80 research outputs found
Quantum Diffusion and Delocalization for Band Matrices with General Distribution
We consider Hermitian and symmetric random band matrices in
dimensions. The matrix elements , indexed by , are independent and their variances satisfy \sigma_{xy}^2:=\E
\abs{H_{xy}}^2 = W^{-d} f((x - y)/W) for some probability density . We
assume that the law of each matrix element is symmetric and exhibits
subexponential decay. We prove that the time evolution of a quantum particle
subject to the Hamiltonian is diffusive on time scales . We
also show that the localization length of the eigenvectors of is larger
than a factor times the band width . All results are uniform in
the size \abs{\Lambda} of the matrix. This extends our recent result
\cite{erdosknowles} to general band matrices. As another consequence of our
proof we show that, for a larger class of random matrices satisfying
for all , the largest eigenvalue of is bounded
with high probability by for any ,
where M \deq 1 / (\max_{x,y} \sigma_{xy}^2).Comment: Corrected typos and some inaccuracies in appendix
The Three Loop Isotopy and Framed Isotopy Invariants of Virtual Knots
This paper introduces two virtual knot theory ``analogues'' of a well-known
family of invariants for knots in thickened surfaces: the Grishanov-Vassiliev
finite-type invariants of order two. The first, called the three loop isotopy
invariant, is an invariant of virtual knots while the second, called the three
loop framed isotopy invariant, is a regular isotopy invariant of framed virtual
knots. The properties of these invariants are investigated at length. In
addition, we make precise the informal notion of ``analogue''. Using this
formal definition, it is proved that a generalized three loop invariant is a
virtual knot theory analogue of a generalization of the Grishanov-Vassiliev
invariants of order two
Zariski density of crystalline representations for any p-adic field
The aim of this article is to prove Zariski density of crystalline
representations in the rigid analytic space associated to the universal
deformation ring of a d-dimensional mod p representation of Gal(\bar{K}/K) for
any d and for any p-adic field K. This is a generalization of the results of
Colmez, Kisin (d=2, K=Q_p), of the author (d=2, any K), of Chenevier (any d,
K=Q_p). A key ingredient for the proof is to construct a p-adic family of
trianguline representations. In this article, we construct (an approximation
of) this family by generalizing Kisin's theory of finite slope subspace X_{fs}
for any d and for any K
Variational energy principle for compressible, baroclinic flow. 1: First and second variations of total kinetic action
The case of a cold gas in the absence of external force fields is considered. Since the only energy involved is kinetic energy, the total kinetic action (i.e., the space-time integral of the kinetic energy density) should serve as the total free-energy functional in this case, and as such should be a local minimum for all possible fluctuations about stable flow. This conjecture is tested by calculating explicit, manifestly covariant expressions for the first and second variations of the total kinetic action in the context of Lagrangian kinematics. The general question of the correlation between physical stability and the convexity of any action integral that can be interpreted as the total free-energy functional of the flow is discussed and illustrated for the cases of rectillinear and rotating shearing flows
Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (I)
Trace formulae for d-regular graphs are derived and used to express the
spectral density in terms of the periodic walks on the graphs under
consideration. The trace formulae depend on a parameter w which can be tuned
continuously to assign different weights to different periodic orbit
contributions. At the special value w=1, the only periodic orbits which
contribute are the non back- scattering orbits, and the smooth part in the
trace formula coincides with the Kesten-McKay expression. As w deviates from
unity, non vanishing weights are assigned to the periodic walks with
back-scatter, and the smooth part is modified in a consistent way. The trace
formulae presented here are the tools to be used in the second paper in this
sequence, for showing the connection between the spectral properties of
d-regular graphs and the theory of random matrices.Comment: 22 pages, 3 figure
- …