Fixed many warnings.

1 files changed, 50 insertions, 41 deletions

diff --git a/stokes.tex b/stokes.tex
index 2a5724f..b2176ce 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -911,7 +911,7 @@ complex matrix $B$.
 
 Introducing replicas to bring the partition function into the numerator of the
 Green function \cite{Livan_2018_Introduction} gives
-\begin{equation} \label{eq:green.replicas}
+\begin{equation} \label{eq:green.replicas} \fl\quad
   G(\sigma)=\lim_{n\to0}\int dz\,dz^*\,(z^{(0)})^\dagger z^{(0)}
     \exp\left\{
     -\sum_\alpha^n\left[(z^{(\alpha)})^\dagger z^{(\alpha)}\sigma
@@ -927,11 +927,13 @@ replica vectors. The replica-symmetric ansatz leaves all replica vectors
 zero, and $\alpha_{\alpha\beta}=\alpha_0\delta_{\alpha\beta}$,
 $\chi_{\alpha\beta}=\chi_0\delta_{\alpha\beta}$. The result is
 \begin{equation}\label{eq:green.saddle}
-  \overline G(\sigma)=N\lim_{n\to0}\int d\alpha_0\,d\chi_0\,d\chi_0^*\,\alpha_0
-  \exp\left\{nN\left[
-    1+\frac18A_0\alpha_0^2-\frac{\alpha_0\sigma}2+\frac12\log(\alpha_0^2-|\chi_0|^2)
-    +\frac14\operatorname{Re}\left(\frac14C_0^*\chi_0^2+\lambda_0^*\chi_0\right)
-  \right]\right\}.
+  \eqalign{
+    \overline G(\sigma)=N\lim_{n\to0}\int &d\alpha_0\,d\chi_0\,d\chi_0^*\,\alpha_0
+    \exp\left\{nN\left[
+      1+\frac18A_0\alpha_0^2-\frac{\alpha_0\sigma}2\right.\right.\cr
+         &\left.\left.+\frac12\log(\alpha_0^2-|\chi_0|^2)+\frac14\operatorname{Re}\left(\frac14C_0^*\chi_0^2+\lambda_0^*\chi_0\right)
+    \right]\right\}.
+  }
 \end{equation}
 
 \begin{figure}
@@ -1256,16 +1258,18 @@ the result can be written, neglecting constant factors,
   \overline\mathcal N\simeq\int dQ\,e^{NS_\mathrm{eff}(Q)}
 \end{equation}
 for an effective action functional of the supermatrix $Q$
-\begin{equation}
-  S_{\mathrm{eff}}=
-  \int d1\,d2\,\operatorname{Tr}\left(
-      \frac14\left[
-        \matrix{\frac14&\frac14\cr\frac14&\frac14}
-      \right]Q^{(p)}(1,2)-\frac p2\left[
-        \matrix{\frac\epsilon2&0\cr0&\frac{\epsilon^*}2}
-      \right](Q(1,1)-I)\delta(1,2)
-    \right)
-    +\frac12\log\det Q
+\begin{equation} \fl
+  \eqalign{
+    S_{\mathrm{eff}}&=
+    \int d1\,d2\,\operatorname{Tr}\left(
+        \frac14\left[
+          \matrix{\frac14&\frac14\cr\frac14&\frac14}
+          \right]Q^{(p)}(1,2)-\frac p2\left[
+          \matrix{\frac\epsilon2&0\cr0&\frac{\epsilon^*}2}
+        \right](Q(1,1)-I)\delta(1,2)
+      \right) \\
+                                &\hspace{28em}+\frac12\log\det Q
+    }
 \end{equation}
 where the exponent in parentheses denotes element-wise exponentiation, and
 \begin{equation}
@@ -1382,23 +1386,28 @@ stationary points with given energies $\epsilon_1$ and $\epsilon_2$ are
   \right]
 \end{equation}
 \begin{equation}
-  S_\mathrm{eff}
-  =\int d1\,d2\,\operatorname{Tr}\left\{
-    \frac14\left[\matrix{1&\frac12&\frac12\cr\frac12&\frac14&\frac14\cr\frac12&\frac14&\frac14}\right]Q^{(p)}(1,2)
-    -\frac p2\left[
-    \matrix{\epsilon_1&0&0\cr0&\frac12\epsilon_2&0\cr0&0&\frac12\epsilon_2^*}
-      \right]\left(
-      Q(1,1)-I
-    \right)\delta(1,2)
-  \right\}+\frac12\det Q
+  \eqalign{
+    S_\mathrm{eff}
+    &=\int d1\,d2\,\operatorname{Tr}\left\{
+      \frac14\left[\matrix{1&\frac12&\frac12\cr\frac12&\frac14&\frac14\cr\frac12&\frac14&\frac14}\right]Q^{(p)}(1,2)
+      \right. \\
+    &\qquad\qquad\left.-\frac p2\left[
+      \matrix{\epsilon_1&0&0\cr0&\frac12\epsilon_2&0\cr0&0&\frac12\epsilon_2^*}
+        \right]\left(
+        Q(1,1)-I
+      \right)\delta(1,2)
+    \right\}+\frac12\det Q
+  }
 \end{equation}
 \begin{equation}
-  0=\frac{\partial S_\mathrm{eff}}{\partial Q(1,2)}
-  =
-    \frac p4\left[\matrix{1&\frac12&\frac12\cr\frac12&\frac14&\frac14\cr\frac12&\frac14&\frac14}\right]\odot Q^{(p-1)}(1,2)
-    -\frac p2\left[
-  \matrix{\epsilon_1&0&0\cr0&\frac12\epsilon_2&0\cr0&0&\frac12\epsilon_2^*}\right]\delta(1,2)
-  +\frac12Q^{-1}(1,2)
+  \eqalign{
+    0=\frac{\partial S_\mathrm{eff}}{\partial Q(1,2)}
+     &=
+      \frac p4\left[\matrix{1&\frac12&\frac12\cr\frac12&\frac14&\frac14\cr\frac12&\frac14&\frac14}\right]\odot Q^{(p-1)}(1,2) \\
+     &\qquad\qquad-\frac p2\left[
+    \matrix{\epsilon_1&0&0\cr0&\frac12\epsilon_2&0\cr0&0&\frac12\epsilon_2^*}\right]\delta(1,2)
+    +\frac12Q^{-1}(1,2)
+  }
 \end{equation}
 where $\odot$ denotes element-wise multiplication.
 \begin{equation}
@@ -1406,8 +1415,8 @@ where $\odot$ denotes element-wise multiplication.
     0
     &=\int d3\,\frac{\partial S_\mathrm{eff}}{\partial Q(1,3)}Q(3,2) \\
     &=\frac p4\int d3\,
-      \left\{\left[\matrix{1&\frac12&\frac12\cr\frac12&\frac14&\frac14\cr\frac12&\frac14&\frac14}\right]\odot Q^{(p-1)}(1,3)\right\}Q(3,2)
-      -\frac p2 \left[
+      \left\{\left[\matrix{1&\frac12&\frac12\cr\frac12&\frac14&\frac14\cr\frac12&\frac14&\frac14}\right]\odot Q^{(p-1)}(1,3)\right\}Q(3,2) \\
+                            &\qquad\qquad\qquad\qquad  -\frac p2 \left[
     \matrix{\epsilon_1&0&0\cr0&\frac12\epsilon_2&0\cr0&0&\frac12\epsilon_2^*}\right]Q(1,2)
     +\frac12I\delta(1,2)
   }
@@ -1478,7 +1487,7 @@ function of $\Delta$ and $\arg\delta$.
   }
 \end{figure}
 
