Almost Sure Frequency Independence of the Dimension of the Spectrum of Sturmian Hamiltonians

We consider the spectrum of discrete Schr\"odinger operators with Sturmian potentials and show that for sufficiently large coupling, its Hausdorff dimension and its upper box counting dimension are the same for Lebesgue almost every value of the frequency.Comment: 12 pages, to appear in Commun. Math. Phy

Independence, Relative Randomness, and PA Degrees

We study pairs of reals that are mutually Martin-L\"{o}f random with respect to a common, not necessarily computable probability measure. We show that a generalized version of van Lambalgen's Theorem holds for non-computable probability measures, too. We study, for a given real $A$, the \emph{independence spectrum} of $A$, the set of all $B$ so that there exists a probability measure $\mu$ so that $\mu\{A,B\} = 0$ and $(A,B)$ is $\mu\times\mu$-random. We prove that if $A$ is r.e., then no $\Delta^0_2$ set is in the independence spectrum of $A$. We obtain applications of this fact to PA degrees. In particular, we show that if $A$ is r.e.\ and $P$ is of PA degree so that $P \not\geq_{T} A$, then $A \oplus P \geq_{T} 0'$

Independence ratio and random eigenvectors in transitive graphs

A theorem of Hoffman gives an upper bound on the independence ratio of regular graphs in terms of the minimum $\lambda_{\min}$ of the spectrum of the adjacency matrix. To complement this result we use random eigenvectors to gain lower bounds in the vertex-transitive case. For example, we prove that the independence ratio of a $3$-regular transitive graph is at least $$q=\frac{1}{2}-\frac{3}{4\pi}\arccos\biggl(\frac{1-\lambda _{\min}}{4}\biggr).$$ The same bound holds for infinite transitive graphs: we construct factor of i.i.d. independent sets for which the probability that any given vertex is in the set is at least $q-o(1)$. We also show that the set of the distributions of factor of i.i.d. processes is not closed w.r.t. the weak topology provided that the spectrum of the graph is uncountable.Comment: Published at http://dx.doi.org/10.1214/14-AOP952 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Lorentz invariant mass and length scales

We show that the standard Lorentz transformations admit an invariant mass (length) scale, such as the Planck scale. In other words, the frame independence of such scale is built-in within those transformations, and one does not need to invoke the principle of relativity for their invariance. This automatically ensures the frame-independence of the spectrum of geometrical operators in quantum gravity. Furthermore, we show that the above predicts a small but measurable difference between the inertial and gravitational mass of any object, regardless of its size or whether it is elementary or composite.Comment: 10 page

A class of rotationally symmetric quantum layers of dimension 4

Under several geometric conditions imposed below, the existence of the discrete spectrum below the essential spectrum is shown for the Dirichlet Laplacian on the quantum layer built over a spherically symmetric hypersurface with a pole embedded in the Euclidean space R4. At the end of this paper, we also show the advantage and independence of our main result comparing with those existent results for higher dimensional quantum layers or quantum tubes.Comment: 12 pages. A slight different version of this paper has appeared in J. Math. Anal. App

Lattice baryons in the 1/N expansion

Results are presented for hadron spectroscopy with gauge groups SU(N) with N=3, 5, 7. Calculations use the quenched approximation. Lattice spacings are matched using the static potential. Meson spectra show independence on N and vacuum-to-hadron matrix elements scale as the square root of N. The baryon spectrum shows the excitation levels of a rigid rotor.Comment: 19 pages, 11 figure