Density of a saturated random packing of congruent circles The problem of the expected density of a saturated random packing of unit circles in the plane can be described as follows.
In a circular region $C$ of a large radius pick a point at random and draw a unit circle centered at the chosen point. Then pick another point in $C$ at random and draw again a unit circle, centered at the chosen point. If this circle overlaps with the previously drawn one, discard it and pick another point in $C$. Continue this process until the probability of finding one more point in $C$ for which the unit circle drawn about it does not overlap with any of the previously drawn circles is zero. The density of such packing of unit circles in $C$ is computed as the sum of the areas of the unit circles divided by the area of $C$. Let $d(C)$ denote the expected value of the density, and let $d$ denote the limit of $d(C)$ as the radius of $C$ tends to infinity.
Numerical experiments indicate that $d$ should be around $0.82$ (see for example https://www.sciencedirect.com/science/article/pii/0021979771903389), but, to the best of my knowledge, the exact value of $d$ is unknown. Are there any rigorous results obtained in this direction? Any reasonably good estimates, perhaps? Any references will be appreciated.
 A: Assuming that you're interested in random sequential addition / adsorption / deposition / packing (the random process described in the text of your answer) and not the process described in the paper you linked, here are a few relevant mathematical papers:
This 2001 paper of Penrose and 2002 paper of Penrose and Yukich show that the limiting density $d$ actually exists (not just in 2 dimensions); however I could not immediately see if their theorems lead to any explicit bounds on $d$. 
This 1997 paper of Caser and Hilhorst give series expansions which lead to rigorous lower bounds for $d$; it seems the best they can do in the case you're interested in was $d>0.328$ (see Eq. 3.21).
(My answer that I linked in the comments above gives links to reviews in the physics literature which contain some discussion of estimates of $d$; I'll just reproduce the estimate $d\approx0.5472\pm0.002$ according to simulations described here).
A: Following user @j.c.'s lead, here is another paper on RSA (Random Sequential Adsorption), which concludes with a density of $0.77$. From the abstract:




Hinrichsen, Einar L., Jens Feder, and Torstein Jøssang. "Random packing of disks in two dimensions." Physical Review A 41.8 (1990): 4199.
  (Journal link.)


          


          

Fig.2: The first iteration step.


