Delocalization in random polymer models

A random polymer model is a one-dimensional Jacobi matrix randomly composed of two finite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer model. It is proven that the Lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there. Large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport, showing that it is almost surely overdiffusive even though the models are known to have pure-point spectrum with exponentially localized eigenstates for almost every configuration of the polymers. Furthermore, the level spacing is shown to be regular at the critical energy

Absence of continuous spectral types for certain nonstationary random models

We consider continuum random Schr\"odinger operators of the type $H_{\omega} = -\Delta + V_0 + V_{\omega}$ with a deterministic background potential $V_0$. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of $-\Delta +V_0$. The models we treat include random surface potentials as well as sparse or slowly decaying random potentials. In particular, we establish absence of absolutely continuous surface spectrum for random potentials supported near a one-dimensional surface (``random tube'') in arbitrary dimension.Comment: 14 pages, 2 figure

A family of Schr\"odinger operators whose spectrum is an interval

By approximation, I show that the spectrum of the Schr\"odinger operator with potential $V(n) = f(n\rho \pmod 1)$ for f continuous and $\rho > 0$, $\rho \notin \N$ is an interval.Comment: Comm. Math. Phys. (to appear

Engineering liver

Interest in “engineering liver” arises from multiple communities: therapeutic replacement; mechanistic models of human processes; and drug safety and efficacy studies. An explosion of micro- and nanofabrication, biomaterials, microfluidic, and other technologies potentially affords unprecedented opportunity to create microphysiological models of the human liver, but engineering design principles for how to deploy these tools effectively toward specific applications, including how to define the essential constraints of any given application (available sources of cells, acceptable cost, and user-friendliness), are still emerging. Arguably less appreciated is the parallel growth in computational systems biology approaches toward these same problems—particularly in parsing complex disease processes from clinical material, building models of response networks, and in how to interpret the growing compendium of data on drug efficacy and toxicology in patient populations. Here, we provide insight into how the complementary paths of engineering liver—experimental and computational—are beginning to interplay toward greater illumination of human disease states and technologies for drug development.National Institutes of Health (U.S.) (UH2TR000496)National Institutes of Health (U.S.) (R01-EB 010246)National Institutes of Health (U.S.) (R01-ES015241)National Institutes of Health (U.S.) (P30-ES002109