8,061 research outputs found
Multi-Scale Jacobi Method for Anderson Localization
A new KAM-style proof of Anderson localization is obtained. A sequence of
local rotations is defined, such that off-diagonal matrix elements of the
Hamiltonian are driven rapidly to zero. This leads to the first proof via
multi-scale analysis of exponential decay of the eigenfunction correlator (this
implies strong dynamical localization). The method has been used in recent work
on many-body localization [arXiv:1403.7837].Comment: 34 pages, 8 figures, clarifications and corrections for published
version; more detail in Section 4.
Optimal Acyclic Hamiltonian Path Completion for Outerplanar Triangulated st-Digraphs (with Application to Upward Topological Book Embeddings)
Given an embedded planar acyclic digraph G, we define the problem of "acyclic
hamiltonian path completion with crossing minimization (Acyclic-HPCCM)" to be
the problem of determining an hamiltonian path completion set of edges such
that, when these edges are embedded on G, they create the smallest possible
number of edge crossings and turn G to a hamiltonian digraph. Our results
include:
--We provide a characterization under which a triangulated st-digraph G is
hamiltonian.
--For an outerplanar triangulated st-digraph G, we define the st-polygon
decomposition of G and, based on its properties, we develop a linear-time
algorithm that solves the Acyclic-HPCCM problem with at most one crossing per
edge of G.
--For the class of st-planar digraphs, we establish an equivalence between
the Acyclic-HPCCM problem and the problem of determining an upward 2-page
topological book embedding with minimum number of spine crossings. We infer
(based on this equivalence) for the class of outerplanar triangulated
st-digraphs an upward topological 2-page book embedding with minimum number of
spine crossings and at most one spine crossing per edge.
To the best of our knowledge, it is the first time that edge-crossing
minimization is studied in conjunction with the acyclic hamiltonian completion
problem and the first time that an optimal algorithm with respect to spine
crossing minimization is presented for upward topological book embeddings
Recent Developments of World-Line Monte Carlo Methods
World-line quantum Monte Carlo methods are reviewed with an emphasis on
breakthroughs made in recent years. In particular, three algorithms -- the loop
algorithm, the worm algorithm, and the directed-loop algorithm -- for updating
world-line configurations are presented in a unified perspective. Detailed
descriptions of the algorithms in specific cases are also given.Comment: To appear in Journal of Physical Society of Japa
Clustering Phase Transitions and Hysteresis: Pitfalls in Constructing Network Ensembles
Ensembles of networks are used as null models in many applications. However,
simple null models often show much less clustering than their real-world
counterparts. In this paper, we study a model where clustering is enhanced by
means of a fugacity term as in the Strauss (or "triangle") model, but where the
degree sequence is strictly preserved -- thus maintaining the quenched
heterogeneity of nodes found in the original degree sequence. Similar models
had been proposed previously in [R. Milo et al., Science 298, 824 (2002)]. We
find that our model exhibits phase transitions as the fugacity is changed. For
regular graphs (identical degrees for all nodes) with degree k > 2 we find a
single first order transition. For all non-regular networks that we studied
(including Erdos - Renyi and scale-free networks) we find multiple jumps
resembling first order transitions, together with strong hysteresis. The latter
transitions are driven by the sudden emergence of "cluster cores": groups of
highly interconnected nodes with higher than average degrees. To study these
cluster cores visually, we introduce q-clique adjacency plots. We find that
these cluster cores constitute distinct communities which emerge spontaneously
from the triangle generating process. Finally, we point out that cluster cores
produce pitfalls when using the present (and similar) models as null models for
strongly clustered networks, due to the very strong hysteresis which
effectively leads to broken ergodicity on realistic time scales.Comment: 13 pages, 11 figure
The effect of colored noise on heteroclinic orbits
The dynamics of a weakly dissipative Hamiltonian system submitted to
stochastic perturbations has been investigated by means of asymptotic methods.
The probability of noise-induced separatrix crossing, which drastically changes
the fate of the system, is derived analytically in the case where noise is an
additive Kubo-Anderson process. This theory shows how the geometry of the
separatrix, as well as the noise intensity and correlation time, affect the
statistics of crossing. Results can be applied to a wide variety of systems,
and are valid in the limit where the noise correlation time scale is much
smaller than the time scale of the undisturbed Hamiltonian dynamics
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