Generalized Lyapunov exponent for the one-dimensional Schrödinger equation with Cauchy disorder: Some exact results.

We consider the one-dimensional Schrödinger equation with a random potential and study the cumulant generating function of the logarithm of the wave function ψ(x), known in the literature as the "generalized Lyapunov exponent"; this is tantamount to studying the statistics of the so-called "finite-size Lyapunov exponent." The problem reduces to that of finding the leading eigenvalue of a certain nonrandom non-self-adjoint linear operator defined on a somewhat unusual space of functions. We focus on the case of Cauchy disorder, for which we derive a secular equation for the generalized Lyapunov exponent. Analytical expressions for the first four cumulants of ln|ψ(x)| for arbitrary energy and disorder are deduced. In the universal (weak-disorder and high-energy) regime, we obtain simple asymptotic expressions for the generalized Lyapunov exponent and for all the cumulants. The large deviation function controlling the distribution of ln|ψ(x)| is also obtained in several limits. As an application, we show that, for a disordered region of size L, the distribution W_{L} of the conductance g exhibits the power-law behavior W_{L}(g)∼g^{-1/2} as g→0.

Dynamics of a tagged monomer: effects of elastic pinning and harmonic absorption.

We study the dynamics of a tagged monomer of a Rouse polymer for different initial configurations. In the case of free evolution, the monomer displays subdiffusive behavior with strong memory of the initial state. In the presence of either elastic pinning or harmonic absorption, we show that the steady state is independent of the initial condition that, however, strongly affects the transient regime, resulting in nonmonotonic behavior and power-law relaxation with varying exponents.

Wigner time-delay distribution in chaotic cavities and freezing transition.

Using the joint distribution for proper time delays of a chaotic cavity derived by Brouwer, Frahm, and Beenakker [Phys. Rev. Lett. 78, 4737 (1997)], we obtain, in the limit of the large number of channels N, the large deviation function for the distribution of the Wigner time delay (the sum of proper times) by a Coulomb gas method. We show that the existence of a power law tail originates from narrow resonance contributions, related to a (second order) freezing transition in the Coulomb gas.

Dimensional crossover in quantum networks: from macroscopic to mesoscopic physics.

We report on magnetoconductance measurements of metallic networks of various sizes ranging from 10 to 10(6) plaquettes, with an anisotropic aspect ratio. Both Altshuler-Aronov-Spivak h/2e periodic oscillations and Aharonov-Bohm h/e periodic oscillations are observed for all networks. For large samples, the amplitude of both oscillations results from the incoherent superposition of contributions of phase coherent regions. When the transverse size becomes smaller than the phase coherent length Lphi, one enters a new regime which is phase coherent (mesoscopic) along one direction and macroscopic along the other, leading to a new size dependence of the quantum oscillations.

Weak localization in multiterminal networks of diffusive wires.

We study the quantum transport through networks of diffusive wires connected to reservoirs in the Landauer-Büttiker formalism. The elements of the conductance matrix are computed by the diagrammatic method. We recover the combination of classical resistances and obtain the weak localization corrections. For arbitrary networks, we show how the Cooperon must be properly weighted over the different wires. Its nonlocality is clearly analyzed. We predict a new geometrical effect that may change the sign of the weak localization correction in multiterminal geometries.