From ce8ff3c8932af48b43a3aacdf6b4f34f100c6d8e Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 25 Oct 2024 15:19:52 +0200
Subject: Fixed mistake in the definiton of the supermatrix of a superoperator,
 Appendix A

---
 marginal.tex | 30 ++++++++++++++++++++++--------
 1 file changed, 22 insertions(+), 8 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index 508b674..8efa79f 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -1772,21 +1772,30 @@ Integrals involving superfields contracted into such operators result in schemat
 \end{equation}
 where the usual role of the determinant is replaced by the superdeterminant.
 The superdeterminant can be defined using the ordinary determinant by writing a
-block version of the matrix $M$: if $\mathbf e(1)=\{1,\bar\theta_1\theta_1\}$ is
+block version of the matrix $M$. If $\mathbf e(1)=\{1,i\bar\theta_1\theta_1\}$ is
 the basis vector of the even subspace of the superspace and $\mathbf
-f(1)=\{\bar\theta_1,\theta_1\}$ is that of the odd subspace, then we can form a
+f(1)=\{i\bar\theta_1,i\theta_1\}$ is that of the odd subspace, dual bases $\mathbf e^\dagger(1)=\{i\bar\theta_1\theta_1,1\}$ and $\mathbf f^\dagger(1)=\{\theta_1,-\bar\theta_1\}$ can be defined by the requirement that
+\begin{align}
+  \int d1\,\mathbf e(1)\mathbf e^\dagger(1)=iI
+  &&
+  \int d1\,\mathbf f(1)\mathbf f^\dagger(1)=iI \\
+  \int d1\,\mathbf e(1)\mathbf f^\dagger(1)=0
+  &&
+  \int d1\,\mathbf f(1)\mathbf e^\dagger(1)=0
+\end{align}
+With such bases and dual bases defined, we can form a
 block representation of $M$ in analogy to the matrix form of an operator in quantum mechanics by
 \begin{equation}
   \int d1\,d2\,\begin{bmatrix}
-    \mathbf e(1)M(1,2)\mathbf e(2)^T
+    \mathbf e(1)M(1,2)\mathbf e^\dagger(2)
     &
-    \mathbf e(1)M(1,2)\mathbf f(2)^T
+    \mathbf e(1)M(1,2)\mathbf f^\dagger(2)
     \\
-    \mathbf f(1)M(1,2)\mathbf e(2)^T
+    \mathbf f(1)M(1,2)\mathbf e^\dagger(2)
     &
-    \mathbf f(1)M(1,2)\mathbf f(2)^T
+    \mathbf f(1)M(1,2)\mathbf f^\dagger(2)
   \end{bmatrix}
-  =\begin{bmatrix}
+  =i\begin{bmatrix}
     A & B \\ C & D
   \end{bmatrix}
 \end{equation}
@@ -1802,7 +1811,12 @@ save for the inverse of $\det D$. Likewise, the supertrace of $M$ is is given by
 \end{equation}
 The same method can be used to calculate the
 superdeterminant and supertrace in arbitrary superspaces, where for $\mathbb R^{N|2D}$ each
-basis has $2^{2D-1}$ elements. For instance, for $\mathbb R^{N|4}$ we have $\mathbf e(1,2)=\{1,\bar\theta_1\theta_1,\bar\theta_2\theta_2,\bar\theta_1\theta_2,\bar\theta_2\theta_1,\bar\theta_1\bar\theta_2,\theta_1\theta_2,\bar\theta_1\theta_1\bar\theta_2\theta_2\}$ and $\mathbf f(1,2)=\{\bar\theta_1,\theta_1,\bar\theta_2,\theta_2,\bar\theta_1\theta_1\bar\theta_2,\bar\theta_2\theta_2\theta_1,\bar\theta_1\theta_1\theta_2,\bar\theta_2\theta_2\theta_1\}$.
+basis has $2^{2D-1}$ elements. For instance, for $\mathbb R^{N|4}$ we have
+\begin{align}
+  &\mathbf e(1,2)=\{1,i\bar\theta_1\theta_1,i\bar\theta_2\theta_2,i\bar\theta_1\theta_2,i\bar\theta_2\theta_1,i\bar\theta_1\bar\theta_2,i\theta_1\theta_2,\bar\theta_1\theta_1\bar\theta_2\theta_2\}\notag \\
+  &\mathbf f(1,2)=\{i\bar\theta_1,i\theta_1,i\bar\theta_2,i\theta_2,\bar\theta_1\theta_1\bar\theta_2,\bar\theta_2\theta_2\theta_1,\bar\theta_1\theta_1\theta_2,\bar\theta_2\theta_2\theta_1\}
+\end{align}
+with the dual bases defined analogously to those above.
 
 \section{BRST symmetry}
 \label{sec:brst}
-- 
cgit v1.2.3-70-g09d2


From 3fdbfe8a8b79f810c173b7eaf657f6fd834d6c0b Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 25 Oct 2024 15:41:22 +0200
Subject: Clarified that eigenvalue integral relies on symmetry of matrix.

---
 marginal.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index 8efa79f..ddd31da 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -141,7 +141,7 @@ at the bottom on the spectrum.
 
 \subsection{The general method}
 
-Consider an $N\times N$ real matrix $A$. An arbitrary function $g$ of the
+Consider an $N\times N$ real symmetric matrix $A$. An arbitrary function $g$ of the
 minimum eigenvalue of $A$ can be expressed using integrals over $\mathbf
 s\in\mathbb R^N$ as
 \begin{equation} \label{eq:λmin}
-- 
cgit v1.2.3-70-g09d2


From 6aad3c973dd44c6e21022e1193f5d6c623d4e23b Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 25 Oct 2024 16:09:14 +0200
Subject: Fixed typo in integral equation for supermatrix multiplication.

