27 research outputs found
Random Walks Along the Streets and Canals in Compact Cities: Spectral analysis, Dynamical Modularity, Information, and Statistical Mechanics
Different models of random walks on the dual graphs of compact urban
structures are considered. Analysis of access times between streets helps to
detect the city modularity. The statistical mechanics approach to the ensembles
of lazy random walkers is developed. The complexity of city modularity can be
measured by an information-like parameter which plays the role of an individual
fingerprint of {\it Genius loci}.
Global structural properties of a city can be characterized by the
thermodynamical parameters calculated in the random walks problem.Comment: 44 pages, 22 figures, 2 table
Complex zeros of real ergodic eigenfunctions
We determine the limit distribution (as ) of complex
zeros for holomorphic continuations \phi_{\lambda}^{\C} to Grauert tubes of
real eigenfunctions of the Laplacian on a real analytic compact Riemannian
manifold with ergodic geodesic flow. If is an
ergodic sequence of eigenfunctions, we prove the weak limit formula
\frac{1}{\lambda_j} [Z_{\phi_{j_k}^{\C}}] \to \frac{i}{\pi} \bar{\partial}
{\partial} |\xi|_g, where [Z_{\phi_{j_k}^{\C}}] is the current of
integration over the complex zeros and where is with respect
to the adapted complex structure of Lempert-Sz\"oke and Guillemin-Stenzel.Comment: Added some examples and references. Also added a new Corollary, and
corrected some typo
Singularities, Lax degeneracies and Maslov indices of the periodic Toda chain
The n-particle periodic Toda chain is a well known example of an integrable
but nonseparable Hamiltonian system in R^{2n}. We show that Sigma_k, the k-fold
singularities of the Toda chain, ie points where there exist k independent
linear relations amongst the gradients of the integrals of motion, coincide
with points where there are k (doubly) degenerate eigenvalues of
representatives L and Lbar of the two inequivalent classes of Lax matrices
(corresponding to degenerate periodic or antiperiodic solutions of the
associated second-order difference equation). The singularities are shown to be
nondegenerate, so that Sigma_k is a codimension-2k symplectic submanifold.
Sigma_k is shown to be of elliptic type, and the frequencies of transverse
oscillations under Hamiltonians which fix Sigma_k are computed in terms of
spectral data of the Lax matrices. If mu(C) is the (even) Maslov index of a
closed curve C in the regular component of R^{2n}, then (-1)^{\mu(C)/2} is
given by the product of the holonomies (equal to +/- 1) of the even- (or odd-)
indexed eigenvector bundles of L and Lmat.Comment: 25 pages; published versio
Nonlinear Diffusion Through Large Complex Networks Containing Regular Subgraphs
Transport through generalized trees is considered. Trees contain the simple
nodes and supernodes, either well-structured regular subgraphs or those with
many triangles. We observe a superdiffusion for the highly connected nodes
while it is Brownian for the rest of the nodes. Transport within a supernode is
affected by the finite size effects vanishing as For the even
dimensions of space, , the finite size effects break down the
perturbation theory at small scales and can be regularized by using the
heat-kernel expansion.Comment: 21 pages, 2 figures include
Autocorrelation function of eigenstates in chaotic and mixed systems
We study the autocorrelation function of different types of eigenfunctions in
quantum mechanical systems with either chaotic or mixed classical limits. We
obtain an expansion of the autocorrelation function in terms of the correlation
length. For localized states, like bouncing ball modes or states living on
tori, a simple model using only classical input gives good agreement with the
exact result. In particular, a prediction for irregular eigenfunctions in mixed
systems is derived and tested. For chaotic systems, the expansion of the
autocorrelation function can be used to test quantum ergodicity on different
length scales.Comment: 30 pages, 12 figures. Some of the pictures are included in low
resolution only. For a version with pictures in high resolution see
http://www.physik.uni-ulm.de/theo/qc/ or http://www.maths.bris.ac.uk/~maab