Update on Overleaf.

1 files changed, 32 insertions, 31 deletions

diff --git a/stokes.tex b/stokes.tex
index 021c865..142b147 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -30,17 +30,17 @@
   In this paper we follow up the study of `complex complex landscapes'
   \cite{Kent-Dobias_2021_Complex}, rugged landscapes of many complex
   variables.  Unlike real landscapes, there is no useful classification of
-  saddles by index. Instead, the spectrum at stationary points determines their
-  tendency to trade topological numbers under analytic continuation of the
-  theory. These trades, which occur at Stokes points, proliferate when the
-  spectrum includes marginal directions and are exponentially suppressed
+  saddles by index. Instead, the spectrum of fluctuations at stationary points determines their
+  topological stability under analytic continuation of the
+  theory. Topological changes, which occur at so-called Stokes points, proliferate when the
+ saddles have marginal (flat) directions and are exponentially suppressed
   otherwise. This gives a direct interpretation of the `threshold' energy---which
-  in the real case separates saddles from minima---where the spectrum of
-  typical stationary points develops a gap. This leads to different consequences
+  in the real case separates saddles from minima--- as the level where the spectrum of
+  the Hessian matrix of stationary points develops a gap. This leads to different consequences
   for the analytic continuation of real landscapes with different structures:
-  the global minima of ``one step replica-symmetry broken'' landscapes lie
-  beyond a threshold and are locally protected from Stokes points, whereas
-  those of ``many step replica-symmetry broken'' lie at the threshold and
+  the global minima of `one step replica-symmetry broken' landscapes lie
+  beyond a threshold, their Hessians are gapped,  and are locally protected from Stokes points, whereas
+  those of ``many step replica-symmetry broken'' have gapless Hessians and
   Stokes points immediately proliferate.
   A new matrix ensemble is found, playing the role that  GUE plays for real landscapes in determining
   the topological nature of saddles.
@@ -50,42 +50,42 @@
 
 \tableofcontents
 
-\section{Preamble}
+\section{Introduction}
 
 Complex landscapes are basically functions of many variables having many minima
 and, inevitably, many saddles of all `indices' (their number of unstable
-directions). Optimization problems require us to find the deepest minima, often
+directions). Optimization theory requires us to find the deepest minima, often
 a difficult task.  For example, particles with a repulsive mutual potential
 enclosed in a box will have many stable configurations, and we are asked to
 find the one with lowest energy.
 
-An aim of complexity theory is to be able to classify these landscapes in
+An aim of complexity studies is to be able to classify these landscapes in
 families having common properties. Two simplifications make the task
-potentially tractable. The first is to consider the limit of many variables. In
-the example of the particles, the limit of many particles (i.e. the
-thermodynamic limit) may be expected to bring about simplifications.  The
-second simplification is of more technical nature:  we consider  functions that
+potentially tractable. The first is to consider the limit of many variables; in
+the example of the particles, the limit of many particles, i.e. the
+thermodynamic limit.  The
+second simplification is of more technical nature:  we often consider  functions that
 contain some random element to them, and we study the average of an ensemble.
-The paradigm of this is the spin-glass, where the interactions are random, and
+The paradigm of this are spin-glasses, where the interactions are random, and
 we are asked to find the ground state energy {\em on average over randomness}.
 
 Spin glass theory gave a surprise: random   landscapes come in two kinds:
 those that have a `threshold level' of energy, below which there are many
 minima but almost no saddles, separated by high barriers, and those that have
 all sorts of saddles all the way down to the lowest energy levels, and local
-minima are separated by relatively small barriers.  The latter are still
+minima are separated by  barriers of sub-extensive energy height.  The latter are still
 complex, but good solutions are easier to find.  This classification is closely
 related to the structure of their Replica Trick solutions.  Armed with this
-solvable (random) example, it was easy to find non-random examples that behave,
-at least approximately,  in these two ways (e.g. sphere packings and the
-travelling salesman problem, belong to first and second classes, respectively).
+solvable random example, it was easy to find non-random examples that behave,
+at least approximately,  in these two ways, for example  sphere packings and the
+travelling salesman problem belong to first and second classes, respectively.
 