---
 marginal.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index ddd31da..fac7c48 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -1759,7 +1759,7 @@ like the super vector $\pmb\phi$ is made up of a linear combination of $N\times
 N$ regular or Grassmann matrices indexed by every nonvanishing combination of
 the Grassmann indices $\bar\theta_1,\theta_1,\bar\theta_2,\theta_2$. Such a supermatrix acts on supervectors by ordinary matrix multiplication and convolution in the Grassmann indices, i.e.,
 \begin{equation}
-  (M\pmb\phi)(1)=\int d1\,M(1,2)\pmb\phi(2)
+  (M\pmb\phi)(1)=\int d2\,M(1,2)\pmb\phi(2)
 \end{equation}
 The identity supermatrix is given by
 \begin{equation}
-- 
cgit v1.2.3-70-g09d2


From 157b8b12bdad4646773d1c596f99af6a1b4d9c9d Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 25 Oct 2024 16:11:21 +0200
Subject: Fixed grammatically bad sentence.

---
 marginal.tex | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index fac7c48..acd3d4a 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -1288,8 +1288,8 @@ $\epsilon$ is increased, the most common type of marginal minimum drifts toward
 points with $\omega_1>\omega_2$.
 
 Multispherical spin glasses may be an interesting platform for testing ideas
-about which among the possible marginal minima can dynamics,
-and cannot. In the limit where $\epsilon=0$ and the configurations of the
+about which among the possible marginal minima can attract dynamics
+and which cannot. In the limit where $\epsilon=0$ and the configurations of the
 two spheres are independent, the minima found dynamically should be marginal on both
 subspaces. Just because technically on the expanded configuration space
 the Cartesian product of a deep stable minimum on one sphere and a marginal minimum on the other is
-- 
cgit v1.2.3-70-g09d2


From d73d3cdc03337fde998db900ed3232151e75f729 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 25 Oct 2024 19:00:16 +0200
Subject: Clarified the definitions of marginal minima and pseudogap in the
 opening paragraphs

---
 marginal.tex | 5 ++++-
 1 file changed, 4 insertions(+), 1 deletion(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index acd3d4a..d9e1d47 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -61,7 +61,10 @@ dynamics would get stuck at a specific energy level, called the threshold
 energy. The threshold energy is the energy level at which level sets of the
 landscape transition from containing mostly saddle points to containing mostly
 minima. The level set associated with this threshold energy contains mostly \emph{marginal
-minima}, or minima that have a pseudogap in the spectrum of their Hessian.
+minima}, or minima whose Hessian matrix has a continuous spectral density over
+all sufficiently small positive eigenvalues. In most circumstances the spectrum
+is \emph{pseudogapped}, which means that the spectral density smoothly
+approaches zero as zero eigenvalue is approached from above.
 
 However, recent work found that the threshold energy is not important even for
 simple gradient descent dynamics \cite{Folena_2020_Rethinking, Folena_2023_On, ElAlaoui_2020_Algorithmic}.
-- 
cgit v1.2.3-70-g09d2


From 35c4e960648856414d3425eddb69881e9028d6f9 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 28 Oct 2024 16:35:14 +0100
Subject: Clarified notation surrounding gradient and Hessian, and standardized
 arXiv bib entries.

---
 marginal.bib | 99 ++++++++++++++++++++++++++----------------------------------
 marginal.tex | 13 ++++++--
 2 files changed, 53 insertions(+), 59 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.bib b/marginal.bib
index 3e9eb8b..784d3f0 100644
--- a/marginal.bib
+++ b/marginal.bib
@@ -239,12 +239,10 @@
 @unpublished{ElAlaoui_2020_Algorithmic,
  author = {El Alaoui, Ahmed and Montanari, Andrea},
  title = {Algorithmic Thresholds in Mean Field Spin Glasses},
- year = {2020},
- month = {Sept},
  url = {http://arxiv.org/abs/2009.11481v1},
  archiveprefix = {arXiv},
  date = {2020-09-24T04:22:42Z},
- eprint = {2009.11481v1},
+ eprint = {2009.11481},
  eprintclass = {cond-mat.stat-mech},
  eprinttype = {arxiv},
  primaryclass = {cond-mat.stat-mech}
@@ -515,29 +513,23 @@
 @unpublished{Huang_2023_Algorithmic,
  author = {Huang, Brice and Sellke, Mark},
  title = {Algorithmic Threshold for Multi-Species Spherical Spin Glasses},
- year = {2023},
- month = {mar},
- url = {http://arxiv.org/abs/2303.12172v2},
+ url = {http://arxiv.org/abs/2303.12172},
  archiveprefix = {arXiv},
- date = {2023-03-21T20:09:08Z},
- eprint = {2303.12172v2},
+ eprint = {2303.12172},
  eprintclass = {math.PR},
- eprinttype = {arxiv},
- urldate = {2024-06-13T13:10:56.404805Z}
+ primaryclass = {math.PR},
+ eprinttype = {arxiv}
 }
 
 @unpublished{Huang_2023_Strong,
  author = {Huang, Brice and Sellke, Mark},
  title = {Strong Topological Trivialization of Multi-Species Spherical Spin Glasses},
- year = {2023},
- month = {aug},
- url = {http://arxiv.org/abs/2308.09677v2},
+ url = {http://arxiv.org/abs/2308.09677},
  archiveprefix = {arXiv},
- date = {2023-08-18T16:56:19Z},
- eprint = {2308.09677v2},
+ eprint = {2308.09677},
  eprintclass = {math.PR},
- eprinttype = {arxiv},
- urldate = {2024-06-13T13:07:13.561947Z}
+ primaryclass = {math.PR},
+ eprinttype = {arxiv}
 }
 
 @article{Huang_2024_Optimization,
@@ -600,13 +592,11 @@
 @unpublished{Kamali_2023_Stochastic,
  author = {Kamali, Persia Jana and Urbani, Pierfrancesco},
  title = {Stochastic Gradient Descent outperforms Gradient Descent in recovering a high-dimensional signal in a glassy energy landscape},
- year = {2023},
- month = {sep},
  url = {http://arxiv.org/abs/2309.04788v2},
- note = {},
  archiveprefix = {arXiv},
- eprint = {2309.04788v2},
+ eprint = {2309.04788},
  eprintclass = {cs.LG},
+ primaryclass = {cs.LG},
  eprinttype = {arxiv}
 }
 
