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-rw-r--r--figs/msg_marg_legend.pdfbin4925 -> 4925 bytes
-rw-r--r--figs/msg_marg_params.pdfbin66167 -> 66014 bytes
-rw-r--r--figs/msg_marg_spectra.pdfbin20075 -> 20075 bytes
-rw-r--r--figures.nb6531
-rw-r--r--marginal.tex191
5 files changed, 3173 insertions, 3549 deletions
diff --git a/figs/msg_marg_legend.pdf b/figs/msg_marg_legend.pdf
index 0bf074f..04cf130 100644
--- a/figs/msg_marg_legend.pdf
+++ b/figs/msg_marg_legend.pdf
Binary files differ
diff --git a/figs/msg_marg_params.pdf b/figs/msg_marg_params.pdf
index 3168bbe..b38d142 100644
--- a/figs/msg_marg_params.pdf
+++ b/figs/msg_marg_params.pdf
Binary files differ
diff --git a/figs/msg_marg_spectra.pdf b/figs/msg_marg_spectra.pdf
index 19eda13..a0a7abc 100644
--- a/figs/msg_marg_spectra.pdf
+++ b/figs/msg_marg_spectra.pdf
Binary files differ
diff --git a/figures.nb b/figures.nb
index 071055a..2ae860b 100644
--- a/figures.nb
+++ b/figures.nb
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"HighlightElements" -> <|
"Label" -> {"XYLabel"}, "Ball" -> {"InterpolatedBall"}|>,
"LayoutOptions" -> <|
"PanelPlotLayout" -> <||>,
- "PlotRange" -> {{-0.0005, 0.0005}, {0., 0.24139520540014045`}},
- "Frame" -> {{False, False}, {False, False}},
- "AxesOrigin" -> {0, 0}, "ImageSize" -> {360, 360/GoldenRatio},
- "Axes" -> {True, True}, "LabelStyle" -> {}, "AspectRatio" ->
- GoldenRatio^(-1), "DefaultStyle" -> {
+ "PlotRange" -> {{-0.0005, 0.0005}, {0., 0.7673108712075356}},
+ "Frame" -> {{True, True}, {True, True}}, "AxesOrigin" -> {0, 0},
+ "ImageSize" -> {360, 360/GoldenRatio}, "Axes" -> {True, True},
+ "LabelStyle" -> {FontFamily -> "Times",
+ GrayLevel[0], FontSize -> 11}, "AspectRatio" -> GoldenRatio^(-1),
+ "DefaultStyle" -> {
Directive[
Opacity[1.],
RGBColor[0.560181, 0.691569, 0.194885],
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"Label" -> {"XYLabel"}, "Ball" -> {"InterpolatedBall"}|>,
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- "ImageSize" -> {360, 360/GoldenRatio}, "Axes" -> {True, True},
- "LabelStyle" -> {}, "AspectRatio" -> GoldenRatio^(-1),
+ "PlotRange" -> {{-0.0005, 0.0005}, {0., 0.7673108712075356}},
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+ "LabelStyle" -> {FontFamily -> "Times",
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diff --git a/marginal.tex b/marginal.tex
index 26a88fe..45d65dc 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -202,12 +202,14 @@ as the emergence of an imaginary part in the function.
As an example, we compute
\begin{equation} \label{eq:large.dev}
- e^{NG_\lambda^*(\mu)}
- =P_{\lambda_\mathrm{min}(B+\mu I)=\lambda^*}
- =\overline{\delta\big(N\lambda^*-N\lambda_\mathrm{min}(B+\mu I)\big)}
+ \begin{aligned}
+ e^{NG_{\lambda^*}(\mu)}
+ &=P\big(\lambda_\mathrm{min}(B+\mu I)=\lambda^*\big) \\
+ &=\overline{\delta\big(N\lambda^*-N\lambda_\mathrm{min}(B+\mu I)\big)}
+ \end{aligned}
\end{equation}
where the overline is the average over $B$, and we have defined the large
-deviation function $G_\sigma(\mu)$.
