276 research outputs found
Scaling symmetry, renormalization, and time series modeling
We present and discuss a stochastic model of financial assets dynamics based
on the idea of an inverse renormalization group strategy. With this strategy we
construct the multivariate distributions of elementary returns based on the
scaling with time of the probability density of their aggregates. In its
simplest version the model is the product of an endogenous auto-regressive
component and a random rescaling factor designed to embody also exogenous
influences. Mathematical properties like increments' stationarity and
ergodicity can be proven. Thanks to the relatively low number of parameters,
model calibration can be conveniently based on a method of moments, as
exemplified in the case of historical data of the S&P500 index. The calibrated
model accounts very well for many stylized facts, like volatility clustering,
power law decay of the volatility autocorrelation function, and multiscaling
with time of the aggregated return distribution. In agreement with empirical
evidence in finance, the dynamics is not invariant under time reversal and,
with suitable generalizations, skewness of the return distribution and leverage
effects can be included. The analytical tractability of the model opens
interesting perspectives for applications, for instance in terms of obtaining
closed formulas for derivative pricing. Further important features are: The
possibility of making contact, in certain limits, with auto-regressive models
widely used in finance; The possibility of partially resolving the long-memory
and short-memory components of the volatility, with consistent results when
applied to historical series.Comment: Main text (17 pages, 13 figures) plus Supplementary Material (16
pages, 5 figures
Nonequilibrium dynamical renormalization group: Dynamical crossover from weak to infinite randomness in the transverse-field Ising chain
In this work we formulate the nonequilibrium dynamical renormalization group
(ndRG). The ndRG represents a general renormalization-group scheme for the
analytical description of the real-time dynamics of complex quantum many-body
systems. In particular, the ndRG incorporates time as an additional scale which
turns out to be important for the description of the long-time dynamics. It can
be applied to both translational invariant and disordered systems. As a
concrete application we study the real-time dynamics after a quench between two
quantum critical points of different universality classes. We achieve this by
switching on weak disorder in a one-dimensional transverse-field Ising model
initially prepared at its clean quantum critical point. By comparing to
numerically exact simulations for large systems we show that the ndRG is
capable of analytically capturing the full crossover from weak to infinite
randomness. We analytically study signatures of localization in both real space
and Fock space.Comment: 15 pages, 4 figures, extended presentation, version as publishe
Techniques of replica symmetry breaking and the storage problem of the McCulloch-Pitts neuron
In this article the framework for Parisi's spontaneous replica symmetry
breaking is reviewed, and subsequently applied to the example of the
statistical mechanical description of the storage properties of a
McCulloch-Pitts neuron. The technical details are reviewed extensively, with
regard to the wide range of systems where the method may be applied. Parisi's
partial differential equation and related differential equations are discussed,
and a Green function technique introduced for the calculation of replica
averages, the key to determining the averages of physical quantities. The
ensuing graph rules involve only tree graphs, as appropriate for a
mean-field-like model. The lowest order Ward-Takahashi identity is recovered
analytically and is shown to lead to the Goldstone modes in continuous replica
symmetry breaking phases. The need for a replica symmetry breaking theory in
the storage problem of the neuron has arisen due to the thermodynamical
instability of formerly given solutions. Variational forms for the neuron's
free energy are derived in terms of the order parameter function x(q), for
different prior distribution of synapses. Analytically in the high temperature
limit and numerically in generic cases various phases are identified, among
them one similar to the Parisi phase in the Sherrington-Kirkpatrick model.
Extensive quantities like the error per pattern change slightly with respect to
the known unstable solutions, but there is a significant difference in the
distribution of non-extensive quantities like the synaptic overlaps and the
pattern storage stability parameter. A simulation result is also reviewed and
compared to the prediction of the theory.Comment: 103 Latex pages (with REVTeX 3.0), including 15 figures (ps, epsi,
eepic), accepted for Physics Report
Techniques of replica symmetry breaking and the storage problem of the McCulloch-Pitts neuron
In this article the framework for Parisi's spontaneous replica symmetry
breaking is reviewed, and subsequently applied to the example of the
statistical mechanical description of the storage properties of a
McCulloch-Pitts neuron. The technical details are reviewed extensively, with
regard to the wide range of systems where the method may be applied. Parisi's
partial differential equation and related differential equations are discussed,
and a Green function technique introduced for the calculation of replica
averages, the key to determining the averages of physical quantities. The
ensuing graph rules involve only tree graphs, as appropriate for a
mean-field-like model. The lowest order Ward-Takahashi identity is recovered
analytically and is shown to lead to the Goldstone modes in continuous replica
symmetry breaking phases. The need for a replica symmetry breaking theory in
the storage problem of the neuron has arisen due to the thermodynamical
instability of formerly given solutions. Variational forms for the neuron's
free energy are derived in terms of the order parameter function x(q), for
different prior distribution of synapses. Analytically in the high temperature
limit and numerically in generic cases various phases are identified, among
them one similar to the Parisi phase in the Sherrington-Kirkpatrick model.
Extensive quantities like the error per pattern change slightly with respect to
the known unstable solutions, but there is a significant difference in the
distribution of non-extensive quantities like the synaptic overlaps and the
pattern storage stability parameter. A simulation result is also reviewed and
compared to the prediction of the theory.Comment: 103 Latex pages (with REVTeX 3.0), including 15 figures (ps, epsi,
eepic), accepted for Physics Report
Dissipative, Entropy-Production Systems across Condensed Matter and Interdisciplinary Classical VS. Quantum Physics
The thematic range of this book is wide and can loosely be described as polydispersive. Figuratively, it resembles a polynuclear path of yielding (poly)crystals. Such path can be taken when looking at it from the first side. However, a closer inspection of the book’s contents gives rise to a much more monodispersive/single-crystal and compacted (than crudely expected) picture of the book’s contents presented to a potential reader. Namely, all contributions collected can be united under the common denominator of maximum-entropy and entropy production principles experienced by both classical and quantum systems in (non)equilibrium conditions. The proposed order of presenting the material commences with properly subordinated classical systems (seven contributions) and ends up with three remaining quantum systems, presented by the chapters’ authors. The overarching editorial makes the presentation of the wide-range material self-contained and compact, irrespective of whether comprehending it from classical or quantum physical viewpoints
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