@@ -627,11 +617,10 @@
 @unpublished{Kent-Dobias_2024_Algorithm-independent,
  author = {Kent-Dobias, Jaron},
  title = {Algorithm-independent bounds on complex optimization through the statistics of marginal optima},
- year = {2024},
  url = {https://arxiv.org/abs/2407.02092},
  archiveprefix = {arXiv},
- eprint = {2407.02092},
- primaryclass = {cond-mat.dis-nn}
+ primaryclass = {cond-mat.dis-nn},
+ eprint = {2407.02092}
 }
 
 @article{Kent-Dobias_2024_Arrangement,
@@ -652,7 +641,6 @@
 @unpublished{Kent-Dobias_2024_Conditioning,
  author = {Kent-Dobias, Jaron},
  title = {Conditioning the complexity of random landscapes on marginal optima},
- year = {2024},
  url = {https://arxiv.org/abs/2407.02082},
  archiveprefix = {arXiv},
  eprint = {2407.02082},
@@ -765,26 +753,23 @@
 @unpublished{Montanari_2023_Solving,
  author = {Montanari, Andrea and Subag, Eliran},
  title = {Solving overparametrized systems of random equations: I. Model and algorithms for approximate solutions},
- year = {2023},
- month = {jun},
  url = {http://arxiv.org/abs/2306.13326v1},
  note = {},
  archiveprefix = {arXiv},
- eprint = {2306.13326v1},
+ eprint = {2306.13326},
  eprintclass = {math.PR},
+ primaryclass = {math.PR},
  eprinttype = {arxiv}
 }
 
 @unpublished{Montanari_2024_On,
  author = {Montanari, Andrea and Subag, Eliran},
  title = {On {Smale}'s 17th problem over the reals},
- year = {2024},
- month = {may},
- url = {http://arxiv.org/abs/2405.01735v1},
- note = {},
+ url = {http://arxiv.org/abs/2405.01735},
  archiveprefix = {arXiv},
- eprint = {2405.01735v1},
+ eprint = {2405.01735},
  eprintclass = {cs.DS},
+ primaryclass = {cs.DS},
  eprinttype = {arxiv}
 }
 
@@ -835,11 +820,9 @@
 @unpublished{Parisi_1995-01_On,
  author = {Parisi, Giorgio},
  title = {On the Statistical Properties of the Large Time Zero Temperature Dynamics of the {SK} Model},
- year = {1995},
- month = {jan},
- url = {http://arxiv.org/abs/cond-mat/9501045v1},
+ url = {http://arxiv.org/abs/cond-mat/9501045},
  archiveprefix = {arXiv},
- eprint = {cond-mat/9501045v1},
+ eprint = {cond-mat/9501045},
  eprinttype = {arxiv}
 }
 
@@ -887,12 +870,11 @@
 @unpublished{Shklovskii_2024_Half,
  author = {Shklovskii, B. I.},
  title = {Half century of {Efros}-{Shklovskii} {Coulomb} gap. Romance with {Coulomb} interaction and disorder},
- year = {2024},
- month = {mar},
- url = {http://arxiv.org/abs/2403.19793v5},
+ url = {http://arxiv.org/abs/2403.19793},
  archiveprefix = {arXiv},
- eprint = {2403.19793v5},
+ eprint = {2403.19793},
  eprintclass = {cond-mat.mtrl-sci},
+ primaryclass = {cond-mat.mtrl-sci},
  eprinttype = {arxiv}
 }
 
@@ -928,15 +910,12 @@
 @unpublished{Subag_2021_TAP,
  author = {Subag, Eliran},
  title = {{TAP} approach for multi-species spherical spin glasses {I}: general theory},
- year = {2021},
- month = {nov},
  url = {http://arxiv.org/abs/2111.07132v1},
  archiveprefix = {arXiv},
- date = {2021-11-13T15:21:40Z},
- eprint = {2111.07132v1},
+ eprint = {2111.07132},
  eprintclass = {math.PR},
- eprinttype = {arxiv},
- urldate = {2024-06-13T13:04:28.790463Z}
+ primaryclass = {math.PR},
+ eprinttype = {arxiv}
 }
 
 @article{Subag_2023_TAP,
@@ -992,25 +971,22 @@
 @unpublished{Urbani_2024_Statistical,
  author = {Urbani, Pierfrancesco},
  title = {Statistical physics of complex systems: glasses, spin glasses, continuous constraint satisfaction problems, high-dimensional inference and neural networks},
- year = {2024},
- month = {may},
- url = {http://arxiv.org/abs/2405.06384v1},
- note = {},
+ url = {http://arxiv.org/abs/2405.06384},
  archiveprefix = {arXiv},
- eprint = {2405.06384v1},
+ eprint = {2405.06384},
  eprintclass = {cond-mat.dis-nn},
+ primaryclass = {cond-mat.dis-nn},
  eprinttype = {arxiv}
 }
 
 @unpublished{Vivo_2024_Random,
  author = {Vivo, Pierpaolo},
  title = {Random Linear Systems with Quadratic Constraints: from Random Matrix Theory to replicas and back},
- year = {2024},
- month = {jan},
- url = {http://arxiv.org/abs/2401.03209v2},
+ url = {http://arxiv.org/abs/2401.03209},
  archiveprefix = {arXiv},
- eprint = {2401.03209v2},
+ eprint = {2401.03209},
  eprintclass = {cond-mat.stat-mech},
+ primaryclass = {cond-mat.stat-mech},
  eprinttype = {arxiv}
 }
 