+deviation function $G_{\lambda^*}(\mu)$.
Using the representation of $\lambda_\mathrm{min}$ defined in \eqref{eq:λmin}, we have
\begin{widetext}
\begin{equation}
@@ -225,10 +227,10 @@ representation, we have
=\overline{\lim_{\beta\to\infty}\lim_{m\to0}\int d\hat\lambda\prod_{\alpha=1}^m\left[d\mathbf s^\alpha\,\delta(N-\|\mathbf s^\alpha\|^2)\right]
\exp\left\{-\beta\sum_{\alpha=1}^m(\mathbf s^\alpha)^T(B+\mu I)\mathbf s^\alpha+\hat\lambda\left[N\lambda^*-(\mathbf s^1)^T(B+\mu I)\mathbf s^1\right]\right\}}
\end{equation}
-having introduced the parameter $\hat\lambda$ in the Fourier representation of
-the $\delta$-function. The whole expression, so transformed, is a simple
+having introduced the auxiliary parameter $\hat\lambda$ in the Fourier representation of
+the $\delta$-function. The whole expression, so transformed, is an
exponential integral linear in the matrix $B$. Taking the average over $B$, we
-have
+find
\begin{equation}
\begin{aligned}
&e^{NG_{\lambda^*}(\mu)}
@@ -241,27 +243,27 @@ have
\end{equation}
\end{widetext}
We make the Hubbard--Stratonovich transformation to the matrix field
-$Q_{ab}=\frac1N\mathbf s_a^T\mathbf s_b$. This gives
+$Q^{\alpha\beta}=\frac1N\mathbf s^\alpha\cdot\mathbf s^\beta$. This gives
\begin{equation}
e^{NG_{\lambda^*}(\mu)}
=\lim_{\beta\to\infty}\lim_{m\to0}\int d\hat\lambda\,dQ\,
- e^{N\mathcal U_\mathrm{GOE}(\hat\lambda,Q\mid\mu,\lambda^*,\beta)}
+ e^{N\mathcal U_\mathrm{GOE}(\hat\lambda,Q\mid\beta,\lambda^*,\mu)}
\end{equation}
where the effective action is given by
\begin{equation}
\begin{aligned}
- &\mathcal U_\textrm{GOE}(\hat\lambda, Q\mid\lambda^*,\mu,\beta)
+ &\mathcal U_\textrm{GOE}(\hat\lambda, Q\mid\beta,\lambda^*,\mu)
=\hat\lambda(\lambda^*-\mu)-m\beta\mu \\
- &+\sigma^2\left[\beta^2\sum_{ab}^mQ_{ab}^2
- +2\beta\hat\lambda\sum_a^mQ_{1a}^2
+ &+\sigma^2\left[\beta^2\sum_{\alpha\gamma}^m(Q^{\alpha\gamma})^2
+ +2\beta\hat\lambda\sum_\alpha^m(Q^{1\alpha})^2
+\hat\lambda^2
\right]+\frac12\log\det Q
\end{aligned}
\end{equation}
-where $Q_{aa}=1$ because of the spherical constraint. We can evaluate this
+and $Q^{\alpha\alpha}=1$ because of the spherical constraint. We can evaluate this
integral using the saddle point method. We make a replica symmetric ansatz for
$Q$, because this is a 2-spin model, but with the first row singled out because
-of its unique coupling with $\hat\lambda$. This gives
+of its unique coupling with $\hat\lambda$. The resulting matrix has the form
\begin{equation} \label{eq:Q.structure}
Q=\begin{bmatrix}
1&\tilde q_0&\tilde q_0&\cdots&\tilde q_0\\
@@ -271,16 +273,17 @@ of its unique coupling with $\hat\lambda$. This gives
\tilde q_0&q_0&q_0&\cdots&1
\end{bmatrix}
\end{equation}
-with $\sum_{ab}Q_{ab}^2=m+2(m-1)\tilde q_0^2+(m-1)(m-2)q_0^2$, $\sum_aQ_{1a}^2=1+(m-1)\tilde q_0^2$,
