Spherical Spin Glass Model with External Field

Abstract

We analyze the free energy and the overlaps in the 2-spin spherical Sherrington Kirkpatrick spin glass model with an external field for the purpose of understanding the transition between this model and the one without an external field. We compute the limiting values and fluctuations of the free energy as well as three types of overlaps in the setting where the strength of the external field goes to zero as the dimension of the spin variable grows. In particular, we consider overlaps with the external field, the ground state, and a replica. Our methods involve a contour integral representation of the partition function along with random matrix techniques. We also provide computations for the matching between different scaling regimes. Finally, we discuss the implications of our results for susceptibility and for the geometry of the Gibbs measure. Some of the findings of this paper are confirmed rigorously by Landon and Sosoe in their recent paper which came out independently and simultaneously.

Appendices

Appendix A Proof of Lemma 3.3

We prove Lemma 3.3. First,

$$\begin{aligned} \langle e^{\beta \eta \mathfrak {M}}\rangle = \frac{1}{\mathcal {Z}_N(h)} \int _{S_{N - 1}} e^{\beta \frac{ \eta }{N} \mathbf{g }\cdot \varvec{\sigma }} e^{\beta \left( \frac{1}{2} \varvec{\sigma }\cdot M\varvec{\sigma }+h \mathbf{g }\cdot \varvec{\sigma }\right) } \mathrm {d}\omega _N(\varvec{\sigma }) = \frac{\mathcal {Z}_N(h+ \eta N^{-1})}{\mathcal {Z}_N(h)}. \end{aligned}$$

Secondly, by definition,

$$\begin{aligned} \langle e^{\beta \eta \mathfrak {O}} \rangle = \frac{1}{\mathcal {Z}_N} \int _{S_{N - 1}} e^{\beta \frac{\eta }{N} (\mathbf{u }_1 \cdot \sigma )^2} e^{\beta \left( \varvec{\sigma }\cdot M\varvec{\sigma }+h \mathbf{g }\cdot \varvec{\sigma }\right) } \mathrm {d}\omega _N(\varvec{\sigma }). \end{aligned}$$

(A.1)

Since

$$\begin{aligned} \frac{1}{2} \varvec{\sigma }\cdot M\varvec{\sigma }+ \frac{\eta }{N} (\mathbf{u }_1\cdot \sigma )^2= \frac{1}{2} \sum _{i=1}^N \lambda _i (\mathbf{u }_i\cdot \sigma )^2+ \frac{\eta }{N} (\mathbf{u }_1\cdot \sigma )^2, \end{aligned}$$

the integral in (A.1) is the same as that of $\mathcal {Z}_N$ with $\lambda _1\mapsto \lambda _1+ \frac{2\eta }{N}$. Finally, using the eigenvalue-eigenvector decomposition $M=O\Lambda O^T$ and changing variables $\frac{1}{\sqrt{N}} O^T\sigma =x$ and $\frac{1}{\sqrt{N}} O^T \tau =y$, we find that

$$\begin{aligned} \langle e^{\eta \mathfrak {R}} \rangle = \frac{J(\frac{\beta N}{2}, \frac{\beta N}{2}; \frac{\eta }{N \beta }, \frac{\sqrt{\beta }h}{\sqrt{2}}, \frac{\sqrt{\beta }h}{\sqrt{2}})}{J(\frac{\beta N}{2}, \frac{\beta N}{2}; 0, \frac{\sqrt{\beta }h}{\sqrt{2}}, \frac{\sqrt{\beta }h}{\sqrt{2}})}. \end{aligned}$$

(A.2)

where we use the notation

$$\begin{aligned}&J(u,v; a,b,c) \\&\quad = (uv)^{\frac{N}{2} - 1}\displaystyle {\int \int } e^{ 2a\sqrt{uv}\sum \limits _{i = 1}^N x_i y_i +u \sum \limits _{i=1}^N \lambda _i x_i^2 + 2b\sqrt{u} \sum \limits _{i=1}^N n_i x_i + v \sum \limits _{i=1}^N \lambda _i y_i^2 + 2c\sqrt{v} \sum \limits _{i=1}^N n_i y_i } \mathrm {d}\Omega ^{\otimes 2}_{N-1}(x, y) . \end{aligned}$$

We evaluate the Laplace transform of J(u, v, a, b, c). Changing of variable as $u = r^2$, $v = s^2$ and $rx \mapsto x$, $sy \mapsto y$, the Laplace transform

$$\begin{aligned} Q(z,w) = \int _0^\infty \int _0^\infty e^{-zu - wv} J(u,v) \mathrm {d}u \mathrm {d}v \end{aligned}$$

becomes a 2-dimensional Gaussian integral which evaluates to

$$\begin{aligned} Q(z,w) = 4 \prod \limits _{i = 1}^N \frac{\pi }{\sqrt{(z - \lambda _i)(w - \lambda _i) - a^2}} e^{\frac{n_i^2((w - \lambda _i)b^2 + 2abc + (z - \lambda _i) c^2}{(z - \lambda _i)(w - \lambda _i) - a^2}}. \end{aligned}$$

The inverse Laplace transform gives a double integral formula for J(u, v).

Appendix B A perturbation argument

The following perturbation lemma is used to obtain (5.7), (5.22) and (6.4).