@@ -1082,3 +1058,14 @@
  issn = {1742-5468}
 }
 
+@unpublished{Kent-Dobias_2024_On,
+ author = {Kent-Dobias, Jaron},
+ title = {On the topology of solutions to random continuous constraint satisfaction problems},
+ url = {http://arxiv.org/abs/2409.12781},
+ archiveprefix = {arXiv},
+ eprint = {2409.12781},
+ eprintclass = {cond-mat.dis-nn},
+ primaryclass = {cond-mat.dis-nn},
+ eprinttype = {arxiv}
+}
+
diff --git a/marginal.tex b/marginal.tex
index d9e1d47..b9fabc9 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -469,8 +469,9 @@ extremizing the Lagrangian
   L(\mathbf x,\pmb\omega)=H(\mathbf x)+\sum_{i=1}^r\omega_ig_i(\mathbf x)
 \end{equation}
 with respect to $\mathbf x$ and the Lagrange multipliers
-$\pmb\omega=\{\omega_1,\ldots,\omega_r\}$. The corresponding gradient and
-Hessian of the energy associated with this constrained extremal problem are
+$\pmb\omega=\{\omega_1,\ldots,\omega_r\}$. To write the gradient and Hessian of the energy, which are necessary to count stationary points, care must be taken to ensure they are constrained to the tangent space of the configuration manifold. For our purposes, the Lagrangian formalism offers a solution: the gradient $\nabla H:\mathbb R^N\times\mathbb R^r\to\mathbb R^N$ and
+Hessian $\operatorname{Hess} H:\mathbb R^N\times\mathbb R^r\to\mathbb R^{N\times N}$ of the energy $H$ can be written as the simple vector derivatives of
+the Lagrangian $L$, with
 \begin{align}
   &\nabla H(\mathbf x,\pmb\omega)
   =\partial L(\mathbf x,\pmb\omega)
@@ -483,7 +484,13 @@ Hessian of the energy associated with this constrained extremal problem are
   \end{aligned}
 \end{align}
 where $\partial=\frac\partial{\partial\mathbf x}$ will always represent the
-derivative with respect to the vector argument $\mathbf x$.
+derivative with respect to the vector argument $\mathbf x$. Note that unlike
+the energy, which is a function of the configuration $\mathbf x$ alone, the
+gradient and Hessian depend also on the Lagrange multipliers $\pmb\omega$. In situations
+with an extensive number of constraints, it is important to take seriously
+contributions of the form $\frac{\partial^2L}{\partial\mathbf
+x\partial\pmb\omega}$ to the Hessian \cite{Kent-Dobias_2024_On}. However, the cases we study here have
+$N^0$ constraints and these contributions appear as finite-$N$ corrections.
 
 The number of
 stationary points in a landscape for a particular function $H$ is found by
-- 
cgit v1.2.3-70-g09d2


From 49e34257f5974cf63ab925f260457a1d5a7be079 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 28 Oct 2024 16:35:59 +0100
Subject: Fixed mistake surrounding the relationship between μ and Lagrange
 multipliers in some cases.
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 marginal.tex | 70 ++++++++++++++++++++++++++++++++++++++----------------------
 1 file changed, 45 insertions(+), 25 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index b9fabc9..442e7e2 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -641,16 +641,31 @@ treat the determinant keeping the absolute value signs, as in previous works
 \cite{Folena_2020_Rethinking, Kent-Dobias_2023_How}. However, other of
 our examples are for models where the same techniques are impossible.
 
-For the cases studied here, fixing the trace results in a relationship
-between $\mu$ and the Lagrange multipliers enforcing the constraints. This is
-because the trace of $\partial\partial H$ is typically an order of $N$ smaller
-than the trace of $\partial\partial g_i$. The result is that
+Finally, the $\delta$-function fixing the trace of the Hessian to $\mu$ in
+\eqref{eq:kac-rice.measure.2} must be addressed. One could treat it using a
+Fourier representation as in (\ref{eq:delta.grad}--\ref{eq:delta.eigen}), but
+this is inconvenient because a term of the form
+$\operatorname{Tr}\partial\partial H(\mathbf x)$ in the exponential integrand
+cannot be neatly captured in superspace representation introduced in the next
+section. However, in the cases we study in this paper a simplification can be made: the trace of $\partial\partial H$ can be separated into two pieces, one
+that is spatially independent and one that is typically small, or
+\begin{equation} \label{eq:mu.star}
+  \operatorname{Tr}\partial\partial H(\mathbf x)=N\mu^*_H+\Delta_H(\mathbf x)
+\end{equation}
+where $\overline{\mu^*_H}=\mu^*$ and $\overline{\Delta_H(\mathbf x)}=O(N^0)$.
+Then fixing the trace of the Hessian to $\mu$ implies that
 \begin{equation}
-  \mu
-  =\frac1N\operatorname{Tr}\operatorname{Hess}H(\mathbf x)
-  =\frac1N\sum_{i=1}^r\omega_i\operatorname{Tr}\partial\partial g_i(\mathbf x)
-  +O(N^{-1})
+  \begin{aligned}
+    \mu
+    &=\frac1N\operatorname{Tr}\operatorname{Hess}H(\mathbf x)
+    =\frac1N\left(\partial\partial H(\mathbf x)+
+    \sum_{i=1}^r\omega_i\operatorname{Tr}\partial\partial g_i(\mathbf x)\right)
+    \\
+    &=\mu^*+\frac1N\sum_{i=1}^r\omega_i\operatorname{Tr}\partial\partial g_i(\mathbf x)
+    +O(N^{-1})
+  \end{aligned}
 \end{equation}
+for typical samples $H$.
 In particular, here we study only cases with quadratic $g_i$, which results in
 a linear expression relating $\mu$ and the $\omega_i$ that is independent of $\mathbf
 x$. Since $H$ contains the disorder of the problem, this simplification means
@@ -720,7 +735,7 @@ functions and the determinant made. The new measures
   \delta\big((\mathbf s_a^\alpha)^T\partial\mathbf g(\mathbf x_a)\big)
   \\
   d\pmb\omega&=\bigg(\prod_{i=1}^rd\omega_i\bigg)
-  \,\delta\bigg(N\mu-\sum_i^r\omega_i\operatorname{Tr}\partial\partial g_i\bigg)
+  \,\delta\bigg(N\mu-\mu^*-\sum_i^r\omega_i\operatorname{Tr}\partial\partial g_i\bigg)
 \end{align}
 collect the individual measures of the various fields embedded in the superfield, along with their constraints.
 \end{widetext}
@@ -765,7 +780,8 @@ unit normal distribution \cite{Crisanti_1993_The}. We focus on marginal minima
 in models with $f'(0)=0$, which corresponds to models without a random external
 field. Such a random field would correspond in each individual sample $H$ to a
 signal, and therefore complicate the analysis by correlating the positions of
-stationary points and the eigenvectors of their Hessians.
+stationary points and the eigenvectors of their Hessians. Here, $\mu^*$ of
+\eqref{eq:mu.star} is zero.
 
