saving work

Manuscript/EPAWTFT.tex

@@ -728,11 +728,12 @@ The UMP partitioning yield the following $\lambda$-dependent Hamiltonian:
 \end{widetext}
 A closed-form expression for the ground-state energy can be obtained but it is cumbersome, so we eschew reporting it.
 The radius of convergence of the UMP series can obtained numerically as a function of $U/t$ and is depicted in Fig.~\ref{fig:RadConv}.
+From it, we clearly see that the UMP series has \titou{always?} a larger radius of convergence than the RMP series \titou{(except maybe at $U/t = 2^+$)}, and that the UMP ground-state series is always convergent as $r_c > 1$ for all $U/t$.
  
 \begin{figure}
 	\includegraphics[width=\linewidth]{RadConv}
 	\caption{
-	Evolution of the radius of convergence $r_c$ associated with the RMP ground state (red), the UMP ground state (blue), and the UMP excited state (orange) as functions of the ration $U/t$.
+	Evolution of the radius of convergence $r_c$ associated with the RMP ground state (red), the UMP ground state (blue), and the UMP excited state (orange) as functions of the ratio $U/t$.
 	\label{fig:RadConv}}
 \end{figure}