Upper bound on the second derivative of the quenched pressure in spin-glass models: weak Griffiths second inequality

The Griffiths inequalities, which characterize that the correlation of ferromagnetic models is always positive, are very important and indispensable in the rigorous analysis of ferromagnetic models. Towards rigorous analyses of finite-dimensional spin-glass models, there are some previous studies [1, 2, 3, 4, 6, 5, 7] which aim to establish the counterpart of the Griffiths inequalities in spin-glass models. These attempts have been partially successful, and, when the distribution function of the interactions is symmetric, the counterpart of the Griffiths first inequality has been obtained in spin-glass models [1, 2]. Besides, it was shown that the ferromagnetic counterpart of the Griffiths second inequality [3, 4] holds on the Nishimori line [8]. However, no counterpart of the Griffiths second inequality has been established for other parameter regions [5]. Rigorous analyses based on the concept of correlation inequalities have not been sufficiently advanced [6, 7].

On the other hand, by focusing only on the distribution of a single interaction among all of the interactions, the previous studies [9, 10] showed that the quenched average of the local energy for Ising models with quenched randomness is always larger than or equal to one in the absence of all the other interactions. Although this is a non-trivial result that holds regardless of any other interaction, an extension to the case with multi variables was not apparent. In this study, we give another derivation of their result [9, 10] and extend it to the case with multi variables. Then, we obtain some correlation inequalities for the Ising models with quenched randomness. Moreover, combining the acquired inequality and our previous work [11] for symmetric distributions, we find that there is a simple upper bound on the second derivative of the quenched pressure with respect to the strength of the randomness. This bound can be regarded as a weak result of the counterpart of the Griffiths second inequality in spin-glass models for general symmetric distributions.

The organization of the paper is as follows. In Sec. II, we define the model and explain the counterpart of the Griffiths inequalities in spin-glass models. In Sec. III, we prove some inequalities for quenched averages in preparation for the next section. Section IV is devoted to obtaining some correlation inequalities for spin glass, which is a systematic extension of the previous study [9, 10]. Furthermore, we provide a non-trivial upper bound on the second derivative of the quenched pressure with respect to the strength of the randomness. Finally, our conclusion is given in Sec. V.

Following Ref. [9], we consider a generic form of the Ising model,

	$\displaystyle H$	$\displaystyle=$	$\displaystyle-\sum_{A\subset{V}}\lambda_{A}J_{A}\sigma_{A},$		(1)
	$\displaystyle\sigma_{A}$	$\displaystyle\equiv$	$\displaystyle\prod_{i\in A}\sigma_{i},$		(2)

where $V$ is the set of sites, the sum over $A$ is over all the subsets of $V$ in which interactions exist, and the lattice structure adopts any form. The probability distribution of a random interaction $J_{A}$ is represented as $P_{A}(J_{A})$. The probability distributions can be generally different from each other, i.e., $P_{A}(x)\neq P_{B}(x)$, and are also allowed to present no randomness, i.e., $P_{A}(J_{A})=\delta(J-J_{A})$. The parameter $\lambda_{A}$ plays a role in controlling the strength of the randomness.

The partition function $Z_{\{J_{A}\}}$ and correlation function $\langle\sigma_{B}\rangle_{\{J_{A}\}}$ for a set of fixed interactions, $\{J_{A}\}$, are given by

	$\displaystyle Z_{\{J_{A}\}}$	$\displaystyle=$	$\displaystyle\Tr\exp\left(\beta\sum_{A\subset{V}}\lambda_{A}J_{A}\sigma_{A}\right),$		(3)
	$\displaystyle\langle\sigma_{B}\rangle_{\{J_{A}\}}$	$\displaystyle=$	$\displaystyle\frac{\Tr\sigma_{B}\exp\left(\beta\sum_{A\subset{V}}\lambda_{A}J_{A}\sigma_{A}\right)}{Z_{\{J_{A}\}}}.$		(4)

The configurational average over the distribution of the interactions is written as

$\displaystyle\mathbb{E}\left[g(\{J_{A}\})\right]=\left(\prod_{A\subset{V}}\int_{-\infty}^{\infty}dJ_{A}P_{A}(J_{A})\right)g(\{J_{A}\}).$

(5)