 The marginal optima of these models can be studied without the methods
 introduced in this paper, and have been in the past \cite{Folena_2020_Rethinking,
@@ -1030,7 +1046,7 @@ for $\mathbf x,\mathbf x'\in\mathbb R^N$ by
   \overline{H_i(\mathbf x)H_j(\mathbf x')}
   =N\delta_{ij}f_i\left(\frac{\mathbf x\cdot\mathbf x'}N\right)
 \end{equation}
-with the functions $f_1$ and $f_2$ not necessarily the same.
+with the functions $f_1$ and $f_2$ not necessarily the same. As for the spherical spin glasses, $\mu^*$ of \eqref{eq:mu.star} is zero.
 
 In this problem, there is an energetic competition between the independent spin
 glass energies on each sphere and their tendency to align or anti-align through
@@ -1374,9 +1390,12 @@ As in the previous sections, we used the method of Lagrange multipliers to analy
     -V_k(\mathbf x)\partial\partial V_k(\mathbf x)\right]+\omega I
   \end{aligned}
 \end{align}
-As in the spherical and multispherical spin glasses, fixing the trace of the Hessian
-is equivalent to constraining the value of the Lagrange
-multiplier $\omega=\mu$.
+Unlike in the spherical and multispherical spin glasses, the value $\mu^*$
+defined in \eqref{eq:mu.star} giving the typical value of
+$\frac1N\operatorname{Tr}\partial\partial H$ is not always zero. Instead
+$\mu^*=-f'(0)$, nonzero where there is a linear term in $V$. Fixing the trace
+of the Hessian is therefore equivalent to setting $\omega=\mu+f'(0)$.
+
 
 The derivation of the marginal complexity for this model is complicated, but
 can be made schematically like that of the derivation of the equilibrium free
@@ -1389,10 +1408,11 @@ $\lambda^*$ is given by
   \begin{aligned}
     \mathcal N(E,\mu,\lambda^*)^n
     &=\int d\hat\beta\,d\hat\lambda\prod_{a=1}^n\lim_{m_a\to0}\prod_{\alpha=1}^{m_a}d\pmb\phi_a^\alpha
-    \exp\left\{
+    \\
+    &\qquad\times\exp\left\{
       \delta^{\alpha1}N(\hat\beta E+\hat\lambda\lambda^*)
       -\frac12\int d1\,d2\,\left[B^\alpha(1,2)\sum_{k=1}^MV_k(\pmb\phi_a^\alpha(1,2))^2
-      -\mu\|\pmb\phi_a^\alpha(1,2)\|^2\right]
+      -\big(\mu+f'(0)\big)\|\pmb\phi_a^\alpha(1,2)\|^2\right]
     \right\}
   \end{aligned}
 \end{equation}
@@ -1509,7 +1529,7 @@ with an effective action
 \begin{equation}
   \begin{aligned}
     &\mathcal S_\mathrm{RSS}(\hat\beta,\hat\lambda,r,d,g,q_0,\tilde q_0\mid\lambda^*,E,\mu,\beta)
-    =\hat\beta E-\mu(r+g+\hat\lambda)
+    =\hat\beta E-\big(\mu+f'(0)\big)(r+g+\hat\lambda)
     +\hat\lambda\lambda^*
     +\frac12\log\left(\frac{d+r^2}{g^2}
     \times\frac{1-2q_0+\tilde q_0^2}{(1-q_0)^2}\right) \\
@@ -1532,7 +1552,7 @@ taking the zero-temperature limit, we find
 \begin{equation}
   \begin{aligned}
     &\mathcal S_\mathrm{RSS}(\hat\beta,\hat\lambda,r,d,g,y,\Delta z\mid\lambda^*,E,\mu,\infty)
-    =\hat\beta E-\mu(r+g+\hat\lambda)
+    =\hat\beta E-\big(\mu+f'(0)\big)(r+g+\hat\lambda)
     +\hat\lambda\lambda^*
     +\frac12\log\left(\frac{d+r^2}{g^2}\times\frac{y^2-2\Delta z}{y^2}\right)
     \\
@@ -2012,9 +2032,9 @@ the replicated count of stationary points can be written
     =\int d\hat\beta\prod_{a=1}^n\,d\pmb\phi_a\,
     \exp\bigg[
       N\hat\beta E \\
-    &\qquad-\frac12\int d1\,\left(
+    &-\frac12\int d1\,\left(
         B(1)\sum_{k=1}^MV_k(\pmb\phi_a(1))^2
-        -\mu\|\pmb\phi_a(1)\|^2
+        -\big(\mu+f'(0)\big)\|\pmb\phi_a(1)\|^2
       \right)
     \bigg]
   \end{aligned}
@@ -2057,9 +2077,9 @@ Making the $M$ independent Gaussian integrals, we find
 \begin{equation}
   \begin{aligned}
     &\mathcal N(E,\mu)^n
-    =\int d\hat\beta\left(\prod_{a=1}^nd\pmb\phi_a\right)
-    \exp\bigg[
-      nN\hat\beta E+\frac\mu2\sum_a^n\int d1\,\|\pmb\phi_a\|^2 \\
+    =\int d\hat\beta\left(\prod_{a=1}^nd\pmb\phi_a\right) \\
+    &\times\exp\bigg[
+      nN\hat\beta E+\frac{\mu+f'(0)}2\sum_a^n\int d1\,\|\pmb\phi_a\|^2 \\
     &\quad-\frac M2\log\operatorname{sdet}\left(
         \delta_{ab}\delta(1,2)+B(1)f\left(\frac{\pmb\phi_a(1)\cdot\pmb\phi_b(2)}N\right)
       \right)
@@ -2080,7 +2100,7 @@ We therefore have
     &\mathcal N(E,\mu)^n
     =\int d\hat\beta\,d\mathbb Q\,
     \exp\bigg\{
-      nN\hat\beta E+N\frac\mu2\operatorname{sTr}\mathbb Q
+      nN\hat\beta E+N\frac{\mu+f'(0)}2\operatorname{sTr}\mathbb Q
     +\frac N2\log\operatorname{sdet}\mathbb Q
     -\frac M2\log\operatorname{sdet}\left[
       \delta_{ab}\delta(1,2)+B(1)f(\mathbb Q_{ab}(1,2))
@@ -2110,7 +2130,7 @@ where the effective action is given by
 \begin{equation}
   \begin{aligned}
     \mathcal S_\mathrm{KR}(\hat\beta,C,R,D,G)
-    &=\hat\beta E+\lim_{n\to0}\frac1n\Bigg(-\mu\operatorname{Tr}(G+R)
+    &=\hat\beta E+\lim_{n\to0}\frac1n\Bigg(-\big(\mu+f'(0)\big)\operatorname{Tr}(G+R)
     +\frac12\log\det\big[G^{-2}(CD+R^2)\big]
     +\alpha\log\det\big[I+G\odot f'(C)\big]
     \\
-- 
cgit v1.2.3-70-g09d2