+The relevant expressions in the effective action produce $\sum_{\alpha\beta}(Q^{\alpha\beta})^2=m+2(m-1)\tilde q_0^2+(m-1)(m-2)q_0^2$, $\sum_\alpha(Q^{1\alpha})^2=1+(m-1)\tilde q_0^2$,
and
\begin{equation}
\log\det Q=(m-2)\log(1-q_0)+\log(1+(m-2)q_0-(m-1)\tilde q_0^2)
\end{equation}
-Inserting these expressions and taking the limit of $m$ to zero, we find
+Inserting these expressions into the effective action and taking the limit of
+$m$ to zero, we arrive at
\begin{equation}
e^{NG_{\lambda^*}(\mu)}
=\lim_{\beta\to\infty}\int d\hat\lambda\,dq_0\,d\tilde q_0\,
- e^{N\mathcal U_\textrm{GOE}(\hat\lambda,q_0,\tilde q_0\mid\mu,\lambda^*,\beta)}
+ e^{N\mathcal U_\textrm{GOE}(\hat\lambda,q_0,\tilde q_0\mid\beta,\lambda^*,\mu)}
\end{equation}
with the effective action
\begin{equation}
@@ -305,32 +308,37 @@ However, taking the limit with $y\neq\tilde y$ results in an expression for the
action that diverges with $\beta$. To cure this, we must take $\tilde y=y$. The result is
\begin{equation}
\begin{aligned}
- \mathcal U_\textrm{GOE}(\hat\lambda,y,\Delta z\mid\mu,\lambda^*,\infty)
- &=\hat\lambda(\lambda^*-\mu)
- +\sigma^2\big[
- \hat\lambda^2-4(y+\Delta z)
- \big] \\
- &\qquad+\frac12\log\left(1+\frac{2\Delta z}{y^2}\right)
+ &\mathcal U_\textrm{GOE}(\hat\lambda,y,\Delta z\mid\infty,\lambda^*,\mu)
+ =\hat\lambda(\lambda^*-\mu) \\
+ &\qquad+\sigma^2\big[
+ \hat\lambda^2+4(y+\Delta z)
+ \big]
+ +\frac12\log\left(1-\frac{2\Delta z}{y^2}\right)
\end{aligned}
\end{equation}
Extremizing this action over the new parameters $y$, $\Delta z$, and $\hat\lambda$, we have
\begin{align}
- \hat\lambda=-\frac1\sigma\sqrt{\frac{(\mu+\lambda^*)^2}{(2\sigma)^2}-1}
+ \hat\lambda&=\frac1\sigma\sqrt{\left(\frac{\mu-\lambda^*}{2\sigma}\right)^2-1}
\\
- y=\frac1{2\sigma}\left(\frac{\mu+\lambda^*}{2\sigma}-\sqrt{\frac{(\mu+\lambda^*)^2}{(2\sigma)^2}-1}\right)
- &\\
- \Delta z=\frac1{4\sigma^2}\left(1-\frac{\mu+\lambda^*}{2\sigma}\left(\frac{\mu+\lambda^*}{2\sigma}-\sqrt{\frac{(\mu+\lambda^*)^2}{(2\sigma)^2}-1}\right)\right)
+ y&=\frac1{2\sigma}\left[
+ \frac{\mu-\lambda^*}{2\sigma}+\sqrt{\left(\frac{\mu-\lambda^*}{2\sigma}\right)^2-1}
+ \right]^{-1}
+ \\
+ \Delta z&=\frac1{4\sigma^2}\left[
+ \left(\frac{\mu-\lambda^*}{2\sigma}\right)^2-1
+ -\frac{\mu-\lambda^*}{2\sigma}\sqrt{\left(\frac{\mu-\lambda^*}{2\sigma}\right)^2-1}
+ \right]
\end{align}
Inserting this solution into $\mathcal S_\infty$ we find
\begin{equation} \label{eq:goe.large.dev}
\begin{aligned}
&G_{\lambda^*}(\mu)
- =\mathop{\textrm{extremum}}_{y,\Delta z,\hat\lambda}
- \mathcal U_\mathrm{GOE}(y,\Delta z,\hat\lambda\mid\mu,\lambda^*,\infty) \\
- &=-\tfrac{\mu+\lambda^*}{2\sigma}\sqrt{\Big(\tfrac{\mu+\lambda^*}{2\sigma}\Big)^2-1}
- +\log\left(
