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authorkurchan.jorge <kurchan.jorge@gmail.com>2020-12-13 10:43:55 +0000
committeroverleaf <overleaf@localhost>2020-12-13 19:55:36 +0000
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@@ -118,7 +118,7 @@ We see from (\ref{cosa}) that at any critical point, $\epsilon=H/N$, the average
Since $H$ is holomorphic, any critical point of $\operatorname{Re}H$ is also a
critical point of $\operatorname{Im}H$. The number of critical points of $H$ is
therefore the same as that of $\operatorname{Re}H$. From each saddle
-emerges a gradient line of $\operatorname{Re}H$, which is also one of constant
+emerge a gradient lines of $\operatorname{Re}H$, which is also one of constant
$\operatorname{Im}H$ and therefore constant phase.
Writing $z=x+iy$, $\operatorname{Re}H$ can be considered a real-valued function
@@ -441,7 +441,7 @@ threshold level, where the system develops a mid-spectrum gap, will play a
crucial role as it does in the real case.
\begin{acknowledgments}
-We wish to thank Alexander Altland for a useful suggestion.
+We wish to thank Alexander Altland, Satya Majumdar and Gregory Schehr for a useful suggestions.
JK-D and JK are supported by the Simons Foundation Grant No.~454943.
\end{acknowledgments}