-What about systems whose variables are not real, but rather complex?  Recalling
+What about systems whose variables are not real, but rather, complex?  Recalling
 the Cauchy--Riemann conditions, we immediately see a difficulty: if our cost is,
 say, the real part of a function of $N$ complex variables, in terms of the
-corresponding $2N$ real variables it has only saddles of index $N$!  Even
+corresponding $2N$ real variables it has only saddles of index $N$.  Even
 worse: often not all saddles are equally interesting, so simply finding the
-lowest is not usually what we need to do (more about this below).  As it turns
+lowest is not usually what we need to do.  As it turns
 out, there is a set of interesting questions to ask, as we describe below. For
 each saddle, there is a `thimble' spanned by the lines along which the cost
 function decreases. The way in which these thimbles fill the complex space is
@@ -104,10 +104,10 @@ proliferation -- or lack of it -- of Stokes lines is the interesting quantity
 in this context.
 
 
-\section{Introduction}
+\section{Analytic continuation in many variables}
 
 Analytic continuation of physical theories is sometimes useful. Some theories
-have a well-motivated hamiltonian or action that nevertheless results in a
+have a well-motivated Hamiltonian or action that nevertheless results in a
 divergent partition function, and can only be properly defined by continuation
 from a parameter regime where everything is well-defined
 \cite{Witten_2011_Analytic}. Others result in oscillatory phase space measures
@@ -132,7 +132,7 @@ study in the computer science of machine learning, the condensed matter theory
 of strange metals, and the high energy physics of black holes. What becomes of
 analytic continuation under these conditions?
 
-\section{Thimble integration and analytic continuation}
+\section{Thimble integration}
 
 \subsection{Decomposition of the partition function into thimbles}
 
@@ -155,9 +155,10 @@ but everything here is general: the action can be complex- or even
 imaginary-valued, and $\Omega$ could be infinite-dimensional. In typical
 contexts, $\Omega$ will be the euclidean real space $\mathbb R^N$ or some
 subspace of this like the sphere $S^{N-1}$ (as in the $p$-spin spherical models
-on which we will focus). In this paper we will consider only analytic
+on which we will treat later). In this paper we will consider only analytic
 continuation of the parameter $\beta$, but any other parameter would work
-equally well, e.g., of some parameter inside the action. The action will have
+equally well, e.g., of some parameter inside the action. The action for real $\beta$ will have,
+along the real direction, 
 some stationary points, e.g., minima, maxima, saddles, and the set of those
 points in $\Omega$ we will call $\Sigma_0$, the set of real stationary points.
 An example action used throughout this section is shown in
@@ -265,7 +266,7 @@ of a collection of pieces called \emph{Lefschetz thimbles}, or just thimbles.
 There is one thimble $\mathcal J_\sigma$ associated with each of the stationary
 points $\sigma\in\Sigma$ of the action, and each is defined by all points that
 approach the stationary point $s_\sigma$ under gradient descent on
-$\operatorname{Re}\beta\mathcal S$.
+$\operatorname{Re}\beta\mathcal S$: each thimble is the basin of attraction of a saddle.
 
 Thimbles guarantee convergent integrals by construction: the value of
 $\operatorname{Re}\beta\mathcal S$ is bounded from below on the thimble $\mathcal J_\sigma$ by its value
@@ -653,7 +654,7 @@ plane. It follows that each stationary point has an equal number of descending
 and ascending directions, e.g., the index of each stationary point is $N$. For
 a stationary point in a real problem this might seem strange, because there are
 clear differences between minima, maxima, and saddles of different index.
-However, for a such a stationary point, its $N$ real Takagi vectors that
+However, for  such a stationary point, its $N$ real Takagi vectors that
 determine its index in the real problem are accompanied by $N$ purely imaginary
 Takagi vectors, pointing into the complex plane and each with the negative
 eigenvalue of its partner. A real minimum on the real manifold therefore has