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Diffstat (limited to '2-point.tex')
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diff --git a/2-point.tex b/2-point.tex index f0fec59..3cd8422 100644 --- a/2-point.tex +++ b/2-point.tex @@ -446,6 +446,15 @@ Because the matrices $C^{00}$, $R^{00}$, and $D^{00}$ are diagonal in this case, \section{Isolated eigenvalue} +The two-point complexity depends on the spectrum at both stationary points +through the determinant of their Hessians, but only on the bulk of the +distribution. As we saw, this bulk is unaffected by the conditions of energy +and proximity. However, these conditions give rise to small-rank perturbations +to the Hessian, which can lead a subextensive number of eigenvalues leaving the +bulk (we will study \emph{one}). + +\cite{Ikeda_2023_Bose-Einstein-like} + \begin{equation} \begin{aligned} \beta F(\beta\mid\mathbf s) @@ -471,6 +480,17 @@ Because the matrices $C^{00}$, $R^{00}$, and $D^{00}$ are diagonal in this case, &=\lim_{m\to0}\int\left[\prod_{a=1}^m d\nu(\pmb\sigma_a,\varsigma_a\mid E_0,\mu_0)\right]\,F(\beta\mid E_1,\mu_1,q,\pmb\sigma_1) \end{aligned} \end{equation} +The minimum eigenvalue of the conditioned Hessian is then given by twice the ground state energy, or +\begin{equation} + \lambda_\text{min}=2\lim_{\beta\to\infty}\frac\partial{\partial\beta}\overline{F(\beta\mid\epsilon_1,\mu_1,\epsilon_2,\mu_2,q)} +\end{equation} +For this calculation, there are three different sets of replicated variables. +Note that, as for the computation of the complexity, the $\pmb\sigma_1$ and +$\mathbf s_1$ replicas are \emph{special}. The first again is the only of the +$\sigma$ replicas constrained to lie at fixed overlap with \emph{all} the +$\mathbf s$ replicas, and the second is the only of the $\mathbf s$ replicas at +which the Hessian is evaluated. + \begin{equation} \mathcal O(\mathbf t)= \sum_a^m\delta(\mathbf t-\pmb\sigma_a)(i\hat{\pmb\sigma}_a\cdot\partial_\mathbf t-\hat\beta_0) @@ -479,6 +499,8 @@ Because the matrices $C^{00}$, $R^{00}$, and $D^{00}$ are diagonal in this case, -\frac12 \delta(\mathbf t-\mathbf s_1)\beta\sum_c^\ell(\mathbf x_c\cdot\partial_{\mathbf t})^2 \end{equation} + + \begin{equation} \begin{aligned} &\frac18\beta^2\int d\mathbf t\,d\mathbf t'\,\delta(\mathbf t-\mathbf s_1)\delta(\mathbf t'-\mathbf s_1)\sum_{ab}^\ell(\mathbf x_a\cdot\partial_{\mathbf t})^2(\mathbf x_b\cdot\partial_{\mathbf t'})^2f\left(\frac{\mathbf t\cdot\mathbf t'}N\right)\\ |