From 405f6727a6915c61e09160fba52dd8832c2207e3 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 29 Oct 2024 10:19:16 +0100
Subject: Slightly modified convention for superbases.

---
 marginal.tex | 18 +++++++++---------
 1 file changed, 9 insertions(+), 9 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index 442e7e2..c8d0b8b 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -1804,26 +1804,26 @@ where the usual role of the determinant is replaced by the superdeterminant.
 The superdeterminant can be defined using the ordinary determinant by writing a
 block version of the matrix $M$. If $\mathbf e(1)=\{1,i\bar\theta_1\theta_1\}$ is
 the basis vector of the even subspace of the superspace and $\mathbf
-f(1)=\{i\bar\theta_1,i\theta_1\}$ is that of the odd subspace, dual bases $\mathbf e^\dagger(1)=\{i\bar\theta_1\theta_1,1\}$ and $\mathbf f^\dagger(1)=\{\theta_1,-\bar\theta_1\}$ can be defined by the requirement that
+f(1)=\{i\bar\theta_1,i\theta_1\}$ is that of the odd subspace, dual bases $\mathbf e^\dagger(1)=\{i\bar\theta_1\theta_1,1\}$ and $\mathbf f^\dagger(1)=\{-\theta_1,\bar\theta_1\}$ can be defined by the requirement that
 \begin{align}
-  \int d1\,\mathbf e(1)\mathbf e^\dagger(1)=iI
+  &\int d1\,e_i^\dagger(1)e_j(1)=i\delta_{ij}
   &&
-  \int d1\,\mathbf f(1)\mathbf f^\dagger(1)=iI \\
-  \int d1\,\mathbf e(1)\mathbf f^\dagger(1)=0
+  \int d1\,f_i^\dagger(1)f_j(1)=i\delta_{ij} \\
+  &\int d1\,e_i^\dagger(1)f_j(1)=0
   &&
-  \int d1\,\mathbf f(1)\mathbf e^\dagger(1)=0
+  \int d1\,f_i^\dagger(1)e_j(1)=0
 \end{align}
 With such bases and dual bases defined, we can form a
 block representation of $M$ in analogy to the matrix form of an operator in quantum mechanics by
 \begin{equation}
   \int d1\,d2\,\begin{bmatrix}
-    \mathbf e(1)M(1,2)\mathbf e^\dagger(2)
+    \mathbf e^\dagger(1)M(1,2)\mathbf e(2)
     &
-    \mathbf e(1)M(1,2)\mathbf f^\dagger(2)
+    \mathbf e^\dagger(1)M(1,2)\mathbf f(2)
     \\
-    \mathbf f(1)M(1,2)\mathbf e^\dagger(2)
+    \mathbf f^\dagger(1)M(1,2)\mathbf e(2)
     &
-    \mathbf f(1)M(1,2)\mathbf f^\dagger(2)
+    \mathbf f^\dagger(1)M(1,2)\mathbf f(2)
   \end{bmatrix}
   =i\begin{bmatrix}
     A & B \\ C & D
-- 
cgit v1.2.3-70-g09d2


From 568324cd4bc0cf2dd6a81464b1c4c700ee7ebfa5 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 29 Oct 2024 10:22:37 +0100
Subject: Another tweak to superbases.

---
 marginal.tex | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index c8d0b8b..88809d2 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -1802,13 +1802,13 @@ Integrals involving superfields contracted into such operators result in schemat
 \end{equation}
 where the usual role of the determinant is replaced by the superdeterminant.
 The superdeterminant can be defined using the ordinary determinant by writing a
-block version of the matrix $M$. If $\mathbf e(1)=\{1,i\bar\theta_1\theta_1\}$ is
+block version of the matrix $M$. If $\mathbf e(1)=\{1,\bar\theta_1\theta_1\}$ is
 the basis vector of the even subspace of the superspace and $\mathbf
-f(1)=\{i\bar\theta_1,i\theta_1\}$ is that of the odd subspace, dual bases $\mathbf e^\dagger(1)=\{i\bar\theta_1\theta_1,1\}$ and $\mathbf f^\dagger(1)=\{-\theta_1,\bar\theta_1\}$ can be defined by the requirement that
+f(1)=\{\bar\theta_1,\theta_1\}$ is that of the odd subspace, dual bases $\mathbf e^\dagger(1)=\{\bar\theta_1\theta_1,1\}$ and $\mathbf f^\dagger(1)=\{-\theta_1,\bar\theta_1\}$ can be defined by the requirement that
 \begin{align}
-  &\int d1\,e_i^\dagger(1)e_j(1)=i\delta_{ij}
+  &\int d1\,e_i^\dagger(1)e_j(1)=\delta_{ij}
   &&
-  \int d1\,f_i^\dagger(1)f_j(1)=i\delta_{ij} \\
+  \int d1\,f_i^\dagger(1)f_j(1)=\delta_{ij} \\
   &\int d1\,e_i^\dagger(1)f_j(1)=0
   &&
   \int d1\,f_i^\dagger(1)e_j(1)=0
@@ -1825,7 +1825,7 @@ block representation of $M$ in analogy to the matrix form of an operator in quan
     &
     \mathbf f^\dagger(1)M(1,2)\mathbf f(2)
   \end{bmatrix}
-  =i\begin{bmatrix}
+  =\begin{bmatrix}
     A & B \\ C & D
   \end{bmatrix}
 \end{equation}
-- 
cgit v1.2.3-70-g09d2


