It now looks like the Immirzi parameter has a critical value which is determined numbertheoretically, not by the Bekenstein-Hawking S = A/4.

It used to be that with LQG the IP appeared multiplicatively in S = A/4 so that the value of the IP had to be ADJUSTED to get the S = A/4 equation right.

But this appears to be no longer the case. The B-H equation (S = A/4 + other terms ) has been derived in LQG without the Immirzi getting in multiplicatively in the leading term. So it is not forced or restricted.

Interestingly however it gets a critical value from other considerations which happens to be around 0.274. This comes out in two recent papers, by Ghosh Perez and by Mitra. I mentioned this in another thread:

The Immirzi comes into the picture when you add a purely quantum correction term involving an index of refinement N. N is the number of spinnetwork links that pass out thru the BH horizon. So if you REFINE the spinnetwork by adding nodes and links to it then intuitively you can be increasing N and adding a kind of quantum hair to the quantum BH state. The Immirzi determines the chemical potential the energy decrease associated with adding one more puncture. (Intuitively, dividing the area and curvature of the horizon up finer and finer.)

I will go get the links to the recent papers by Ghosh Perez and by Mitra.

http://arxiv.org/abs/1107.1320 Black hole entropy and isolated horizons thermodynamics
Amit Ghosh, Alejandro Perez
(Submitted on 7 Jul 2011)
We present a statistical mechanical calculation of the thermodynamical properties of (non rotating) isolated horizons. The introduction of Planck scale allows for the definition of an universal horizon temperature (independent of the mass of the black hole) and a well-defined notion of energy (as measured by suitable local observers) proportional to the horizon area in Planck units. The microcanonical and canonical ensembles associated with the system are introduced. Black hole entropy and other thermodynamical quantities can be consistently computed in both ensembles and results are in agreement with Hawking's semiclassical analysis for all values of the Immirzi parameter.
5 pages

http://arxiv.org/abs/1107.4605 Area law for black hole entropy in the SU(2) quantum geometry approach
P. Mitra
(Submitted on 22 Jul 2011)
SU(2) loop quantum gravity can be considered with the level k of the Chern-Simons theory held fixed. Then the black hole entropy is proportional to the area without any logarithmic correction. However, if k is made large, there is the -(3/2)log A correction.
5 pages

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MTD2 spotted the paper by Parthasarathi Mitra and logged it onto our bibliography a couple of days ago. It seems like an important paper to me, in a similar spirit and of comparable importance to the one by Amit Ghosh and Alejandro Perez.
Note that "quantum geometry" as used in the the Mitra paper is just a synonym for Loop Quantum Gravity. It was Ashtekar's preferred name for the Loop approach. Also notice that Mitra's result is not the same. There is a tension. But they have an important similarity in that the Immirzi does not enter multiplicatively in the leading term and so does not need to be adjusted in order to make S = A/4. That is how I read it anyway. See what you think.

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EDIT: brief mention of today's posting by Christian Corda at the Prato ITP http://arxiv.org/pdf/1107.5334
Part of the re-understanding of BH entropy that is happening in various schools of thought not just Loop.
He comes at it from a different direction but happens to find Loop corroboration, citing Dreyer 2003 at the top of page 7 and Ghosh Mitra 2005 at the bottom of page 8 and also another LQG BH paper I didn't know of, his reference [19] J. Zhang 2008.
He has the entropy depend on an "overtone number" N, in subleading term. They also have a refinement number N that they let run to infinity, as he does, affecting the subleading terms: "quantum hair" as Alejandro Perez says. Not the same as Corda's N but curiously analogous.

You mentioned Bianchi Perini Magliaro. I'll take a look. Could you tell me where they give a value of the IP? My eyes get tired scanning. And do they let the IP run from that critical value down to zero? Could be interesting. I vaguely remember reading that BPM paper but have forgotten detail.

As its reference [19], Corda's paper cites a 2008 one by Jingyi Zhang published in Physical Letters B. It was new to me so I post the abstract here for a closer look. I see it is another one where the author comes in from a different direction and happens to find a quantum correction to the entropy which agrees with what LQG found. Nice to get some confirmation from another approach:

http://arXiv.org/abs/0806.2441 Black hole quantum tunnelling and black hole entropy correction
Jingyi Zhang
(Submitted on 15 Jun 2008)
Parikh-Wilczek tunnelling framework, which treats Hawking radiation as a tunnelling process, is investigated again. As the first order correction, the log-corrected entropy-area relation naturally emerges in the tunnelling picture if we consider the emission of a spherical shell. The second order correction of the emission rate for the Schwarzschild black hole is calculated too. In this level, the result is still in agreement with the unitary theory, however, the entropy of the black hole will contain three parts: the usual Bekenstein-Hawking entropy, the logarithmic term and the inverse area term. In our results the coefficient of the logarithmic term is -1. Apart from a coefficient, our correction to the black hole entropy is consistent with that of loop quantum gravity.
9 pages; Physics Letters B Volume 668, Issue 5, 23 October 2008, Pages 353-356