We prove localization and probabilistic bounds on the minimum level spacing for a random block Anderson model without monotonicity. Using a sequence of narrowing energy windows and associated Schur complements, we obtain detailed probabilistic inform
ation about the microscopic structure of energy levels of the Hamiltonian, as well as the support and decay of eigenfunctions.
A new KAM-style proof of Anderson localization is obtained. A sequence of local rotations is defined, such that off-diagonal matrix elements of the Hamiltonian are driven rapidly to zero. This leads to the first proof via multi-scale analysis of expo
nential decay of the eigenfunction correlator (this implies strong dynamical localization). The method has been used in recent work on many-body localization [arXiv:1403.7837].
For a one-dimensional spin chain with random local interactions, we prove that many-body localization follows from a physically reasonable assumption that limits the amount of level attraction in the system. The construction uses a sequence of local
unitary transformations to diagonalize the Hamiltonian and connect the exact many-body eigenfunctions to the original basis vectors.
We present a phase-space model that simulates Pleistocene ice volume changes based on Earths orbital parameters. Terminations in the model are triggered by a combination of ice volume and orbital forcing and agree well with age estimates for Late Ple
istocene terminations. The average phase at which model terminations begin is approximately 90 +/- 90 degrees before the maxima in all three orbital cycles. The large variability in phase is likely caused by interactions between the three cycles and ice volume. Unlike previous ice volume models, this model produces an orbitally driven increase in 100-kyr power during the mid-Pleistocene transition without any change in model parameters. This supports the hypothesis that Pleistocene variations in the 100-kyr power of glacial cycles could be caused, at least in part, by changes in Earths orbital parameters, such as amplitude modulation of the 100-kyr eccentricity cycle, rather than changes within the climate system.
We give a survey and unified treatment of functional integral representations for both simple random walk and some self-avoiding walk models, including models with strict self-avoidance, with weak self-avoidance, and a model of walks and loops. Our r
epresentation for the strictly self-avoiding walk is new. The representations have recently been used as the point of departure for rigorous renormalization group analyses of self-avoiding walk models in dimension 4. For the models without loops, the integral representations involve fermions, and we also provide an introduction to fermionic integrals. The fermionic integrals are in terms of anti-commuting Grassmann variables, which can be conveniently interpreted as differential forms.
