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| -rw-r--r-- | frsb_kac-rice.tex | 19 |
1 files changed, 11 insertions, 8 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index ec71085..680cc89 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -25,9 +25,11 @@ \author{Jaron Kent-Dobias \& Jorge Kurchan} \maketitle \begin{abstract} - We derive the general solution for the computation of stationary points of + We derive the general solution for counting the stationary points of mean-field complex landscapes. It incorporates Parisi's solution - for the ground state, as it should. + for the ground state, as it should. Using this solution, we count the + stationary points of two models: one with multi-step replica symmetry + breaking, and one with full replica symmetry breaking. \end{abstract} \section{Introduction} @@ -410,7 +412,7 @@ the value of the stability $\mu$ in all replicas to the value $\mu^*$. \item For $\mu^*<\mu_m$, this amounts to fixing the index density. Since the overwhelming majority of saddles have a semicircle distribution, the fluctuations are rarer than exponential. - \item For the gapped case $\mu^*>\mu_m$, there is a an exponentially small + \item For the gapped case $\mu^*>\mu_m$, there is an exponentially small probability that $r=1,2,...$ eigenvalues detach from the semicircle in such a way that the index is in fact $N {\cal{I}}=r$. We shall not discuss these subextensive index fluctuations in this paper, the interested reader @@ -697,7 +699,7 @@ $\hat\beta=\tilde\beta$, and $C=\tilde Q$. {\em Therefore a $(k-1)$RSB ansatz in Kac--Rice will predict the correct ground state energy for a model whose equilibrium state at small temperatures is $k$RSB } Moreover, there is an -exact correspondance between the saddle parameters of each. If the equilibrium +exact correspondence between the saddle parameters of each. If the equilibrium is given by a Parisi matrix with parameters $x_1,\ldots,x_k$ and $q_1,\ldots,q_k$, then the parameters $\hat\beta$, $r_d$, $d_d$, $\tilde x_1,\ldots,\tilde x_{k-1}$, and $c_1,\ldots,c_{k-1}$ for the @@ -1346,9 +1348,10 @@ global overlap between different states is at most $1/2$. At $\hat \beta>\hat \subsection{\textit{R} and \textit{D}: response functions} The matrix field $R$ is related to responses of the stationary points to -perturbations of the tensors $J$. One adds to the Hamiltonian a random term $\varepsilon \tilde H_p = \varepsilon \sum_{i_1,...,i_p} \tilde J_{i_1,...,i_p} s_{i_1}...s_{i_p}$, where the $\tilde J$ are -random Gaussian uncorrelated with the $J$'s. -The response to these is: +perturbations of the tensors $J$. One adds to the Hamiltonian a random term +$\varepsilon \tilde H_p = \varepsilon \sum_{i_1,...,i_p} \tilde J_{i_1,...,i_p} +s_{i_1}...s_{i_p}$, where the $\tilde J$ are random Gaussian uncorrelated with +the $J$'s. ] The response to these is: \begin{equation} \begin{aligned} & \overline{ \frac{\partial \langle \tilde H_p \rangle_{\tilde J} } {\partial \varepsilon} } @@ -1454,7 +1457,7 @@ saddles at this transition point. The authors would like to thank Valentina Ros for helpful discussions. \paragraph{Funding information} -J K-D and J K are supported by the Simons Foundation Grant No. 454943. +JK-D and JK are supported by the Simons Foundation Grant No.~454943. \printbibliography |