From 738d0a77ae244cce8e5f9b6faaba14a277bda330 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 21 Oct 2022 17:03:57 +0200
Subject: More citations.

---
 frsb_kac-rice_letter.tex | 26 +++++++++++++-------------
 1 file changed, 13 insertions(+), 13 deletions(-)

(limited to 'frsb_kac-rice_letter.tex')

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
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