We construct a tight lower bound for the site percolation threshold on an infinite graph, which becomes exact for an infinite tree. The bound is given by the inverse of the maximal eigenvalue of the Hashimoto matrix used to count nonbacktracking walks on the original graph. Our bound always exceeds the inverse spectral radius of the graph’s adjacency matrix, and it is also generally tighter than the existing bound in terms of the maximum degree. We give a constructive proof for existence of such an eigenvalue in the case of a connected infinite quasitransitive graph, a graph-theoretic analog of a translationally invariant system.

We examine both Voyager and Advanced Composition Explorer magnetic field measurements at frequencies that characterize the inertial range and evaluate the anisotropy of the fluctuations as they relate to both the compressive component and underlying wavevector anisotropy of the turbulence. The magnetic fluctuation anisotropy as it relates to the compressive component is directly dependent upon both the plasma beta of the thermal proton component and the ratio of magnetic fluctuation magnitude to the strength of the mean magnetic field. This has been seen before at 1 au. The magnetic fluctuation anisotropy in the plane perpendicular to the mean magnetic field, which is a measure of the anisotropy of the underlying wavevector distribution, should depend on the angle between the mean magnetic field and the radial direction and should be confined to values between one and the index of the power spectrum, which is typically 5/3. Our results show that the average of this anisotropy exceeds the value of the spectral index and is out of bounds with the theory. Although the results are suggestive of past analyses, we find that spherical expansion of the turbulence may offer at least a partial explanation of the apparent amplification of this measured anisotropy.