14,929 research outputs found
Fractal Strings and Multifractal Zeta Functions
For a Borel measure on the unit interval and a sequence of scales that tend
to zero, we define a one-parameter family of zeta functions called multifractal
zeta functions. These functions are a first attempt to associate a zeta
function to certain multifractal measures. However, we primarily show that they
associate a new zeta function, the topological zeta function, to a fractal
string in order to take into account the topology of its fractal boundary. This
expands upon the geometric information garnered by the traditional geometric
zeta function of a fractal string in the theory of complex dimensions. In
particular, one can distinguish between a fractal string whose boundary is the
classical Cantor set, and one whose boundary has a single limit point but has
the same sequence of lengths as the complement of the Cantor set. Later work
will address related, but somewhat different, approaches to multifractals
themselves, via zeta functions, partly motivated by the present paper.Comment: 32 pages, 9 figures. This revised version contains new sections and
figures illustrating the main results of this paper and recent results from
others. Sections 0, 2, and 6 have been significantly rewritte
Algorithmic information and incompressibility of families of multidimensional networks
This article presents a theoretical investigation of string-based generalized
representations of families of finite networks in a multidimensional space.
First, we study the recursive labeling of networks with (finite) arbitrary node
dimensions (or aspects), such as time instants or layers. In particular, we
study these networks that are formalized in the form of multiaspect graphs. We
show that, unlike classical graphs, the algorithmic information of a
multidimensional network is not in general dominated by the algorithmic
information of the binary sequence that determines the presence or absence of
edges. This universal algorithmic approach sets limitations and conditions for
irreducible information content analysis in comparing networks with a large
number of dimensions, such as multilayer networks. Nevertheless, we show that
there are particular cases of infinite nesting families of finite
multidimensional networks with a unified recursive labeling such that each
member of these families is incompressible. From these results, we study
network topological properties and equivalences in irreducible information
content of multidimensional networks in comparison to their isomorphic
classical graph.Comment: Extended preprint version of the pape
Topological Subsystem Codes
We introduce a family of 2D topological subsystem quantum error-correcting
codes. The gauge group is generated by 2-local Pauli operators, so that 2-local
measurements are enough to recover the error syndrome. We study the
computational power of code deformation in these codes, and show that
boundaries cannot be introduced in the usual way. In addition, we give a
general mapping connecting suitable classical statistical mechanical models to
optimal error correction in subsystem stabilizer codes that suffer from
depolarizing noise.Comment: 16 pages, 11 figures, explanations added, typos correcte
Relaxation dynamics of the toric code in contact with a thermal reservoir: Finite-size scaling in a low temperature regime
We present an analysis of the relaxation dynamics of finite-size topological
qubits in contact with a thermal bath. Using a continuous-time Monte Carlo
method, we explicitly compute the low-temperature nonequilibrium dynamics of
the toric code on finite lattices. In contrast to the size-independent bound
predicted for the toric code in the thermodynamic limit, we identify a
low-temperature regime on finite lattices below a size-dependent crossover
temperature with nontrivial finite-size and temperature scaling of the
relaxation time. We demonstrate how this nontrivial finite-size scaling is
governed by the scaling of topologically nontrivial two-dimensional classical
random walks. The transition out of this low-temperature regime defines a
dynamical finite-size crossover temperature that scales inversely with the log
of the system size, in agreement with a crossover temperature defined from
equilibrium properties. We find that both the finite-size and
finite-temperature scaling are stronger in the low-temperature regime than
above the crossover temperature. Since this finite-temperature scaling competes
with the scaling of the robustness to unitary perturbations, this analysis may
elucidate the scaling of memory lifetimes of possible physical realizations of
topological qubits.Comment: 14 Pages, 13 figure
Dynamical correlations in the escape strategy of Influenza A virus
The evolutionary dynamics of human Influenza A virus presents a challenging
theoretical problem. An extremely high mutation rate allows the virus to
escape, at each epidemic season, the host immune protection elicited by
previous infections. At the same time, at each given epidemic season a single
quasi-species, that is a set of closely related strains, is observed. A
non-trivial relation between the genetic (i.e., at the sequence level) and the
antigenic (i.e., related to the host immune response) distances can shed light
into this puzzle. In this paper we introduce a model in which, in accordance
with experimental observations, a simple interaction rule based on spatial
correlations among point mutations dynamically defines an immunity space in the
space of sequences. We investigate the static and dynamic structure of this
space and we discuss how it affects the dynamics of the virus-host interaction.
Interestingly we observe a staggered time structure in the virus evolution as
in the real Influenza evolutionary dynamics.Comment: 14 pages, 5 figures; main paper for the supplementary info in
arXiv:1303.595
Universality and Critical Phenomena in String Defect Statistics
The idea of biased symmetries to avoid or alleviate cosmological problems
caused by the appearance of some topological defects is familiar in the context
of domain walls, where the defect statistics lend themselves naturally to a
percolation theory description, and for cosmic strings, where the proportion of
infinite strings can be varied or disappear entirely depending on the bias in
the symmetry. In this paper we measure the initial configurational statistics
of a network of string defects after a symmetry-breaking phase transition with
initial bias in the symmetry of the ground state. Using an improved algorithm,
which is useful for a more general class of self-interacting walks on an
infinite lattice, we extend the work in \cite{MHKS} to better statistics and a
different ground state manifold, namely , and explore various different
discretisations. Within the statistical errors, the critical exponents of the
Hagedorn transition are found to be quite possibly universal and identical to
the critical exponents of three-dimensional bond or site percolation. This
improves our understanding of the percolation theory description of defect
statistics after a biased phase transition, as proposed in \cite{MHKS}. We also
find strong evidence that the existence of infinite strings in the Vachaspati
Vilenkin algorithm is generic to all (string-bearing) vacuum manifolds, all
discretisations thereof, and all regular three-dimensional lattices.Comment: 62 pages, plain LaTeX, macro mathsymb.sty included, figures included.
also available on
http://starsky.pcss.maps.susx.ac.uk/groups/pt/preprints/96/96011.ps.g
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