Structure of quantum disordered wave functions: weak localization, far tails, and mesoscopic transport

We report on the comprehensive numerical study of the fluctuation and correlation properties of wave functions in three-dimensional mesoscopic diffusive conductors. Several large sets of nanoscale samples with finite metallic conductance, modeled by an Anderson model with different strengths of diagonal box disorder, have been generated in order to investigate both small and large deviations (as well as the connection between them) of the distribution function of eigenstate amplitudes from the universal prediction of random matrix theory. We find that small, weak localization-type, deviations contain both diffusive contributions (determined by the bulk and boundary conditions dependent terms) and ballistic ones which are generated by electron dynamics below the length scale set by the mean free path ell. By relating the extracted parameters of the functional form of nonperturbative deviations (``far tails'') to the exactly calculated transport properties of mesoscopic conductors, we compare our findings based on the full solution of the Schrodinger equation to different approximative analytical treatments. We find that statistics in the far tail can be explained by the exp-log-cube asymptotics (convincingly refuting the log-normal alternative), but with parameters whose dependence on ell is linear and, therefore, expected to be dominated by ballistic effects. It is demonstrated that both small deviations and far tails depend explicitly on the sample size--the remaining puzzle then is the evolution of the far tail parameters with the size of the conductor since short-scale physics is supposedly insensitive to the sample boundaries.Comment: 13 pages, 9 embedded EPS figures, expanded discussion (with extra one figure) on small size effec

Hamiltonian Pseudo-rotations of Projective Spaces

The main theme of the paper is the dynamics of Hamiltonian diffeomorphisms of ${\mathbb C}{\mathbb P}^n$ with the minimal possible number of periodic points (equal to $n+1$ by Arnold's conjecture), called here Hamiltonian pseudo-rotations. We prove several results on the dynamics of pseudo-rotations going beyond periodic orbits, using Floer theoretical methods. One of these results is the existence of invariant sets in arbitrarily small punctured neighborhoods of the fixed points, partially extending a theorem of Le Calvez and Yoccoz and Franks to higher dimensions. The other is a strong variant of the Lagrangian Poincar\'e recurrence conjecture for pseudo-rotations. We also prove the $C^0$-rigidity of pseudo-rotations with exponentially Liouville mean index vector. This is a higher-dimensional counterpart of a theorem of Bramham establishing such rigidity for pseudo-rotations of the disk.Comment: 38 pages; final version (with minor revisions and updated references); published Online First in Inventiones mathematica

Fragility and Persistence of Leafwise Intersections

In this paper we study the question of fragility and robustness of leafwise intersections of coisotropic submanifolds. Namely, we construct a closed hypersurface and a sequence of Hamiltonians $C^0$-converging to zero such that the hypersurface and its images have no leafwise intersections, showing that some form of the contact type condition on the hypersurface is necessary in several persistence results. In connection with recent results in continuous symplectic topology, we also show that $C^0$-convergence of hypersurfaces, Hamiltonian diffeomorphic to each other, does not in general force $C^0$-convergence of the characteristic foliations.Comment: 17 pages, 3 figures; we removed one of our results (a refinement of Moser's theorem on leafwise intersections) and its proof, since a stronger theorem is proved in arXiv:1408.457