From 43ed1805b695086eb1eb7218fc483557a6df82be Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 12 Mar 2021 16:38:17 +0100
Subject: Changed some notation to be more clear.

---
 bezout.tex | 32 +++++++++++++-------------------
 1 file changed, 13 insertions(+), 19 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index eaa98ce..458b448 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -51,7 +51,7 @@ review see \cite{Castellani_2005_Spin-glass}) defined by the energy
   H_0 = \frac1{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p},
 \end{equation}
 where $J$ is a symmetric tensor whose elements are real Gaussian variables and
-$z\in\mathbb R^N$ is constrained to the sphere $z^2=N$. This problem has been
+$z\in\mathbb R^N$ is constrained to the sphere $z^Tz=N$. This problem has been
 studied in the algebra \cite{Cartwright_2013_The} and probability literature
 \cite{Auffinger_2012_Random, Auffinger_2013_Complexity}.  It has been attacked
 from several angles: the replica trick to compute the Boltzmann--Gibbs
@@ -68,7 +68,7 @@ In this paper we extend the study to complex variables: we shall take
 $z\in\mathbb C^N$ and $J$ to be a symmetric tensor whose elements are
 \emph{complex} normal, with $\overline{|J|^2}=p!/2N^{p-1}$ and
 $\overline{J^2}=\kappa\overline{|J|^2}$ for complex parameter $|\kappa|<1$. The
-constraint remains $z^2=N$.
+constraint remains $z^Tz=N$.
 
 The motivations for this paper are of two types. On the practical side, there
 are indeed situations in which complex variables appear naturally in disordered
@@ -98,27 +98,26 @@ of Lagrange multipliers: introducing $\epsilon\in\mathbb C$, our energy is
 \begin{equation} \label{eq:constrained.hamiltonian}
   H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right).
 \end{equation}
-One might balk at the constraint $z^2=N$---which could appropriately be called
-a \emph{hyperbolic} constraint---by comparison with $|z|^2=N$. The reasoning
+One might balk at the constraint $z^Tz=N$---which could appropriately be called
+a \emph{hyperbolic} constraint---by comparison with $z^\dagger z=N$. The reasoning
 behind the choice is twofold.
 
 First, we seek draw conclusions from our model that would be applicable to
 generic holomorphic functions without any symmetry. Samples of $H_0$ nearly
 provide this, save for a single anomaly: the value of the energy and its
 gradient at any point $z$ correlate along the $z$ direction, with
-$\overline{H_0\partial_iH_0}\propto\kappa(z^2)^{p-1}z_i$ and
-$\overline{H_0(\partial_iH_0)^*}\propto|z|^{2(p-1)}z_i$. Besides being a
+$\overline{H_0\partial_iH_0}\propto \overline{H_0(\partial_iH_0)^*}\propto z_i$. Besides being a
 spurious correlation, in each sample there is also a `radial' gradient of
 magnitude proportional to the energy, since $z\cdot\partial H_0=pH_0$. This
 anomalous direction must be neglected if we are to draw conclusions about
-generic functions, and the constraint surface $z^2=N$ is the unique surface
+generic functions, and the constraint surface $z^Tz=N$ is the unique surface
 whose normal is parallel to $z$ and which contains the configuration space of
 the real $p$-spin model as a subspace.
 
 Second, taking the constraint to be the level set of a holomorphic function
 means the resulting configuration space is a \emph{bone fide} complex manifold,
 and therefore permits easy generalization of the integration techniques
-referenced above. The same cannot be said for the space defined by $|z|^2=N$,
+referenced above. The same cannot be said for the space defined by $z^\dagger z=N$,
 which is topologically the $(2N-1)$-sphere and cannot admit a complex
 structure.
 
@@ -128,10 +127,10 @@ related problems have similar properties but are concerned with subspaces on
 which the energy is bounded. (In fact, identifying the appropriate subspace on
 which to define one's model often requires the study of critical points in the
 whole space.) Where it might be a problem, we introduce the additional
-constraint $|z|^2\leq Nr^2$. The resulting configuration space is a complex
+constraint $z^\dagger z\leq Nr^2$. The resulting configuration space is a complex
 manifold with boundary. We shall see that the `radius' $r$ proves an insightful
 knob in our present problem, revealing structure as it is varied. Note
-that---combined with the constraint $z^2=N$---taking $r=1$ reduces the problem
+that---combined with the constraint $z^Tz=N$---taking $r=1$ reduces the problem
 to that of the ordinary $p$-spin.
 
 The critical points are of $H$ given by the solutions to the set of equations
@@ -211,8 +210,7 @@ be averaged independently. The $\delta$-functions are converted to exponentials
 by the introduction of auxiliary fields $\hat z=\hat x+i\hat y$.  The average
 of those factors over $J$ can then be performed. A generalized
 Hubbard--Stratonovich allows a change of variables from the $4N$ original
-and auxiliary fields to eight bilinears defined by $Na=|z|^2$, $N\hat a=|\hat
-z|^2$, $N\hat c=\hat z^2$, $Nb=\hat z^*z$, and $Nd=\hat zz$ (and their
+and auxiliary fields to eight bilinears defined by $Na=z^\dagger z$, $N\hat a=\hat z^\dagger\hat z$, $N\hat c=\hat z^T\hat z$, $Nb=\hat z^\dagger z$, and $Nd=\hat z^Tz$ (and their
 conjugates). The result, to leading order in $N$, is
 \begin{equation} \label{eq:saddle}
     \overline{\mathcal N}(\kappa,\epsilon,r)
@@ -244,10 +242,7 @@ where $\theta=\frac12\arg\kappa$ and
 \begin{equation}
   C_{\pm}=\frac{a^p(1+p(a^2-1))\mp a^2|\kappa|}{a^{2p}\pm(p-1)a^p(a^2-1)|\kappa|-a^2|\kappa|^2}.
 \end{equation}
-This leaves a single parameter, $a$, which dictates the magnitude of $|z|^2$,
-or alternatively the magnitude $y^2$ of the imaginary part. The latter vanishes
-as $a\to1$, where (as we shall see) one recovers known results for the real
-$p$-spin.
+This leaves a single parameter, $a$, which dictates the norm of $z$. 
 
 The Hessian $\partial\partial H=\partial\partial H_0-p\epsilon I$ is equal to
 the unconstrained Hessian with a constant added to its diagonal. The eigenvalue
@@ -385,9 +380,8 @@ in supersymmetric quiver theories \cite{Manschot_2012_From}.
   \includegraphics{fig/complexity.pdf}
   \caption{
     The complexity of the 3-spin model at $\epsilon=0$ as a function of
-    the maximum `radius' $r=|z_{\mathrm{max}}|/\sqrt N$ at several values of
-    $\kappa$. The dashed line shows $\frac12\log(p-1)$, while the dotted shows
-    $\log(p-1)$.
+    the maximum `radius' $r$ at several values of $\kappa$. The dashed line
+    shows $\frac12\log(p-1)$, while the dotted shows $\log(p-1)$.
   } \label{fig:complexity}
 \end{figure}
 
-- 
cgit v1.2.3-54-g00ecf

