x_{e,3}$ are some real numbers such that $|x_{e,i}| \le |y_e|$ for $i =1,2,3$. By (41), we have

$$\begin{aligned} |J| \le 2 \dfrac{\sup _{|x_e| \le |y_e|} |\hat{F}^{(k-1)}(x_{e})|}{(k-1)!} |y_e|^k + \dfrac{\sup _{|x_e| \le |y_e|} |\hat{F}^{(k)}(x_{e})|}{(k-1)!} |y_e|^{k-1}, \end{aligned}$$

(43)

and it follows from (42) that

$$\begin{aligned} |J| \le \sup _{|x_e| \le |y_e|} |\hat{F}^{(k)}(x_{e})| \Big ( \dfrac{|y_e|^{k+1}}{k!} + \dfrac{|y_e|^{k-1}}{(k-1)!} \Big ). \end{aligned}$$

(44)

For any $K \ge 1$, taking the expectation on $|y_e| \ge K$ for (43), we have

$$\begin{aligned} {\mathbb {E}}[|J| {\mathbb {I}}(|y_e| \ge K)]&\le \hat{I}_1, \end{aligned}$$

(45)

where

$$\begin{aligned} \hat{I}_1= \dfrac{2}{(k-1)!} {\mathbb {E}}\Big [ \Big ( \sup _{|x_e| \le |y_e|} |\hat{F}^{(k-1)}(x_{e})| + \sup _{|x_e| \le |y_e|} |\hat{F}^{(k)}(x_{e})| \Big ) |y_e|^{k} {\mathbb {I}}( |y_e| \ge K)\Big ]. \end{aligned}$$

Similarly, taking expectation on $|y_e| \le K$ for (44) gives that

$$\begin{aligned} {\mathbb {E}}[|J|&{\mathbb {I}}(|y_e| \le K)] \nonumber \\&\le {\mathbb {E}}\Big [ \sup _{|x_e| \le |y_e|} |\hat{F}^{(k)}(x_{e})| \Big ( \dfrac{|y_e|^{k+1}}{k!} + \dfrac{|y_e|^{k-1} }{(k-1)!} \Big ){\mathbb {I}}(|y_e| \le K) \Big ] \le \hat{I}_2, \end{aligned}$$

(46)

where

$$\begin{aligned} \hat{I}_2 =K {\mathbb {E}}\Big [ \sup _{|x_e| \le |y_e|} |\hat{F}^{(k)}(x_{e})| \Big ( \dfrac{|y_e|^{k}}{k!} + \dfrac{|y_e|^{k-2}}{(k-1)!} \Big ) {\mathbb {I}}(|y_e| \le K) \Big ]. \end{aligned}$$

Since ${\mathbb {E}}[y_e^n]={\mathbb {E}}[g^n]$ for $n=1, \ldots ,k$, we have ${\mathbb {E}}[y_e^{n+1}/(n+1)!] = {\mathbb {E}}[y_e^{n}/n!]$ for $n=1 \ldots , k-1$ and ${\mathbb {E}}[y_e]=0$. Therefore,

$$\begin{aligned} {\mathbb {E}}[y_e \hat{F}(y_e) - \hat{F}'(y_e) ] = {\mathbb {E}}[J]. \end{aligned}$$

(47)

We then deduce from (45) and (46) that

$$\begin{aligned} |{\mathbb {E}}[y_e \hat{F}(y_e) - \hat{F}'(y_e) ]|&= |{\mathbb {E}}[J {\mathbb {I}}(|y_e| \ge K)] + {\mathbb {E}}[J {\mathbb {I}}(|y_e| < K)]| \le \hat{I}_1+ \hat{I}_2, \end{aligned}$$

as desired in (10). Finally, thanks to (44) and (47), we get (12). $\square$