- \tfrac{\mu+\lambda^*}{2\sigma}+\sqrt{\Big(\tfrac{\mu+\lambda^*}{2\sigma}\Big)^2-1}
- \right)
+ =\mathop{\textrm{extremum}}_{\hat\lambda,y,\Delta z}
+ \mathcal U_\mathrm{GOE}(\hat\lambda,y,\Delta z\mid\infty,\lambda^*,\mu) \\
+ &=-\frac{\mu-\lambda^*}{2\sigma}\sqrt{\left(\frac{\mu-\lambda^*}{2\sigma}\right)^2-1} \\
+ &\hspace{5em}-\log\left[
+ \frac{\mu-\lambda^*}{2\sigma}-\sqrt{\left(\frac{\mu-\lambda^*}{2\sigma}\right)^2-1}
+ \right]
\end{aligned}
\end{equation}
This function is plotted in Fig.~\ref{fig:large.dev} for $\lambda^*=0$. For $\mu<2\sigma$ $G_{0}(\mu)$ has an
@@ -1376,7 +1384,7 @@ taking the zero-temperature limit, we find
for $\alpha=\frac32$ and $f(q)=q^2+q^3$. The ground state energy
$E_\mathrm{gs}$ and the threshold energy $E_\mathrm{th}$ are marked on the
plot.
- }
+ } \label{fig:ls.complexity}
\end{figure}
\section{Conclusions}
@@ -1572,24 +1580,28 @@ fixing the trace of the Hessian, this gives
\end{align}
so that the differential form of the symmetry is
\begin{equation}
- \mathcal D=\bar{\pmb\eta}\frac\partial{\partial\mathbf x}
- -i\hat\beta\bar{\pmb\eta}\frac\partial{\partial\hat{\mathbf x}}
- -i\hat{\mathbf x}\frac\partial{\partial\pmb\eta}
+ \mathcal D=\bar{\pmb\eta}\cdot\frac\partial{\partial\mathbf x}
+ -i\hat\beta\bar{\pmb\eta}\cdot\frac\partial{\partial\hat{\mathbf x}}
+ -i\hat{\mathbf x}\cdot\frac\partial{\partial\pmb\eta}
\end{equation}
The Ward identities associated with this symmetry give rise to relationships among the order parameters. These identities are
\begin{align}
- 0=\frac1N\mathcal D\langle\mathbf x_a^T\pmb\eta_b\rangle
- =\frac1N\left[
- \langle\bar{\pmb\eta}_a^T\pmb\eta_b\rangle-
- i\langle\mathbf x_a^T\hat{\mathbf x}_b\rangle
- \right]
- =G_{ab}+R_{ab} \\
- 0=\frac iN\mathcal D\langle\hat{\mathbf x}_a^T\pmb\eta_b\rangle
- =\frac1N\left[
- \hat\beta\langle\bar{\pmb\eta}_a^T\pmb\eta_b\rangle
- +\langle\hat{\mathbf x}_a^T\hat{\mathbf x}_b\rangle
- \right]
- =\hat\beta G_{ab}+D_{ab}
+ \begin{aligned}
+ 0&=\frac1N\mathcal D\langle\mathbf x_a\cdot\pmb\eta_b\rangle
+ =\frac1N\left[
+ \langle\bar{\pmb\eta}_a\cdot\pmb\eta_b\rangle-
+ i\langle\mathbf x_a\cdot\hat{\mathbf x}_b\rangle
+ \right] \\
+ &=G_{ab}+R_{ab}
+ \end{aligned} \\
+ \begin{aligned}
+ 0&=\frac iN\mathcal D\langle\hat{\mathbf x}_a\cdot\pmb\eta_b\rangle
+ =\frac1N\left[
+ \hat\beta\langle\bar{\pmb\eta}_a\cdot\pmb\eta_b\rangle
+ +\langle\hat{\mathbf x}_a\cdot\hat{\mathbf x}_b\rangle
+ \right] \\
+ &=\hat\beta G_{ab}+D_{ab}
+ \end{aligned}
\end{align}
These identities establish $G_{ab}=-R_{ab}$ and $D_{ab}=\hat\beta R_{ab}$,
allowing elimination of the matrices $G$ and $D$ in favor of $R$. Fixing the
@@ -1600,74 +1612,85 @@ trace to $\mu$ explicitly breaks this symmetry, and the simplification is lost.
