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| -rw-r--r-- | frsb_kac-rice_letter.tex | 26 |
1 files changed, 13 insertions, 13 deletions
diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex index b3c288a..21807cb 100644 --- a/frsb_kac-rice_letter.tex +++ b/frsb_kac-rice_letter.tex @@ -128,8 +128,8 @@ and the entropy at zero temperature, any hierarchical structure in the equilibrium should be reflected in the complexity. The complexity is calculated using the Kac--Rice formula, which counts the -stationary points using a $\delta$-function weighted by a Jacobian. The count -is given by +stationary points using a $\delta$-function weighted by a Jacobian +\cite{Kac_1943_On, Rice_1939_The}. The count is given by \begin{equation} \begin{aligned} \mathcal N(E, \mu) @@ -156,17 +156,17 @@ flat directions. When $\mu<\mu_m$, the stationary points are saddles with indexed fixed to within order one (fixed macroscopic index). It's worth reviewing the complexity for the best-studied case of the pure model -for $p\geq3$. Here, because the covariance is a homogeneous polynomial, $E$ and -$\mu$ cannot be fixed separately, and one implies the other: $\mu=pE$. -Therefore at each energy there is only one kind of stationary point. When the -energy reaches $E_\mathrm{th}=-\mu_m/p$, the population of stationary points -suddenly shifts from all saddles to all minima, and there is an abrupt -percolation transition in the topology of constant-energy slices of the -landscape. This behavior of the complexity can be used to explain a rich -variety of phenomena in the equilibrium and dynamics of the pure models: the -threshold energy $E_\mathrm{th}$ corresponds to the average energy at the -dynamic transition temperature, and the asymptotic energy reached by slow aging -dynamics. +for $p\geq3$ \cite{Cugliandolo_1993_Analytical}. Here, because the covariance +is a homogeneous polynomial, $E$ and $\mu$ cannot be fixed separately, and one +implies the other: $\mu=pE$. Therefore at each energy there is only one kind of +stationary point. When the energy reaches $E_\mathrm{th}=-\mu_m/p$, the +population of stationary points suddenly shifts from all saddles to all minima, +and there is an abrupt percolation transition in the topology of +constant-energy slices of the landscape. This behavior of the complexity can be +used to explain a rich variety of phenomena in the equilibrium and dynamics of +the pure models: the threshold energy $E_\mathrm{th}$ corresponds to the +average energy at the dynamic transition temperature, and the asymptotic energy +reached by slow aging dynamics. Things become much less clear in even the simplest mixed models. For instance, one mixed model known to have a replica symmetric complexity was shown to |