From f824dea3df7492fecfa95d34b33900a533bfd699 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 29 Oct 2024 10:24:54 +0100
Subject: Changed also convention for superbasis in R^N|4

---
 marginal.tex | 13 +++++++++++--
 1 file changed, 11 insertions(+), 2 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index 88809d2..85743c3 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -1843,8 +1843,17 @@ The same method can be used to calculate the
 superdeterminant and supertrace in arbitrary superspaces, where for $\mathbb R^{N|2D}$ each
 basis has $2^{2D-1}$ elements. For instance, for $\mathbb R^{N|4}$ we have
 \begin{align}
-  &\mathbf e(1,2)=\{1,i\bar\theta_1\theta_1,i\bar\theta_2\theta_2,i\bar\theta_1\theta_2,i\bar\theta_2\theta_1,i\bar\theta_1\bar\theta_2,i\theta_1\theta_2,\bar\theta_1\theta_1\bar\theta_2\theta_2\}\notag \\
-  &\mathbf f(1,2)=\{i\bar\theta_1,i\theta_1,i\bar\theta_2,i\theta_2,\bar\theta_1\theta_1\bar\theta_2,\bar\theta_2\theta_2\theta_1,\bar\theta_1\theta_1\theta_2,\bar\theta_2\theta_2\theta_1\}
+  &\mathbf e(1,2)=\{
+    1,\bar\theta_1\theta_1,\bar\theta_2\theta_2,
+    \bar\theta_1\theta_2,\bar\theta_2\theta_1,
+    \bar\theta_1\bar\theta_2,\theta_1\theta_2,
+    \bar\theta_1\theta_1\bar\theta_2\theta_2
+  \}\notag \\
+  &\mathbf f(1,2)=\{
+    \bar\theta_1,\theta_1,\bar\theta_2,\theta_2,
+    \bar\theta_1\theta_1\bar\theta_2,\bar\theta_2\theta_2\theta_1,
+    \bar\theta_1\theta_1\theta_2,\bar\theta_2\theta_2\theta_1
+  \}
 \end{align}
 with the dual bases defined analogously to those above.
 
-- 
cgit v1.2.3-70-g09d2


From 9cba972037904577a402062c35193ce05f0eb2ea Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 29 Oct 2024 10:46:26 +0100
Subject: Restored paragraph on difference between this work and Müller et al.
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 marginal.tex | 19 +++++++++++++++++++
 1 file changed, 19 insertions(+)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index 85743c3..3902aa8 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -1705,6 +1705,25 @@ self-similarity and stochastic stability of minima have
 recently been suggested as a route to understanding this problem, but this approach is
 still in its infancy \cite{Urbani_2024_Statistical}.
 
+The title of our paper and that of \citeauthor{Muller_2006_Marginal} suggest
+they address the same topic, but this is not the case
+\cite{Muller_2006_Marginal}. That work differs in three important and
+fundamental ways. First, it describes minima of the TAP free energy and
+involves peculiarities specific to the TAP. Second, it describes dominant
+minima which happen to be marginal, not a condition for finding subdominant marginal minima. Finally, it
+focuses on minima with a single soft direction (which are the typical minima of
+the low temperature Sherrington--Kirkpatrick TAP free energy), while we aim to
+avoid such minima in favor of ones that have a pseudogap (which we argue are relevant
+to out-of-equilibrium dynamics). The fact that the typical minima studied by
+\citeauthor{Muller_2006_Marginal} are not marginal in this latter sense may
+provide an intuitive explanation for the seeming discrepancy between the proof
+that the low-energy Sherrington--Kirkpatrick model cannot be sampled
+\cite{ElAlaoui_2022_Sampling} and the proof that a message passing algorithm
+can find near-ground states \cite{Montanari_2021_Optimization}: the algorithm
+finds the atypical low-lying states that are marginal in the sense considered
+here but cannot find the typical ones that are marginal in the sense of
+\citeauthor{Muller_2006_Marginal}.
+
 \begin{acknowledgements}
   JK-D is supported by a \textsc{DynSysMath} Specific Initiative of the INFN.
 \end{acknowledgements}
-- 
cgit v1.2.3-70-g09d2


From 8afe75733c423d131e1cbed04e12930cfacbd256 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 29 Oct 2024 10:51:10 +0100
Subject: Tweaked wording in paragraph on relationship with work of Müller et
 al.
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 marginal.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index 3902aa8..faa45ba 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -1721,7 +1721,7 @@ that the low-energy Sherrington--Kirkpatrick model cannot be sampled
 \cite{ElAlaoui_2022_Sampling} and the proof that a message passing algorithm
 can find near-ground states \cite{Montanari_2021_Optimization}: the algorithm
 finds the atypical low-lying states that are marginal in the sense considered
-here but cannot find the typical ones that are marginal in the sense of
+here but cannot find the typical ones considered by
 \citeauthor{Muller_2006_Marginal}.
 
 \begin{acknowledgements}
-- 
cgit v1.2.3-70-g09d2


From cae62fde30c9da6a91daf478ecd318366d2a9d1a Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 29 Oct 2024 10:56:09 +0100
Subject: Changed mentions of "companion" paper to "related work" or similar

---
 marginal.tex | 7 +++----
 1 file changed, 3 insertions(+), 4 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index faa45ba..32f1fd5 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -119,7 +119,7 @@ continuous spectrum, we enforce the condition that the spectrum has a
 pseudogap, and is therefore marginal. We demonstrate the method on the
 spherical spin glasses, where it is unnecessary but instructive, and on extensions of
 the spherical models where the technique is more useful.
-In a companion paper, we compare the marginal complexity with the performance
+In a related work, we compare the marginal complexity with the performance
 of gradient descent and approximate message passing algorithms \cite{Kent-Dobias_2024_Algorithm-independent}.
 