-\subsection{Pure $p$-spin: is analytic continuation possible?}
+\subsection{Pure {\it p}-spin: is analytic continuation possible?}
 
 \begin{equation}
   \eqalign{
@@ -1489,7 +1498,7 @@ function of $\Delta$ and $\arg\delta$.
       e^{-\beta\mathcal S(s_\sigma)} \\
     &\simeq\sum_{k=0}^D\int d\epsilon\,\mathcal N_\mathrm{typ}(\epsilon,k)
       \left(\frac{2\pi}\beta\right)^{D/2}i^k
-      \left(|\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}\bigm|\mathcal S(s_\sigma)=N\epsilon,k_\sigma=k\right) e^{-\beta N\epsilon}
+      |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12} e^{-\beta N\epsilon}
   }
 \end{equation}
 Following Derrida \cite{Derrida_1991_The},
@@ -1503,17 +1512,17 @@ governs things at large $|\beta|$, not its total. This gives two terms to the ty
 \begin{equation}
   Z_\mathrm{typ}=Z_A+Z_B
 \end{equation}
-\begin{eqnarray}
+\begin{eqnarray} \fl
   Z_A
-  &\simeq\sum_{k=0}^D\int d\epsilon\,\overline\mathcal N(\epsilon,k)
+  \simeq\sum_{k=0}^D\int d\epsilon\,\overline\mathcal N(\epsilon,k)
     \left(\frac{2\pi}\beta\right)^{D/2}i^k
     |\det\operatorname{Hess}\mathcal S|^{-\frac12} e^{-\beta N\epsilon}
     =\int d\epsilon\,e^{Nf_A(\epsilon)}
-    \\
+    \\ \fl
   Z_B
-  &\simeq\sum_{k=0}^D\int d\epsilon\,\eta(\epsilon,k)\overline\mathcal N(\epsilon,k)^{1/2}
+  \simeq\sum_{k=0}^D\int d\epsilon\,\eta(\epsilon,k)\overline\mathcal N(\epsilon,k)^{1/2}
     \left(\frac{2\pi}\beta\right)^{D/2}i^k
-    |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}e^{-\beta N\epsilon}
+    |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}e^{-\beta N\epsilon} \\
     =\int d\epsilon\,\tilde\eta(\epsilon)e^{Nf_B(\epsilon)}
 \end{eqnarray}
 for
@@ -1577,7 +1586,7 @@ large-$|\beta|$ saddle-point used to evaluate the thimble integrals. Taking the
 thimbles to the next order in $\beta$ may reveal more explicitly where Stokes
 points become important.
 
-\section{The $p$-spin spherical models: numerics}
+\section{The {\it p}-spin spherical models: numerics}
 
 To study Stokes lines numerically, we approximated them by parametric curves.
 If $z_0$ and $z_1$ are two stationary points of the action with