+
+\subsubsection*{Phase boundaries at $\mu > 0$}
+
+As a first step in the determination of the phase boundaries as functions of all three parameters $\beta, \, M^2$ and $\mu$,
+in Fig.~\ref{muphases} we plot the observables $\langle U \rangle$, $\langle |\phi|^2 \rangle$ and $\langle n \rangle$ as functions
+of $\beta$ and $M^2$ for four different values of the chemical potential $\mu=0.0,\, 0.5,\, 1.0$ and $1.5$.
+
+The phase-transition from the confining phase to the Coulomb phase shown in Fig.~\ref{phasediagram}
+is characterized by a rapid increase of $\langle U \rangle$ across the transition but does not give rise to
+significant changes in the other observables (compare the top row of plots in Fig.~\ref{muphases}).
+This behavior persists also at finite $\mu$ and the
+confinement-Coulomb transition can only be seen in the $\langle U \rangle$-plots.
+
+The transition between the Higgs- and the confinig phase is characterized by a strong first order discontinuity in all observables
+(except for $\langle n \rangle = 0$ at $\mu = 0$), a feature that persists for all our values of $\mu$. Also the transition between the Higgs- and the
+Coulomb phase is seen in all observables. It is obvious from the plots, that with increasing $\mu$ all three transitions become more pronounced in
+all variables they are seen in, and the Higgs-Coulomb transition might even change from crossover to first order. Still, the shown results
+have to be considered preliminary and more detailed studies will be necessary to draw final conclusions.
+
+\subsubsection*{Dual occupation numbers}
+
+The dual reformulation of lattice field theories makes it possible to look at the same physics from a different perspective
+by studying the dynamics of the dual degrees of freedom instead of the conventional ones.
+This being a feature we find especially interesting about the dual formulation, we here present an example where a transition
+manifests itself as the condensation of dual variables.
+
+Let us first look at the transition using the standard observables. In Fig.~\ref{occutrans_plaq} we
+plot the plaquette expectation value $\langle U \rangle$ and the corresponding susceptibility $\chi_U$
+as function of the chemical potential, for two different volumes $12^3\times60$ and $16^3\times60$.
+We see that for the larger volume the transition is shifted slightly towards lower chemical potential,
+but the volume dependence seems to be reasonably small. The parameters $\beta$ and $M^2$ are
+fixed to $\beta=0.75$ and $M^2=5.73$. Increasing the chemical potential takes us from the confining-
+to the Higgs phase where we cross the phase boundary
+at some critical value of $\mu$, which is $\mu_c\simeq2.65$
+for the larger of the two lattices. Below the critical value of the chemical potential both
+$\langle U \rangle$ and $\chi_U$ are independent of $\mu$, which is characteristic for a Silver Blaze type of transition \cite{cohen}.
+At $\mu_c$ a strong first order transition signals the entry into the Higgs phase.
+
+In Fig.~\ref{occutrans} we have a look at the same transition, by now showing typical configurations of the dual variables
+just below (top) and above (bottom) the critical chemical potential $\mu_c$.
+In particular we show snapshots of the occupation numbers of all dual link variables $\bar{j}$, $\bar{l}$, $j$,
+$l$ and dual plaquette variables $p$. Here blue links/plaquettes depict positive occupation numbers,
+green links/plaquettes depict negative occupation numbers and links/plaquettes with $0$-occupation
+are not shown. It can be seen that below $\mu_c$ links and plaquettes are hardly occupied,
+while above $\mu_c$ their occupation is abundant. In that sense the Silver Blaze transition of Fig.~\ref{occutrans_plaq}
+can be understood as a condensation phenomenon of the dual variables, which is a new perspective on the underlying
+physics we gained from the dual reformulation of the problem.
+
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