In this appendix we derive an expression for the asymptotic spectral density in
the two-sphere multispherical spin glass that we describe in Section
-\ref{sec:multispherical}. \cite{Livan_2018_Introduction}
+\ref{sec:multispherical}. We use a typical approach of employing replicas to
+compute the resolvent \cite{Livan_2018_Introduction}. The resolvent for the
+Hessian of the multispherical model is given by an integral over $\mathbf
+y=[\mathbf y^{(1)},\mathbf y^{(2)}]\in\mathbb R^{2N}$ as
+\begin{widetext}
\begin{equation}
\begin{aligned}
- &G(\lambda)
- =\lim_{n\to0}\int\|\mathbf y_1\|^2\,\prod_{a=1}^nd\mathbf y_a\,
+ G(\lambda)
+ &=\lim_{n\to0}\int\|\mathbf y_1\|^2\,\prod_{a=1}^nd\mathbf y_a\,
\exp\left\{
- -\frac12\mathbf y_a^T(\operatorname{Hess}H(\mathbf x,\pmb\omega)+\lambda I)\mathbf y_a
+ -\frac12\mathbf y_a^T(\operatorname{Hess}H(\mathbf x,\pmb\omega)-\lambda I)\mathbf y_a
\right\} \\
&
- =\lim_{n\to0}\int\big(\|\mathbf y_1^{(1)}\|^2+\|\mathbf y_1^{(2)}\|^2\big)\,\prod_{a=1}^nd\mathbf y_a\, \\
- &\times\exp\left\{
+ =\lim_{n\to0}\int\big(\|\mathbf y_1^{(1)}\|^2+\|\mathbf y_1^{(2)}\|^2\big)\,\prod_{a=1}^nd\mathbf y_a\,
+ \exp\left\{
-\frac12\begin{bmatrix}\mathbf y_a^{(1)}\\\mathbf y_a^{(2)}\end{bmatrix}^T
\left(
\begin{bmatrix}
\operatorname{Hess}H_1(\mathbf x^{(1)},\omega_1) & -\epsilon \\
-\epsilon & \operatorname{Hess}H_2(\mathbf x^{(2)},\omega_2)
\end{bmatrix}
- +\lambda I
+ -\lambda I
\right)\begin{bmatrix}\mathbf y_a^{(1)}\\\mathbf y_a^{(2)}\end{bmatrix}
- \right\} \\
+ \right\}
\end{aligned}
\end{equation}
-If $Y_{ab}^{(ik)}=\frac1N\mathbf y_a^{(i)}\cdot\mathbf y_b^{(j)}$ is the matrix
-of overlaps of the $\mathbf y$, then a short and standard calculation yields
+If $Y_{ab}^{(ij)}=\frac1N\mathbf y_a^{(i)}\cdot\mathbf y_b^{(j)}$ is the matrix
+of overlaps of the vectors $\mathbf y$, then a short and standard calculation involving the average over $H$ and the change of variables from $\mathbf y$ to $Y$ yields
\begin{equation}
- G(\lambda)=N\lim_{n\to0}\int dY\,(Y_{11}^{(11)}+Y_{11}^{(22)})\,
+ \overline{G(\lambda)}=N\lim_{n\to0}\int dY\,\big(Y_{11}^{(11)}+Y_{11}^{(22)}\big)\,
e^{nN\mathcal S(Y)}
\end{equation}
-for
