Abstract:
This is a colloquium-style introduction to two electronic processes in a carbon monolayer (graphene), each having an analogue in relativistic quantum mechanics. Both processes couple electron-like and hole-like states, through the action of either a superconducting pair potential or an electrostatic potential. The first process, Andreev reflection, is the electron-to-hole conversion at the interface with a superconductor. The second process, Klein tunneling, is the tunneling through a p-n junction. Existing and proposed experiments on Josephson junctions and bipolar junctions in graphene are discussed from a unified perspective. CONTENTS: I. INTRODUCTION II. BASIC PHYSICS OF GRAPHENE (Dirac equation; Time reversal symmetry; Boundary conditions; Pseudo-diffusive dynamics) III. ANDREEV REFLECTION (Electron-hole conversion; Retro-reflection vs. specular reflection; Dirac-Bogoliubov-de Gennes equation; Josephson junctions; Further reading) IV. KLEIN TUNNELING (Absence of backscattering; Bipolar junctions; Magnetic field effects; Further reading) V. ANALOGIES (Mapping between NS and p-n junction; Retro-reflection vs. negative refraction; Valley-isospin dependent quantum Hall effect; Pseudo-superconductivity)

Abstract:
This is a comprehensive review of the random-matrix approach to the theory of phase-coherent conduction in mesocopic systems. The theory is applied to a variety of physical phenomena in quantum dots and disordered wires, including universal conductance fluctuations, weak localization, Coulomb blockade, sub-Poissonian shot noise, reflectionless tunneling into a superconductor, and giant conductance oscillations in a Josephson junction.

Abstract:
This is a cursory overview of applications of concepts from random matrix theory (RMT) to quantum electronics and classical & quantum optics. The emphasis is on phenomena, predicted or explained by RMT, that have actually been observed in experiments on quantum wires, quantum dots, disordered wave guides, and chaotic resonators. Topics considered include universal conductance fluctuations, weak localization and coherent backscattering, sub-Poissonian shot noise and open transmission channels, non-Gaussian conductance and thermopower distributions, mesoscopic superconductivity, grey-body radiation, and chaotic laser cavities.

Abstract:
Recent experiments on conduction between a semiconductor and a superconductor have revealed a variety of new mesoscopic phenomena. Here is a review of the present status of this rapidly developing field. A scattering theory is described which leads to a conductance formula analogous to Landauer's formula in normal-state conduction. The theory is used to identify features in the conductance which can serve as "signatures" of phase-coherent Andreev reflection, i.e. for which the phase coherence of the electrons and the Andreev-reflected holes is essential. The applications of the theory include a quantum point contact, quantum dot, weak localization, universal conductance fluctuations, shot noise, and reflectionless tunneling. This review is based on lectures at the Les Houches summer school, Session LXI, 1994.

Abstract:
Carl Gustav Hempel (1905-1997) formulated the dilemma that carries his name in an attempt to determine the boundaries of physics. Where does physics go over into metaphysics? The purpose of this contribution is to indicate how a recently developed field of research, the physics of computation, might offer a new answer to that old question: The boundary between physics and metaphysics is the boundary between what can and what cannot be computed in the age of the universe.

Abstract:
We compute the quantum correction due to weak localization for transport properties of disordered quasi-one-dimensional conductors, by integrating the Dorokhov-Mello-Pereyra-Kumar equation for the distribution of the transmission eigenvalues. The result is independent of sample length or mean free path, and has a universal dependence on the symmetry index of the ensemble of scattering matrices. This result generalizes the theory of weak localization for the conductance to all linear statistics on the transmission eigenvalues. ***Accepted for publication in Physical Review B.****

Abstract:
The smoothed correlation function for the eigenvalues of large hermitian matrices, derived recently by Brezin and Zee [Nucl. Phys. B402 (1993) 613], is generalized to all random-matrix ensembles of Wigner-Dyson type. Submitted to Nuclear Physics B[FS].

Abstract:
The distribution function of the local density of states is computed exactly for the Wigner-Dyson ensemble of random Hamiltonians. In the absence of time-reversal symmetry, precise agreement is obtained with the "supersymmetry" theory by Efetov and Prigodin of the NMR lineshape in disordered metal particles. Upon breaking time-reversal symmetry, the variance of the Knight shift in the smallest particles is reduced by a universal factor of 2/3. ***To be published in Physical Review B.****

Abstract:
A theory is presented for the universal reduction of shot noise by Coulomb repulsion, which was observed in computer simulations of a disordered non-degenerate electron gas by Gonzalez et al. (cond-mat/9803372). The universality of the reduction below the uncorrelated value is explained as a feature of the high-voltage regime of space-charge limited conduction. The reduction factor depends on the dimensionality d of the density of states, being close but not quite equal to 1/d in two and three dimensions.

Abstract:
Recent developments are reviewed in the scaling theory of phase-coherent conduction through a disordered wire. The Dorokhov-Mello-Pereyra-Kumar equation for the distribution of transmission eigenvalues has been solved exactly, in the absence of time-reversal symmetry. Comparison with the previous prediction of random-matrix theory shows that this prediction was highly accurate --- but not exact: The repulsion of the smallest eigenvalues was overestimated by a factor of two. This factor of two resolves several disturbing discrepancies between random-matrix theory and microscopic calculations, notably in the magnitude of the universal conductance fluctuations in the metallic regime, and in the width of the log-normal conductance distribution in the insulating regime. ***To be published as a "Brief Review" in Modern Physics Letters B.****