DOS at $\omega=0$ can be written as $$\begin{aligned}
N_{\mathbf{i}}(\omega=0) \approx - \beta G_{\mathbf{i}}(\mathbf{j}=0,
\tau=\beta/2) / \pi.
\label{eq:a0}\end{aligned}$$ In Fig. \[Fig:local\_DOS\_Nf\_contour\] we present a contour plot for $N^{f}_{\mathbf{i}}(0)$ at $\beta t = 20$. At sites far from the magnetic defect, the local DOS vanishes, as expected in the spin-singlet phase. Near the impurity, there is a large DOS, supporting the picture of broken singlets around the defect, in accordance with the local magnetic susceptibility results, discussed above. As displayed in the contour plot of Fig. \[Fig:local\_DOS\_Nc\_contour\], precisely at the impurity site the conduction electron DOS, $N^{c}_{\mathbf{i}}(0)$, is large, owing to the absence of a partner $f$ electron.

![DOS of sites at distance (a) $|\mathbf{i}| = 1$, (b) $\sqrt{2}$ and (c) $4 \sqrt{2}$ from the depleted site, as well as (d) the regular (undepleted) PAM, both for $\beta t= 20$ and $V/t=1.20$. At short distances, the singlet gap present in the undepleted PAM is partially filled in, so that there is nonvanishing $f$ spectral weight at $\omega=0$.[]{data-label="Fig:local_DOS_Nf"}](DOS_L160_v4.pdf)

To provide an independent check on the validity of Eq., we performed a direct inversion of Eq., using the Maximum Entropy Method[@Jarrell96]. The local DOS is displayed in Fig. \[Fig:local\_DOS\_Nf\] for sites at (a) $|\mathbf{i}|= 