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diff --git a/marginal.tex b/marginal.tex index 5c3e033..7df99d5 100644 --- a/marginal.tex +++ b/marginal.tex @@ -607,16 +607,57 @@ which gives $\mu_m(E)=2\sqrt{f''(1)}$ independent of $E$, as we presaged above. that the marginal complexity in these models is simply the ordinary complexity evaluated at a fixed trace of the Hessian. -\subsection{Twin spherical spin glasses} - -$\Omega=S^{N-1}\times S^{N-1}$ -\begin{equation} - H(\pmb\sigma)=H_1(\pmb\sigma^{(1)})+H_2(\pmb\sigma^{(2)})+\epsilon\pmb\sigma^{(1)}\cdot\pmb\sigma^{(2)} -\end{equation} -\begin{equation} - \overline{H_s(\pmb\sigma_1)H_s(\pmb\sigma_2)} - =Nf_s\left(\frac{\pmb\sigma_1\cdot\pmb\sigma_2}N\right) -\end{equation} +\subsection{Multispherical spin glasses} + +The multispherical models are a simple extension of the spherical ones, where +the configuration space is taken to be the union of more than one hypersphere. +Here we consider the specific case where the configuration space is the union +of two $(N-1)$-spheres, with $\Omega=S^{N-1}\times S^{N-1}$, and where the +energy is given by +\begin{equation} + H(\mathbf x)=H_1(\mathbf x^{(1)})+H_2(\mathbf x^{(2)})+\epsilon\mathbf x^{(1)}\cdot\mathbf x^{(2)} +\end{equation} +for $\mathbf x=[\mathbf x^{(1)},\mathbf x^{(2)}]$ for components $\mathbf +x^{(1)},\mathbf x^{(2)}\in\mathbb R^N$. Each individual sphere energy $H_s$ is +taken to be a centered Gaussian random function with a covariance given in the +usual spherical way by +\begin{equation} + \overline{H_s(\pmb\sigma_1)H_p(\pmb\sigma_2)} + =N\delta_{sp}f_s\left(\frac{\pmb\sigma_1\cdot\pmb\sigma_2}N\right) +\end{equation} +with the functions $f_1$ and $f_2$ not necessarily the same. In this problem, +there is an energetic competition between the independent spin glass energies +on each sphere and their tendency to align or anti-align through the +interaction term. + +Because the energy is Gaussian, properties of the Hessian are once again +statistically independent of those of the energy and gradient. However, unlike +the previous example of the spherical models, the spectrum of the Hessian at +different points in the configuration space has different shapes. This appears +in this problem through the presence of a configuration space defined by +multiple constraints, and therefore multiple Lagrange multipliers are necessary +to ensure they are all fixed. +\begin{align} + H(\mathbf x) + +\frac12\omega^{(1)}\big(\|\mathbf x^{(1)}\|^2-N\big) + +\frac12\omega^{(2)}\big(\|\mathbf x^{(2)}\|^2-N\big) + \\ + \nabla H(\mathbf x,\pmb\omega) + =\partial H(\mathbf x)+\begin{bmatrix} + \omega^{(1)}\mathbf x^{(1)} \\ + \omega^{(2)}\mathbf x^{(2)} + \end{bmatrix} + \\ + \operatorname{Hess}H(\mathbf x,\pmb\omega) + =\partial\partial H(\mathbf x)+\begin{bmatrix} + \omega^{(1)}I&0 \\ + 0&\omega^{(2)}I + \end{bmatrix} +\end{align} +Like in the spherical model, fixing the trace of the Hessian to $\mu$ is +equivalent to a constraint on the Lagrange multipliers. However, in this case +it corresponds to $\mu=\omega^{(1)}+\omega^{(2)}$, and therefore they are not +uniquely fixed by the trace. \begin{widetext} \begin{equation} @@ -769,19 +810,29 @@ The first step to evaluate this expression is to linearize the dependence on the \big(V^k(\pmb\phi_{a\alpha}(1,2))-v_{a\alpha}^k(1,2)\big) \right] \end{equation} -where we have introduced auxiliary fields $\hat v$. +where we have introduced auxiliary fields $\hat v$. With this inserted into the +integral, all other instances of $V$ are