Lemma B.1

Let I be a closed interval of ${\mathbb {R}}$. Let G(z; N) be a sequence of random $C^{4}$-functions for $z \in I$. Let $\epsilon = \epsilon (N) := N^{-\delta }$ for some $\delta > 0$ and assume that

$$\begin{aligned} G(z; N)= G_0(z; N) + G_1(z; N) \epsilon + G_2(z; N) \epsilon ^2 + {\mathcal {O}}\left( \epsilon ^3\right) \end{aligned}$$

(B.1)

and

$$\begin{aligned} G'(z; N) = G'_0(z; N) + G'_1(z; N) \epsilon + G'_2(z; N) \epsilon ^2 + {\mathcal {O}}\left( \epsilon ^3\right) \end{aligned}$$

(B.2)

for random $C^{4}$-functions $G_k(z; N)$. Suppose that

$$\begin{aligned} G_k^{(\ell )}(z;N) = {\mathcal {O}}\left( 1\right) \end{aligned}$$

(B.3)

uniformly for $z\in I$ for all $k=0,1,2$, $0\le \ell \le 4$ and also assume that there is a $\gamma _0 \in I$ satisfying

$$\begin{aligned} G_0'(\gamma _0;N)=0, \qquad |G_0''(\gamma _0;N) |\ge C > 0 \end{aligned}$$

(B.4)

for a positive constant C. Then there is a critical point $\gamma =\gamma (N)$ of G(z; N) admitting the asymptotic expansion

$$\begin{aligned} \gamma = \gamma _0+ \gamma _1 \epsilon + \gamma _2 \epsilon ^2 + {\mathcal {O}}\left( \epsilon ^3\right) \end{aligned}$$

(B.5)

where

$$\begin{aligned} \gamma _1 = -\frac{G_1'(\gamma _0;N)}{G_0''(\gamma _0;N)}, \qquad \gamma _2 = -\frac{G_2'(\gamma _0;N) + G_1''(\gamma _0;N)\gamma _1 + \frac{1}{2} G_0'''(\gamma _0;N)\gamma _1^2}{G_0''(\gamma _0;N)}. \end{aligned}$$

(B.6)

Furthermore,

$$\begin{aligned} \begin{aligned} G(\gamma ;N) = G_0(\gamma _0;N) + G_1(\gamma _0;N) \epsilon + \left( \frac{1}{2} G_1'(\gamma _0;N)\gamma _1 + G_2(\gamma _0;N) \right) \epsilon ^2 + {\mathcal {O}}\left( \epsilon ^3\right) . \end{aligned} \end{aligned}$$

(B.7)

Proof

This lemma is standard when G(z; N) is deterministic. The proof for the random G(z; N) does not change. For simplicity, we suppress the dependence on N in the notations; for example we write $G_0(z)$ instead of $G_0(z;N)$. In order to prove (B.5), it is enough to show that for any $0< t < \delta$, $G'(\gamma _+)G'(\gamma _-) < 0$ with $\gamma _\pm = \gamma _0 + \gamma _1 \epsilon + \gamma _2 \epsilon ^2 \pm \epsilon ^3N^t$. From the Taylor expansion,

$$\begin{aligned} \begin{aligned} G'(\gamma _\pm )&= G'_0(\gamma _0) + (G_0''(\gamma )\gamma _1 + G_1'(\gamma _0)) \epsilon \\&\quad + \left( G_0''(\gamma _0)\gamma _2 + G_2'(\gamma _0) + G_1''(\gamma _0)\gamma _1 + \frac{1}{2} G_0'''(\gamma _0)\gamma _1^2 \right) \epsilon ^2 \\&\quad \pm G''_0(\gamma _0) \epsilon ^3 N^t + {\mathcal {O}}\left( \epsilon ^3\right) . \end{aligned} \end{aligned}$$

(B.8)

The definitions of $\gamma _0, \gamma _1$, and $\gamma _2$ imply that

$$\begin{aligned} \begin{aligned} G'(\gamma _\pm )&= \pm G''_0(\gamma _0) \epsilon ^3 N^t + {\mathcal {O}}\left( \epsilon ^3\right) \end{aligned} \end{aligned}$$

(B.9)

Thus, $G'(\gamma _+)G'(\gamma _-) <0$ for all large enough N and we obtain (B.5). The Eq. (B.7) follows from

$$\begin{aligned} \begin{aligned} G(\gamma )&= G_0(\gamma ) + G_1(\gamma ) \epsilon + G_2(\gamma ) \epsilon ^2 + {\mathcal {O}}\left( \epsilon ^3\right) = G_0(\gamma _0) + (G_0'(\gamma _0)\gamma _1 + G_1(\gamma _0)) \epsilon \\&\quad + \left( G_0'(\gamma _0)\gamma _2 + \frac{1}{2} G_0''(\gamma _0)\gamma _1^2 + G_1'(\gamma _0)\gamma _1 + G_2(\gamma _0) \right) \epsilon ^2 + {\mathcal {O}}\left( \epsilon ^3\right) , \end{aligned} \end{aligned}$$

(B.10)

together with $G'_0(\gamma _0) = 0$ and (B.6). $\square$

Remark B.2

Here, we consider the asymptotic expansion of G(z) up to the third order term. One can also consider the case where the expansion is up to the second order, then (B.7) is still valid up to the second order.