For example, the quenched average of the correlation function is obtained as

$\displaystyle\mathbb{E}\left[\langle\sigma_{B}\rangle_{\{J_{A}\}}\right]$

$\displaystyle=$

$\displaystyle\left(\prod_{A\subset{V}}\int_{-\infty}^{\infty}dJ_{A}P_{A}(J_{A})\right)\frac{\Tr\sigma_{B}\exp\left(\beta\sum_{A\subset{V}}\lambda_{A}J_{A}\sigma_{A}\right)}{Z_{\{J_{A}\}}}.$

(6)

In addition, the quenched pressure $P$ is defined as

$\displaystyle P$

$\displaystyle=$

$\displaystyle\mathbb{E}\left[\log Z_{\{J_{A}\}}\right].$

(7)

Recent studies showed [1, 2], when the probability distribution of a random interaction $J_{B}$ is symmetric, $P_{B}(J_{B})=P_{B}(-J_{B})$, the first derivative of the quenched pressure with respect to the strength of the randomness is always positive:

$\displaystyle\frac{1}{\beta}\frac{\partial P}{\partial\lambda_{B}}$

$\displaystyle=$

$\displaystyle\mathbb{E}\left[J_{B}\langle\sigma_{B}\rangle_{\{J_{A}\}}\right]\geq 0,$

(8)

which is regarded as the counterpart of the Griffiths first inequality in spin-glass models.

Next, we consider the counterpart of the Griffiths second inequality in spin-glass models. Previous studies [2, 3, 4, 5] have focused on the second derivative of the quenched pressure with respect to the strength of the randomness,

$\displaystyle\frac{1}{\beta^{2}}\frac{\partial^{2}P}{\partial\lambda_{B}\partial\lambda_{C}}$

$\displaystyle=$

$\displaystyle\mathbb{E}\left[J_{B}J_{C}(\langle\sigma_{B}\sigma_{C}\rangle_{\{J_{A}\}}-\langle\sigma_{B}\rangle_{\{J_{A}\}}\langle\sigma_{C}\rangle_{\{J_{A}\}})\right].$

(9)

In ferromagnetic models, the Griffiths second inequality means that the second derivative of the pressure with respect to the ferromagnetic interactions is always positive. Fortunately, on the Nishimori line, a similar relation also holds in spin-glass models and Eq.(9) is always positive [3, 4], which is the ferromagnetic counterpart of the Griffiths second inequality on the Nishimori line in spin-glass models.

In the case of symmetric distributions, however, studies on the counterpart of the Griffiths second inequality have not satisfactorily progressed. When $P_{B}(J_{B})$ and $P_{C}(J_{C})$ follow the symmetric Gaussian distributions with the variance $\Lambda_{B}^{2}$ and $\Lambda_{C}^{2}$, respectively, by integration by parts, Eq. (9) is deformed to

$\displaystyle\frac{1}{\beta^{2}}\frac{\partial^{2}P}{\partial\lambda_{B}\partial\lambda_{C}}$

$\displaystyle=$

$\displaystyle\frac{\Lambda_{B}^{2}}{\beta}\frac{\partial}{\partial\lambda_{C}}\mathbb{E}\left[-\langle\sigma_{B}\rangle_{\{J_{A}\}}^{2}\right].$

(10)

This means that, if Eq. (9) is negative for $\sigma_{B}\neq\sigma_{C}$, the overlap expectation $\mathbb{E}\left[\langle\sigma_{B}\rangle_{\{J_{A}\}}^{2}\right]$ is monotonic non decreasing with the system size [2]. The overlap expectation $\mathbb{E}\left[\langle\sigma_{B}\rangle_{\{J_{A}\}}^{2}\right]$ tends to increase with increasing randomness. Then, as the counterpart of the Griffiths second inequality, it is expected that Eq. (9) is always negative for general symmetric distributions; however, an explicit counterexample exists [5] and it was shown that Eq. (9) takes both positive and negative values depending on the details of the model. Thus, the counterpart of the Griffiths second inequality has not been established even for symmetric distributions in spin-glass models and it is an important problem to investigate when Eq. (9) is negative.

