Statistical description of the black hole degeneracy spectrum

We use mathematical methods based on generating functions to study the statistical properties of the black hole degeneracy spectrum in loop quantum gravity. In particular we will study the persistence of the observed effective quantization of the entropy as a function of the horizon area. We will show that this quantization disappears as the area increases despite the existence of black hole configurations with a large degeneracy. The methods that we describe here can be adapted to the study of the statistical properties of the black hole degeneracy spectrum for all the existing proposals to define black hole entropy in loop quantum gravity.

Figure 1

Plot of the black hole degeneracy spectrum $D(a)$, in units of ${10}^{19}$, for a range of area values (in units of $4\pi \gamma {\ell }_P^2$, where $\gamma$ is the Immirzi parameter and ${\ell }_P$ the Planck length). The band structure can be traced all the way back to the smaller values of the area.

Figure 2

Peak corresponding to $p=150$ in units of ${10}^{14}$. By plotting the values of $D_{*}(a|p)$ one gets, for the largest degeneracy values, a roughly symmetric shape. The values corresponding to the largest degeneracies can be readily seen in the plot. Notice, however, that the peak extends to the left much farther than it does to the right. This phenomenon can be easily seen in a plot of the logarithm of $D_{*}(a|p)$ that also displays a distinct subpeak structure to be discussed later (see Fig. 10).

Figure 3

The upper part of the figure shows the exact value of the entropy $S_{*}(a)$ for areas below 18 (in units of $4\pi \gamma {\ell }_P^2$) in the case in which the projection constraint is ignored. The lower part represents the staircase approximation to the entropy given by $S_{*}^{\mathrm{stairs}}(a)=\log (\sum _{p=0}^{\infty }{\alpha }_{*p}\theta (a-\mu ·p))$.

Figure 4

Comparison between the exact shape of the step $\sum _{a^{\prime }\le a}{D}_{*}(a^{\prime }|120)$ and the corresponding smoothed Gaussian approximation $\frac{{\alpha }_{*120}}{2}(1+\mathrm{erf}(\frac{a-{\mu }_{*120}}{\sqrt{2}{\sigma }_{*120}}))$. The vertical band is centered around the value ${\mu }_{*120}$ and has a width of $2{\sigma }_{*120}$.

Figure 5

The upper part of the figure shows the exact value of the entropy $S_{*}(a)$ for areas below 18 (in units of $4\pi \gamma {\ell }_P^2$) in the case in which the projection constraint is ignored. The lower part represents the Gaussian approximation $S_{*}^{\mathrm{gauss}}(a)=\log (1+\sum _{p=1}^{\infty }\frac{{\alpha }_{*p}}{2}(1+\mathrm{erf}(\frac{a-\mu p}{\sqrt{2p}\sigma })))$.

Figure 6

The upper part of the figure shows the exact value of the entropy $S(a)$ for areas below 18 (in units of $4\pi \gamma {\ell }_P^2$) in the case in which the projection constraint is taken into account. The lower part represents the Gaussian approximation $S^{\mathrm{gauss}}(a)=\log (1+\sum _{p=1}^{\infty }\frac{{\alpha }_p}{2}(1+\mathrm{erf}(\frac{a-\mu p}{\sqrt{2p}\sigma })))$. The staircase structure decays more slowly now as discussed in the text.

Figure 7

Comparison between the exact values of the entropy $S_{*}$ and their Gaussian approximation for two different area intervals. The necessary computations are made possible by the introduction of the master generating functions (2.3). We can see that the Gaussian approximation works remarkably well in both cases. The fading of the oscillations in the entropy for the largest area interval is evident. Similar plots for area intervals around 70 show an essentially linear growth.

Figure 8

Values of $\chi (\alpha ,\beta )$ for $1\le \alpha$, $\beta \le 80$. The abscissa corresponds to $\beta$. Darker colors represent the largest (better) values of $\chi (\alpha ,\beta )$. The best choice is $\alpha =3$, $\beta =2$ for which $\chi (3,2)\approx 35$.

Figure 9

Values of $\pi {\gamma }_{\mathrm{stairs}}(\alpha ,\beta )$ for $1\le \alpha$, $\beta \le 80$. Darker colors represent the largest (better) values of $\pi {\gamma }_{\mathrm{stairs}}(\alpha ,\beta )$. The best value corresponds to $\alpha =3$, $\beta =2$ but other choices give values very close to $\pi {\gamma }_{\mathrm{stairs}}(3,2)$. A similar picture can be drawn for $\pi {\gamma }_{\mathrm{gauss}}(\alpha ,\beta )$ that essentially coincides with this one for $\alpha >\beta$. When $\alpha <\beta$ the behavior changes because $\pi {\gamma }_{\mathrm{gauss}}(\alpha ,\beta )$ becomes larger than $\pi \gamma$ or even complex. This does not change our conclusions about the goodness of $P(3,2)$.

Figure 10

Detail of the peak $P_1=3K+2N=130$. This logarithmic plot clearly shows the existence of several subpeaks labeled by the counter $P_2=5K+4N$. Notice the asymmetric shape (mentioned in Fig. 2).

Figure 11

We represent here the regions relevant to the discussion of the convergence of the functional series (c2). We also plot the position of the relevant singularities of the Laplace transform (there is an infinite number of other isolated singularities also located on the hyperbola which are not relevant for the analysis presented in the paper).

Figure 12

Comparison between the absolutes values of the Laplace transforms $\mathcal{L}(\exp S_{*},\pi \gamma +iy)$ (continuous line) and $\mathcal{L}(\exp S_{*}^{\mathrm{gauss}},\pi \gamma +iy)$ (dashed line).