+where the effective action $\mathcal S$ is given by
\begin{equation}
\begin{aligned}
&\mathcal S(Y)
- =\frac1n\sum_{ab}\left[
+ =\lim_{n\to0}\frac1n\left\{
+ \frac14\sum_{ab}^n\left[
\sigma_1^2(Y_{ab}^{(11)})^2
+\sigma_2^2(Y_{ab}^{(22)})^2
- \right]+\frac12\log\det\begin{bmatrix}
- Y^{(11)}&Y^{(12)}\\Y^{(12)}&Y^{(22)}
- \end{bmatrix}\\
- &+\frac1n\sum_a^n\left[
+ \right]
+ +\frac12\sum_a^n\left[
2\epsilon Y_{aa}^{(12)}
- -\omega_1Y_{aa}^{(11)}
- -\omega_2Y_{aa}^{(22)}
- +\lambda(Y_{aa}^{(11)}
- +Y_{aa}^{(22)})
+ +(\lambda-\omega_1)Y_{aa}^{(11)}
+ +(\lambda-\omega_2)Y_{aa}^{(22)}
\right]
+ +\frac12\log\det\begin{bmatrix}
+ Y^{(11)}&Y^{(12)}\\Y^{(12)}&Y^{(22)}
+ \end{bmatrix}
+ \right\}
\end{aligned}
\end{equation}
-Making the replica symmetric ansatz $Y_{ab}^{(ij)}=y^{(ij)}\delta_{ab}$ yields
+\end{widetext}
+Making the replica symmetric ansatz $Y_{ab}^{(ij)}=y^{(ij)}\delta_{ab}$ for
+each of the matrices $Y^{(ij)}$ yields
\begin{equation}
\begin{aligned}
- &\mathcal S(y)
- =
- \sigma_1^2(y^{(11)})^2
- +\sigma_2^2(y^{(22)})^2
- +\frac12\log(
+ \mathcal S(y)
+ &=
+ \frac14\left[\sigma_1^2(y^{(11)})^2
+ +\sigma_2^2(y^{(22)})^2\right]+\epsilon y^{(12)}
+ \\
+ &
+ \qquad+\frac12\left[(\lambda-\omega_1)y^{(11)}
+ +(\lambda-\omega_2)y^{(22)}\right] \\
+ &
+ \qquad+\frac12\log(
y^{(11)}y^{(22)}-y^{(12)}y^{(12)}
- )\\
- &+2\epsilon y^{(12)}
- -\omega_1y^{(11)}
- -\omega_2y^{(22)}
- +\lambda(y^{(11)}
- +y^{(22)})
+ )
\end{aligned}
\end{equation}
+while the average resolvent becomes
\begin{equation}
\overline{G(\lambda)}
=N(y^{(11)}+y^{(22)})
\end{equation}
+for $y^{(11)}$ and $y^{(22)}$ evaluated at a saddle point of $\mathcal S$. The
+spectral density at large $N$ is then given by the discontinuity in its
+imaginary point on the real axis, or
\begin{equation}
\rho(\lambda)
=\frac1{i\pi N}
@@ -1840,7 +1863,9 @@ $\hat\beta$, $c_0$, $r$, and $r_0$ is
\end{aligned}
\end{equation}
When $f(0)=0$ as in the cases directly studied in this work, this further
-simplifies as $c_0=r_0=0$.
+simplifies as $c_0=r_0=0$. Extremizing this expression with respect to the
+order parameters $\hat\beta$ and $r$ produces the red line of dominant minima
+shown in Fig.~\ref{fig:ls.complexity}.
\end{widetext}