On the other hand, it was shown that, for $P_{B}(J_{B})=P_{B}(-J_{B})$, the first derivative of the quenched pressure with respect to the strength of the randomness has the following upper bound [9],

$\displaystyle\frac{1}{\beta}\frac{\partial P}{\partial\lambda_{B}}=\mathbb{E}\left[J_{B}\langle\sigma_{B}\rangle_{\{J_{A}\}}\right]\leq\mathbb{E}\left[J_{B}\tanh(\beta\lambda_{B}J_{B})\right].$

(11)

We note that Eq. (11) is independent of any other interaction, which is a non-trivial result. The proof of Eq. (11) was obtained by focusing only on the distribution of a single interaction among all of the interactions; however, it is not clear how to extend it to the case with multi variables. In the following sections, we give another proof of Eq. (11) and extend it to the case with multi variables. Then, we obtain some correlation inequalities in the Ising models with quenched randomness. Furthermore, in Sec. IV, we show that Eq. (9) has a non-trivial positive upper bound.

In this section, we prove three inequalities for expectations, which play an important role in the next section.

We consider the case that the distribution functions of $J_{B}$ and $J_{C}$ satisfy the following relations:

	$\displaystyle P_{B}(-J_{B})$	$\displaystyle=$	$\displaystyle\exp(-2\beta_{\text{NL},B}J_{B})P_{B}(J_{B}),$		(12)
	$\displaystyle P_{C}(-J_{C})$	$\displaystyle=$	$\displaystyle\exp(-2\beta_{\text{NL},C}J_{C})P_{C}(J_{C}),$		(13)

where $\beta_{\text{NL},B}$ and $\beta_{\text{NL},C}$ are allowed to be any real values. For example, in the case of the Gaussian distribution

$\displaystyle P_{B}(J_{B})$

$\displaystyle=$

$\displaystyle\frac{1}{\sqrt{2\pi\mu^{2}}}\exp\left(-\frac{(J_{B}-J_{0})^{2}}{2\mu^{2}}\right),$

(14)

and the binary distribution

$\displaystyle P_{B}(J_{B})$

$\displaystyle=$

$\displaystyle p\delta(J_{B}-J)+(1-p)\delta(J_{B}+J),$

(15)

$\beta_{\text{NL},B}$ is given as follows, respectively,

	$\displaystyle\beta_{\text{NL},B}$	$\displaystyle=$	$\displaystyle\frac{J_{0}}{\mu^{2}},$		(16)
	$\displaystyle\beta_{\text{NL},B}$	$\displaystyle=$	$\displaystyle\frac{p}{1-p}.$		(17)

We note that we do not impose any constraint on all the other interactions than $J_{B}$ and $J_{C}$. In addition, for $f(\{J_{A}\})$, when we focus on the interaction $J_{B}$, we denote $f(\{J_{A}\})$ as $f(J_{B},\{J_{A}\}\setminus J_{B})$. Similarly, when we are interested in $J_{B}$ and $J_{C}$, we represent $f(\{J_{A}\})$ as $f(J_{B},J_{C},\{J_{A}\}\setminus\{J_{B},J_{C}\})$

Our first result in this section is as follows.

Using Lemma 3.1, we give another derivation of the inequality for the local energy[9, 10, 11],

$\displaystyle\mathbb{E}\left[-f(\{J_{A}\})J_{B}\langle\sigma_{B}\rangle_{\{J_{A}\}}\right]\geq\mathbb{E}\left[-f(\{J_{A}\})J_{B}\left(\tanh(\beta\lambda_{B}J_{B})+\frac{1-e^{-\beta_{\text{NL},B}J_{B}}}{\sinh(2\beta\lambda_{B}J_{B})}\right)\right],$

(24)

where $f(\{J_{A}\})$ satisfies

	$\displaystyle f(\{J_{A}\})$	$\displaystyle\geq$	$\displaystyle 0,$		(25)
	$\displaystyle f(J_{B},\{J_{A}\}\setminus J_{B})$	$\displaystyle=$	$\displaystyle f(-J_{B},\{J_{A}\}\setminus J_{B}).$		(26)

We note that, for $\beta_{\text{NL},B}=0$ and $f(\{J_{A}\})=1$, this inequality coincides with Eq. (11).

Our second result is a extension of Lemma. 3.1 to two-variable case.

Our third result is another extension of Lemma. 3.1 to two-variable case.

We have proved three lemmas for expectations in Sec III. In this section, using the acquired inequalities, we obtain some correlation inequalities in spin-glass models which is an extension of Eq. (24) to the case with multi variables.