This paper shows that on regular trees, the random appearance of infinite connected clusters happens at a lower probability than the appearance of extended quantum states. In other words, if you randomly keep each edge with probability p, then there is a range of p values just above the percolation threshold where infinite clusters exist but those clusters almost surely do not support the kind of extended states that correspond to absolutely continuous spectrum.

The authors study Bernoulli bond percolation on an infinite regular tree of degree at least three. In this model each edge is kept independently with probability p. The percolation threshold p_c is the smallest p at which an infinite cluster appears; for the (k+1)-regular tree the threshold is p_c = 1/k. For each such tree they prove there is an explicit small amount epsilon_k so that for p in the interval (p_c, p_c + epsilon_k) the adjacency operator of every open cluster almost surely has no absolutely continuous spectral component. The adjacency operator is the matrix that describes hopping between neighboring vertices and is the standard object used to study quantum or wave propagation on a graph.

At a high level the proof uses analytic properties of the resolvent, a complex-valued function that encodes the spectrum of an operator. On a tree the resolvent obeys a recursion that leads, after conditioning on cluster survival, to a fixed-point law for the forward resolvent. The authors compare the effect of two simple finite side-branch configurations. These produce distinct fractional-linear (Möbius) maps whose combined action gives a contraction. Near the percolation threshold the probability of seeing many infinite child branches is small, and after a smoothing argument on the hyperbolic plane this yields a contradiction unless the imaginary part of the resolvent boundary values vanishes almost everywhere. That vanishing rules out any absolutely continuous component of the spectrum.