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I want to construct a scale invariant random packing of the plane with circles.

Here is a way to construct a rotationally invariant, but not scale invariant random packing of the plane with circles: Suppose I have a Poisson point process in the plane (with say intensity 1 and time parametrised). After some fixed timestep $t_0$ I temporarily stop the PPP. Then start to grow a circle from each point (with the same speed for all circles) until the circle touches another circle then stop. This is also called the random Poisson lilypond model. Now, we continue the Poisson point process and for every point arriving we grow a circle around the point until it touches one of the other circles. Continuing this we get a random packing of the plane with circles. The packing must be rotationally invariant since it was construct rotationally invariant. However, it is quite clear that it is not scale-invariant.

Is there a way to construct a random circle-packing which is also scale invariant?

Ideas: Maybe one can change this construction slightly. For example one could do it on the hyperbolic plane, in the disc and do inversion, on the sphere and do stereographic projection or something along those lines.

Further motivation is provided by the interfaces in the Ising model which have a scaling limit which is even conformally invariant. But the interfaces have a structure which is very difficult to think about so I was wondering whether one could find some of these properties just with circles. Here is an example of how such a circle packing could look. From arXiv:cond-mat/0203252v2

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I do not think any scale-invariant random packing of circles in the plane exists, regardless of other symmetries.

Suppose there is such a distribution $D$ of random packings. With probability $1$, the origin will lie in the interior of a circle in the packing; let $X$ be the random variable equal to the radius of this circle when we sample a packing $P$ from $D$.

Now, because $D$ is scale-invariant, the distribution of $X$ is equal to the distribution of $cX$ for any positive real number $c$ (since that is the result of scaling $P$ by a factor of $c$).

But it is easy to see that any such $X$ must be $0$ with probability $1$ - if there were a closed bounded interval of the positive reals which $X$ had probability $\epsilon>0$ of lying in, then we could find infinitely many disjoint multiples of it which would also have to have probability $\epsilon$, forcing the PDF to integrate to more than $1$.

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    $\begingroup$ This uses the (reasonable) implicit assumption that the packing is also translation-invariant. I'm not sure whether this was intended by the question - I don't think it's very hard to formulate a rotation-invariant, scale-invariant, but not translation-invariant version. $\endgroup$ – Will Sawin Jan 22 at 18:20
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    $\begingroup$ So it does, thanks for pointing that out. It's not obvious to me how one would get a scale-invariant packing even without the translation or rotation assumptions - do you see a construction, or a sketch of what one would look like? Naively I'd expect to run into similar issues as in the above answer even without translation invariance. $\endgroup$ – RavenclawPrefect Jan 22 at 18:28
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    $\begingroup$ Take the construction in the original question, but make the density of the Poisson process proportional to the inverse square distance from the origin (so it's scale-invariant) and the growth rate proportional to the distance from the origin (again, scale-invariant). $\endgroup$ – Will Sawin Jan 22 at 18:44
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    $\begingroup$ Amazing, so with translation invariance the answer is no and without translation invariance it is yes! Do you have a guess what happens with the Ising model interfaces? Here we do have translation invariance, but of course not a way to define the radius (conformal radius?), but if we can define the area couldn't we make the same argument to prove that the CLE does not exist? en.wikipedia.org/wiki/Conformal_loop_ensemble $\endgroup$ – Frederik Ravn Klausen Jan 22 at 19:04
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    $\begingroup$ Also in the scale invariant construction it is only scale invariant when scaling while keeping one point fixed. Maybe it is possible to prove that one cannot do general scaling. $\endgroup$ – Frederik Ravn Klausen Jan 23 at 15:07

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