replaced by $v$, and the only +remaining dependence on the disorder is from the term $\hat vV$ arising from +the Fourier representation of the Dirac $\delta$ function. This term is linear in $V$, and therefore the random functions can be averaged over to produce \begin{equation} - -\sum_{ab}\sum_{\alpha\gamma}\sum_k\frac12\int d1\,d2\,d3\,d4\, - \hat v_{a\alpha}^kf\big(\pmb\phi_{a\alpha}(1,2)^T\pmb\phi_{b\gamma}(3,4)\big)\hat v_{b\gamma}^k + \overline{ + \exp\left[ + i\sum_{a\alpha k}\int d1\,d2\,\hat v_{a\alpha}^k(1,2) + V^k(\pmb\phi_{a\alpha}(1,2)) + \right] + } + = + -\frac N2\sum_{ab}^n\sum_{\alpha\gamma}^{m_a}\sum_k^{\alpha N}\int d1\,d2\,d3\,d4\, + \hat v_{a\alpha}^k(1,2)f\big(\pmb\phi_{a\alpha}(1,2)^T\pmb\phi_{b\gamma}(3,4)\big)\hat v_{b\gamma}^k(3,4) \end{equation} -We're now quadratic in the $v$ and $\hat v$ with the kernel +The entire integrand is now quadratic in the $v$ and $\hat v$ with the kernel \begin{equation} \begin{bmatrix} B_{a\alpha}(1,2)\delta(1,3)\delta(2,4)\delta_{ab}\delta_{\alpha\gamma} & i\delta(1,3)\,\delta(2,4) \delta_{ab}\delta_{\alpha\gamma}\\ i\delta(1,3)\,\delta(2,4) \delta_{ab}\delta_{\alpha\gamma}& f\big(\pmb\phi_{a\alpha}(1,2)^T\pmb\phi_{b\gamma}(3,4)\big) \end{bmatrix} \end{equation} -Upon integration, this results in a term in the effective action of the form +The integration over the $v$ and $\hat v$ results in a term in the effective action of the form \begin{equation} -\frac M2\log\operatorname{sdet}\left( \delta(1,3)\,\delta(2,4) \delta_{ab}\delta_{\alpha\gamma} @@ -836,6 +887,55 @@ typical pairs of stationary points have no overlap. This gives \end{align} We further take a planted replica symmetric structure for the matrix $Q$, identical to that in \eqref{eq:Q.structure}. +\begin{equation} + \begin{aligned} + \mathcal S + =-\frac\alpha2\log\left[ + \frac{ + (f'(1)d-\hat\beta-f''(1)(r^2-g^2+q_0^2\beta^2-\tilde q_0^2\beta(\beta+\hat\lambda)+\beta\hat\lambda+\frac12\hat\lambda^2)))(f(1)-f(0))+(1-rf'(1))^2) + }{ + (1+gf'(1))^2 + } + \right] \\ + +\frac{\alpha f(0)}2\frac{ + \hat\beta-df'(1)+(r^2-g^2+q_0^2\beta^2-\tilde q_0^2\beta(\beta+\lambda)+\lambda\beta+\frac12\lambda^2)f''(1) + }{ + (f'(1)d-\hat\beta- (r^2-g^2+q_0^2\beta^2-\tilde q_0^2\beta(\beta+\hat\lambda)+\beta\hat\lambda+\frac12\hat\lambda^2)f''(1))(f(1)-f(0))+(1-rf'(1))^2 + } + \\ + -\frac\alpha2\log\left(\frac{1-f'(1)(2\beta(1-q_0)+\lambda-(1-2q_0+\tilde q_0^2)\beta(\beta+\lambda)f'(1))}{(1-(1-q_0)\beta f'(1))^2}\right) + +\frac12\log\frac{d+r^2}{g^2}\frac{1-2q_0+\tilde q_0^2}{(1-q_0)^2} + -\mu(r+g)-\frac12\mu\hat\lambda+\hat\beta E+\hat\lambda\lambda^* + \end{aligned} +\end{equation} +\begin{equation} + \begin{aligned} + \mathcal S_\infty + =-\frac\alpha2\log\left[ + \frac{ + (f'(1)d-\hat\beta-f''(1)(r^2-g^2+2y_0\hat\lambda+\Delta z+\frac12\hat\lambda^2)))(f(1)-f(0))+(1-rf'(1))^2) + }{ + (1+gf'(1))^2 + } + \right] \\ + +\frac{\alpha f(0)}2\frac{ + \hat\beta-df'(1)+(r^2-g^2+2y_0\hat\lambda+\Delta z+\frac12\lambda^2)f''(1) + }{ + (f'(1)d-\hat\beta- (r^2-g^2+2y_0\hat\lambda+\Delta z+\frac12\lambda^2)f''(1))(f(1)-f(0))+(1-rf'(1))^2 + } + \\ + -\frac\alpha2\log\left( + \frac{ + 1-(2y_0+\hat\lambda)f'(1)+(y_0^2-\Delta z)f'(1)^2 + }{(1-y_0f'(1))^2} + \right) + +\frac12\log\frac{d+r^2}{g^2}\frac{y_0^2-\Delta z}{y_0^2} + -\mu(r+g)-\frac12\mu\hat\lambda+\hat\beta E+\hat\lambda\lambda^* + \end{aligned} +\end{equation} +\begin{equation} + \Sigma_{\lambda^*}(E,\mu)=\operatorname{extremum}_{\hat\beta,r,d,g,y_0,\Delta z,\hat\lambda}\mathcal S_\infty +\end{equation} \end{widetext} \appendix |