According to wikipedia:
"In physics, self-organized criticality (SOC) is a property of (classes of) dynamical systems which have a critical point as an attractor. Their macroscopic behaviour thus displays the spatial and/or temporal scale-invariance characteristic of the critical point of a phase transition, but without the need to tune control parameters to precise values."
The asymptotic safe point pretty much fits this description, apparently.
In relation to Quantum Gravity, the only thing I could find was this:
Self-organized criticality in quantum gravity
Mohammad H. Ansari, Lee Smolin
(Submitted on 27 Dec 2004 (v1), last revised 18 May 2005 (this version, v5))
We study a simple model of spin network evolution motivated by the hypothesis that the emergence of classical space-time from a discrete microscopic dynamics may be a self-organized critical process. Self organized critical systems are statistical systems that naturally evolve without fine tuning to critical states in which correlation functions are scale invariant. We study several rules for evolution of frozen spin networks in which the spins labelling the edges evolve on a fixed graph. We find evidence for a set of rules which behaves analogously to sand pile models in which a critical state emerges without fine tuning, in which some correlation functions become scale invariant.
Perhaps this is a clue that AS is really related to spin networks?