 An outline of this paper follows. In Section \ref{sec:eigenvalue} we introduce the technique for conditioning on
@@ -1652,7 +1652,7 @@ to determine the marginal stability continue to hold even in non-Gaussian cases
 where the complexity and the condition to fix the minimum eigenvalue are
 tangled together.
 
-In our companion paper, we use a sum of squared random functions model to explore the relationship between the marginal
+In a related paper, we use a sum of squared random functions model to explore the relationship between the marginal
 complexity and the performance of two generic algorithms:
 gradient descent and approximate message passing
 \cite{Kent-Dobias_2024_Algorithm-independent}. We show that the range of
@@ -1678,8 +1678,7 @@ We have introduced a method for conditioning complexity on the marginality of
 stationary points. This method is general, and permits
 conditioning without first needing to understand the statistics of the Hessian at stationary points. We used our approach to study marginal complexity in three
 different models of random landscapes, showing that the method works and can be
-applied to models whose marginal complexity was not previously known. In our
-companion paper, we further show that marginal complexity in the third
+applied to models whose marginal complexity was not previously known. In related work, we further show that marginal complexity in the third
 model of sums of squared random functions can be used to effectively bound
 algorithmic performance \cite{Kent-Dobias_2024_Algorithm-independent}.
 
-- 
cgit v1.2.3-70-g09d2


From 8af129d0bda10cb76bf4cf9e7fbcb40febe7d338 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 29 Oct 2024 10:58:52 +0100
Subject: Clarified conditions for decomposition of action

Added language to prevent implying that Gaussianity is sufficient to see
the decomposition of the effective action into two loosly connected
terms.
---
 marginal.tex | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index 32f1fd5..cf9db4c 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -956,9 +956,9 @@ the contributions from the marginal pieces of the calculation, and is given by
 \end{equation}
 \end{widetext}
 The fact that the complexity can be split into two relatively independent
-pieces in this way is a characteristic of the Gaussian nature of the spherical
+pieces in this way is a characteristic of the isotropic and Gaussian nature of the spherical
 spin glass. In Section \ref{sec:least.squares} we will study a model whose
-energy is not Gaussian and where such a decomposition is impossible.
+energy is isotropic but not Gaussian and where such a decomposition is impossible.
 
 There are some dramatic simplifications that emerge from the structure of this
 particular problem. First, notice that the dependence on the parameters $X$ and $\hat X$ are purely quadratic.
-- 
cgit v1.2.3-70-g09d2


From 8b2218dbcb8b05059312fa7741e5510c0ca077f2 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 29 Oct 2024 13:46:30 +0100
Subject: Grammer tweak

---
 marginal.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index cf9db4c..3e89bd7 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -61,7 +61,7 @@ dynamics would get stuck at a specific energy level, called the threshold
 energy. The threshold energy is the energy level at which level sets of the
 landscape transition from containing mostly saddle points to containing mostly
 minima. The level set associated with this threshold energy contains mostly \emph{marginal
-minima}, or minima whose Hessian matrix has a continuous spectral density over
+minima}, or minima whose Hessian matrix have a continuous spectral density over
 all sufficiently small positive eigenvalues. In most circumstances the spectrum
 is \emph{pseudogapped}, which means that the spectral density smoothly
 approaches zero as zero eigenvalue is approached from above.
-- 
cgit v1.2.3-70-g09d2


From 79309cd70841ecc1a0c34776ab962785c1e6ebce Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 29 Oct 2024 13:59:40 +0100
Subject: Wording tweak.

---
 marginal.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index 3e89bd7..1f1143c 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -669,7 +669,7 @@ for typical samples $H$.
 In particular, here we study only cases with quadratic $g_i$, which results in
 a linear expression relating $\mu$ and the $\omega_i$ that is independent of $\mathbf
 x$. Since $H$ contains the disorder of the problem, this simplification means
-that the effect of fixing the trace is independent of the disorder and only
+that the effect of fixing the trace is largely independent of the disorder and mostly
 depends on properties of the constraint manifold.
 
 \subsection{Superspace representation}
-- 
cgit v1.2.3-70-g09d2


From d944977b38915e8fe51ef1cca68b95d0a8107217 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 29 Oct 2024 14:00:55 +0100
Subject: Wording tweak.

---
 marginal.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index 1f1143c..30d3dce 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -1394,7 +1394,7 @@ Unlike in the spherical and multispherical spin glasses, the value $\mu^*$
 defined in \eqref{eq:mu.star} giving the typical value of
 $\frac1N\operatorname{Tr}\partial\partial H$ is not always zero. Instead
 $\mu^*=-f'(0)$, nonzero where there is a linear term in $V$. Fixing the trace
-of the Hessian is therefore equivalent to setting $\omega=\mu+f'(0)$.
+of the Hessian is therefore equivalent to constraining the value of the Lagrange multiplier $\omega=\mu+f'(0)$.
 
 
 The derivation of the marginal complexity for this model is complicated, but
-- 
cgit v1.2.3-70-g09d2


From 77cf86b193f24630890990e105fe40730d353fd0 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 29 Oct 2024 14:49:46 +0100
Subject: Fixed spelling mistake.

---
 marginal.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index 30d3dce..7a81a74 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -1245,7 +1245,7 @@ asymptotic behavior of the overlaps. These take the form
 $q^{ij}_0=q^{ij}_d-y^{ij}_0\beta^{-1}-z^{ij}_0\beta^{-2}$. Notice that in this case, the asymptotic behavior of the
 off-diagonal elements is to approach the value of the diagonal rather than to approach one.
 We also require $\tilde q^{ij}_d=q^{ij}_d-\tilde y^{ij}_d\beta^{-1}-\tilde
-z^{ij}_d\beta^{-2}$, i.e., that the tilde diagonal terms also approache the
+z^{ij}_d\beta^{-2}$, i.e., that the tilde diagonal terms also approach the
 same diagonal value as the untilde terms, but with potentially different rates.
 
 As before, in order for the logarithmic term to stay finite, there are necessary
-- 
cgit v1.2.3-70-g09d2