Localization criteria for Anderson models on locally finite graphs

We prove spectral and dynamical localization for Anderson models on locally finite graphs using the fractional moment method. Our theorems extend earlier results on localization for the Anderson model on \ZZ^d. We establish geometric assumptions for the underlying graph such that localization can be proven in the case of sufficiently large disorder

Competition between fluctuations and disorder in frustrated magnets

We investigate the effects of impurities on the nature of the phase transition in frustrated magnets, in d=4-epsilon dimensions. For sufficiently small values of the number of spin components, we find no physically relevant stable fixed point in the deep perturbative region (epsilon << 1), contrarily to what is to be expected on very general grounds. This signals the onset of important physical effects.Comment: 4 pages, 3 figures, published versio

The role of intracellular zinc release in aging, oxidative stress, and Alzheimer's disease

Brain aging is marked by structural, chemical, and genetic changes leading to cognitive decline and impaired neural functioning. Further, aging itself is also a risk factor for a number of neurodegenerative disorders, most notably Alzheimer's disease (AD). Many of the pathological changes associated with aging and aging-related disorders have been attributed in part to increased and unregulated production of reactive oxygen species (ROS) in the brain. ROS are produced as a physiological byproduct of various cellular processes, and are normally detoxified by enzymes and antioxidants to help maintain neuronal homeostasis. However, cellular injury can cause excessive ROS production, triggering a state of oxidative stress that can lead to neuronal cell death. ROS and intracellular zinc are intimately related, as ROS production can lead to oxidation of proteins that normally bind the metal, thereby causing the liberation of zinc in cytoplasmic compartments. Similarly, not only can zinc impair mitochondrial function, leading to excess ROS production, but it can also activate a variety of extra-mitochondrial ROS-generating signaling cascades. As such, numerous accounts of oxidative neuronal injury by ROS-producing sources appear to also require zinc. We suggest that zinc deregulation is a common, perhaps ubiquitous component of injurious oxidative processes in neurons. This review summarizes current findings on zinc dyshomeostasis-driven signaling cascades in oxidative stress and age-related neurodegeneration, with a focus on AD, in order to highlight the critical role of the intracellular liberation of the metal during oxidative neuronal injury. © 2014 McCord and Aizenman

Regulation of pro-apoptotic phosphorylation of Kv2.1 K⁺ channels

Caspase activity during apoptosis is inhibited by physiological concentrations of intracellular K+. To enable apoptosis in injured cortical and hippocampal neurons, cellular loss of this cation is facilitated by the insertion of Kv2.1 K+ channels into the plasma membrane via a Zn2+ /CaMKII/SNARE-dependent process. Pro-apoptotic membrane insertion of Kv2.1 requires the dual phosphorylation of the channel by Src and p38 at cytoplasmic N- and C- terminal residues Y124 and S800, respectively. In this study, we investigate if these phosphorylation sites are mutually co-regulated, and whether putative N- and C-terminal interactions, possibly enabled by Kv2.1 intracellular cysteine residues C73 and C710, influence the phosphorylation process itself. Studies were performed with recombinant wild type and mutant Kv2.1 expressed in Chinese hamster ovary (CHO) cells. Using immunoprecipitated Kv2.1 protein and phospho-specific antibodies, we found that an intact Y124 is required for p38 phosphorylation of S800, and, importantly, that Src phosphorylation of Y124 facilitates the action of the p38 at the S800 residue. Moreover, the actions of Src on Kv2.1 are substantially decreased in the non-phosphorylatable S800A channel mutant. We also observed that mutations of either C73 or C710 residues decreased the p38 phosphorylation at S800 without influencing the actions of Src on tyrosine phosphorylation of Kv2.1. Surprisingly, however, apoptotic K+ currents were suppressed only in cells expressing the Kv2.1(C73A) mutant but not in those transfected with Kv2.1(C710A), suggesting a possible structural alteration in the C-terminal mutant that facilitates membrane insertion. These results show that intracellular N-terminal domains critically regulate phosphorylation of the C-terminal of Kv2.1, and vice versa, suggesting possible new avenues for modifying the apoptotic insertion of these channels during neurodegenerative processes

Spin Glass Computations and Ruelle's Probability Cascades

We study the Parisi functional, appearing in the Parisi formula for the pressure of the SK model, as a functional on Ruelle's Probability Cascades (RPC). Computation techniques for the RPC formulation of the functional are developed. They are used to derive continuity and monotonicity properties of the functional retrieving a theorem of Guerra. We also detail the connection between the Aizenman-Sims-Starr variational principle and the Parisi formula. As a final application of the techniques, we rederive the Almeida-Thouless line in the spirit of Toninelli but relying on the RPC structure.Comment: 20 page

The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion

For independent nearest-neighbour bond percolation on Z^d with d >> 6, we prove that the incipient infinite cluster's two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the two-point function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of n^{-3/2}, plus an error term of order n^{-3/2-\epsilon} with \epsilon >0. This is a strong statement that the critical exponent delta is given by delta =2.Comment: 56 pages, 3 Postscript figures, in AMS-LaTeX, with graphicx, epic, and xr package

No quasi-long-range order in strongly disordered vortex glasses: a rigorous proof

The paper contains a rigorous proof of the absence of quasi-long-range order in the random-field O(N) model for strong disorder in the space of an arbitrary dimensionality. This result implies that quasi-long-range order inherent to the Bragg glass phase of the vortex system in disordered superconductors is absent as the disorder or external magnetic field is strong.Comment: 3 pages, Revte

A non-perturbative method of calculation of Green functions

A new method for non-perturbative calculation of Green functions in quantum mechanics and quantum field theory is proposed. The method is based on an approximation of Schwinger-Dyson equation for the generating functional by exactly soluble equation in functional derivatives. Equations of the leading approximation and the first step are solved for $\phi^4_d$-model. At $d=1$ (anharmonic oscillator) the ground state energy is calculated. The renormalization program is performed for the field theory at $d=2,3$. At $d=4$ the renormalization of the coupling involves a trivialization of the theory.Comment: 13 pages, Plain LaTex, no figures, some discussion of results for anharmonic oscillator and a number of references are added, final version published in Journal of Physics

On Bernoulli Decompositions for Random Variables, Concentration Bounds, and Spectral Localization

As was noted already by A. N. Kolmogorov, any random variable has a Bernoulli component. This observation provides a tool for the extension of results which are known for Bernoulli random variables to arbitrary distributions. Two applications are provided here: i. an anti-concentration bound for a class of functions of independent random variables, where probabilistic bounds are extracted from combinatorial results, and ii. a proof, based on the Bernoulli case, of spectral localization for random Schroedinger operators with arbitrary probability distributions for the single site coupling constants. For a general random variable, the Bernoulli component may be defined so that its conditional variance is uniformly positive. The natural maximization problem is an optimal transport question which is also addressed here