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Diffstat (limited to 'frsb_kac-rice.tex')
| -rw-r--r-- | frsb_kac-rice.tex | 38 |
1 files changed, 16 insertions, 22 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index f3b0558..e02cab8 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -67,13 +67,11 @@ and therefore its spectrum is given by the Wigner semicircle with radius $\sqrt{ \end{equation} and the spectrum of $\operatorname{Hess}H$ is this shifted by $\mu$, or $\rho(\lambda-\mu)$. -The parameter $\mu$ fixes the spectrum of the hessian. By manipulating it, one -can decide to find the complexity of saddles of a certain macroscopic index, or -of minima with a certain harmonic stiffness. When $\mu$ is taken to be within -the range $\pm\sqrt{4f''(1)}=\pm\mu_m$, the critical points are constrained to have -index $\frac12N(1-\mu/\mu_m)$. When $\mu>\mu_m$, the critical -points are minima whose sloppiest eigenvalue is $\mu-\mu_m$. Finally, -when $\mu=\mu_m$, the critical points are marginal minima. +The parameter $\mu$ fixes the spectrum of the hessian. When it is an integration variable, +and one restricts the domain of all integrations to compute saddles of a certain macroscopic index, or +of minima with a certain harmonic stiffness, its value is the `softest' mode that adapts to change the Hessian \cite{Fyodorov_2007_Replica}. When it is fixed, then the restriction of the index of saddles is `payed' by the realization of the eigenvalues of the Hessian, usually a +`harder' mode. + \subsection{What to expect?} @@ -88,7 +86,7 @@ have, in the absence of coupling, the same dependence, but are stretched to one \begin{equation} \Sigma_1(H_1)= \Sigma_o(H_1/\alpha_1) \qquad ; \qquad \Sigma_2(H_2)= \Sigma_o(H_2/\alpha_2) \end{equation} -Each system has a ground state energy $E_{gs}^{1,2}$, a threshold energy $E_{thres}^{1,2}$ (a well-defined notion, since we are considering pure p-spins), abd the corresponding limit values $X^{1,2}_{gs}=\left. \frac{d \Sigma_1}{dE_{1,2}}\right|_{E^{gs}_{12}}$ +Each system has a ground state energy $E_{gs}^{1,2}$, a threshold energy $E_{thres}^{1,2}$ (a well-defined notion, since we are considering pure p-spins), the corresponding limit values $X^{1,2}_{gs}=\left. \frac{d \Sigma_1}{dE_{1,2}}\right|_{E^{gs}_{12}}$ and $X^{1,2}_{thres}=\left. \frac{d \Sigma_1}{dE_{1,2}}\right|_{E^{thres}_{12}}$ Considering the cartesian product of both systems, we have, in terms of the total energy $H=H_1+H_2$ three regimes: @@ -98,12 +96,10 @@ $H=H_1+H_2$ three regimes: & & X_1 \equiv \frac{d \Sigma_1}{dE_1}= X_2 \equiv \frac{d \Sigma_2}{dE_2} \end{eqnarray} \item {\bf Semi-frozen} -As we go down in energy, one of the systems (say, the first) reaches its ground state, -At lower temperatures, the first system is thus frozen, while the second is not, -so that $X_1=X_1^{gs}> X_2$. The lowest energy is such that both systems are frozen. +As we go down in energy, one of the systems (say, the first) reaches its frozen phase, + the first system is thus concentrated in a few states of $O(1)$ energy, while the second is not, so that $X_1=X_1^{gs}> X_2$. The lowest energy is reached when systems are frozen. \item {\bf Semi-threshold } As we go up from the unfrozen upwards in energy, -the second system reaches its threshold $X_2^{thres}$. At higher energies minima are extremely rare, -so the second system remains stuck at its threshold for higher energies. +the second system reaches its threshold $X_2^{thres}$. At higher energies minima are extremely rare, so the minima of the second system remain stuck at its threshold for higher energies. \item{\bf Both systems reach their thresholds} There essentially no more minima above that. \end{itemize} @@ -111,13 +107,12 @@ Consider now two combined vectors $({\bf s},{\bf \hat s})$ and $({\bf s}',{\bf \ chosen at the same energies.\\ $\bullet$ Their normalized overlap is close to one when both subsystems are frozen, -close to a half in the semifrozen phase, and zero at all higher energies.\\ +between zero and one in the semifrozen phase, and zero at all higher energies.\\ $\bullet$ In phases where one or both systems are stuck in their thresholds (and only in those), the -minima are exponentially subdominant with respect to saddles. +minima are exponentially subdominant with respect to saddles, because a saddle is found by releasing the constraint of staying on the threshold. + -$\bullet$ {\bf note that the same reasoning leads us to the conclusion that -minima of two total energies such that one of the systems is frozen have nonzero overlaps} \section{Main result} @@ -285,7 +280,7 @@ $F$ is a $k-1$ RSB ansatz with all eigenvalues scaled by $y$ and shifted by $z$. -\section{Kac--Rice} +\section{Kac-Rice} \cite{Auffinger_2012_Random, BenArous_2019_Geometry} @@ -475,10 +470,9 @@ The odd and even fermion numbers decouple, so we can neglect all odd terms in $\ The variables $\bar \theta \theta$ and $\bar \theta ' \theta'$ play the role of `times' in a superspace treatment. We have a long experience of -making an ansatz for replicated quantum problems, which naturally involve a (Matsubara) time. The analogy strongly -suggests that only the diagonal ${\bf Q}_{aa}$ depend on the $\theta$'s. This boils down the ansatz \ref{ansatz} - -Not surprisingly, this ansatz closes, as we shall see. That it closes under Hadamard products is simple. +making an ansatz for replicated quantum problems, which naturally involve a (Matsubara) time. The dependence on this time only holds for diagonal replica elements, a consequence of ultrametricity. The analogy strongly +suggests that only the diagonal ${\bf Q}_{aa}$ depend on the $\theta$'s. This boils down the ansatz \ref{ansatz}. +Not surprisingly, and for the same reason as in the quantum case, this ansatz closes, as we shall see.For example, consider the convolution: \begin{equation} \begin{aligned} |