347 research outputs found
Zipf's Law for Atlas Models
A set of data with positive values follows a Pareto distribution if the
log-log plot of value versus rank is approximately a straight line. A Pareto
distribution satisfies Zipf's law if the log-log plot has a slope of -1. Since
many types of ranked data follow Zipf's law, it is considered a form of
universality. We propose a mathematical explanation for this phenomenon based
on Atlas models and first-order models, systems of positive continuous
semimartingales with parameters that depend only on rank. We show that the
stable distribution of an Atlas model will follow Zipf's law if and only if two
natural conditions, conservation and completeness, are satisfied. Since Atlas
models and first-order models can be constructed to approximate systems of
time-dependent rank-based data, our results can explain the universality of
Zipf's law for such systems. However, ranked data generated by other means may
follow non-Zipfian Pareto distributions. Hence, our results explain why Zipf's
law holds for word frequency, firm size, household wealth, and city size, while
it does not hold for earthquake magnitude, cumulative book sales, the intensity
of solar flares, and the intensity of wars, all of which follow non-Zipfian
Pareto distributions.Comment: Accepted for publication by the Applied Probability Trust
(http://www.appliedprobability.org) in the Journal of Applied Probability
57.4 (December 2020
A Rank-Based Approach to Zipf's Law
An Atlas model is a rank-based system of continuous semimartingales for which
the steady-state values of the processes follow a power law, or Pareto
distribution. For a power law, the log-log plot of these steady-state values
versus rank is a straight line. Zipf's law is a power law for which the slope
of this line is -1. In this note, rank-based conditions are found under which
an Atlas model will follow Zipf's law. An advantage of this rank-based approach
is that it provides information about the dynamics of systems that result in
Zipf's law.Comment: 6 page
Empirical Methods for Dynamic Power Law Distributions in the Social Sciences
This paper introduces nonparametric econometric methods that characterize
general power law distributions under basic stability conditions. These methods
extend the literature on power laws in the social sciences in several
directions. First, we show that any stationary distribution in a random growth
setting is shaped entirely by two factors - the idiosyncratic volatilities and
reversion rates (a measure of cross-sectional mean reversion) for different
ranks in the distribution. This result is valid regardless of how growth rates
and volatilities vary across different economic agents, and hence applies to
Gibrat's law and its extensions. Second, we present techniques to estimate
these two factors using panel data. Third, we show how our results offer a
structural explanation for a generalized size effect in which higher-ranked
processes grow more slowly than lower-ranked processes on average. Finally, we
employ our empirical methods using panel data on commodity prices and show that
our techniques accurately describe the empirical distribution of relative
commodity prices. We also show the existence of a generalized "size" effect for
commodities, as predicted by our econometric theory.Comment: 33 pages, 7 figures. arXiv admin note: text overlap with
arXiv:1601.0409
Adiabatic potentials using multiple radio frequencies
Adiabatic radio frequency (RF) potentials are powerful tools for creating
advanced trapping geometries for ultra-cold atoms. While the basic theory of RF
trapping is well understood, studies of more complicated setups involving
multiple resonant frequencies in the limit where their effects cannot be
treated independently are rare. Here we present an approach based on Floquet
theory and show that it offers significant corrections to existing models when
two RF frequencies are near degenerate. Furthermore it has no restrictions on
the dimension, the number of frequencies or the orientation of the RF fields.
We show that the added degrees of freedom can, for example, be used to create a
potential that allows for easy creation of ring vortex solitons.Comment: 9 Pages, 6 Figure
Permutation-Weighted Portfolios and the Efficiency of Commodity Futures Markets
A market portfolio is a portfolio in which each asset is held at a weight
proportional to its market value. Functionally generated portfolios are
portfolios for which the logarithmic return relative to the market portfolio
can be decomposed into a function of the market weights and a process of
locally finite variation, and this decomposition is convenient for
characterizing the long-term behavior of the portfolio. A permutation-weighted
portfolio is a portfolio in which the assets are held at weights proportional
to a permutation of their market values, and such a portfolio is functionally
generated only for markets with two assets (except for the identity
permutation). A reverse-weighted portfolio is a portfolio in which the asset
with the greatest market weight is assigned the smallest market weight, the
asset with the second-largest weight is assigned the second-smallest, and so
forth. Although the reverse-weighted portfolio in a market with four or more
assets is not functionally generated, it is still possible to characterize its
long-term behavior using rank-based methods. This result is applied to a market
of commodity futures, where we show that the reverse price-weighted portfolio
substantially outperforms the price-weighted portfolio from 1977-2018.Comment: 18 pages, 8 figures, 2 table
Asset Price Distributions and Efficient Markets
We explore a decomposition in which returns on a large class of portfolios
relative to the market depend on a smooth non-negative drift and changes in the
asset price distribution. This decomposition is obtained using general
continuous semimartingale price representations, and is thus consistent with
virtually any asset pricing model. Fluctuations in portfolio relative returns
depend on stochastic time-varying dispersion in asset prices. Thus, our
framework uncovers an asset pricing factor whose existence emerges from an
accounting identity universal across different economic and financial
environments, a fact that has deep implications for market efficiency. In
particular, in a closed, dividend-free market in which asset price dispersion
is relatively constant, a large class of portfolios must necessarily outperform
the market portfolio over time. We show that price dispersion in commodity
futures markets has increased only slightly, and confirm the existence of
substantial excess returns that co-vary with changes in price dispersion as
predicted by our theory.Comment: 45 pages, 6 figures, 3 table
The Rank Effect for Commodities
We uncover a large and significant low-minus-high rank effect for commodities
across two centuries. There is nothing anomalous about this anomaly, nor is it
clear how it can be arbitraged away. Using nonparametric econometric methods,
we demonstrate that such a rank effect is a necessary consequence of a
stationary relative asset price distribution. We confirm this prediction using
daily commodity futures prices and show that a portfolio consisting of
lower-ranked, lower-priced commodities yields 23% higher annual returns than a
portfolio consisting of higher-ranked, higher-priced commodities. These excess
returns have a Sharpe ratio nearly twice as high as the U.S. stock market yet
are uncorrelated with market risk. In contrast to the extensive literature on
asset pricing factors and anomalies, our results are structural and rely on
minimal and realistic assumptions for the long-run behavior of relative asset
prices.Comment: 25 pages, 10 figures, 1 tabl
A Statistical Model of Inequality
This paper develops a nonparametric statistical model of wealth distribution
that imposes little structure on the fluctuations of household wealth. In this
setting, we use new techniques to obtain a closed-form household-by-household
characterization of the stable distribution of wealth and show that this
distribution is shaped entirely by two factors - the reversion rates (a measure
of cross-sectional mean reversion) and idiosyncratic volatilities of wealth
across different ranked households. By estimating these factors, our model can
exactly match the U.S. wealth distribution. This provides information about the
current trajectory of inequality as well as estimates of the distributional
effects of progressive capital taxes. We find evidence that the U.S. wealth
distribution might be on a temporarily unstable trajectory, thus suggesting
that further increases in top wealth shares are likely in the near future. For
capital taxes, we find that a small tax levied on just 1% of households
substantially reshapes the distribution of wealth and reduces inequality.Comment: 46 pages, 9 tables, 8 figure
The Rank Effect
We decompose returns for portfolios of bottom-ranked, lower-priced assets
relative to the market into rank crossovers and changes in the relative price
of those bottom-ranked assets. This decomposition is general and consistent
with virtually any asset pricing model. Crossovers measure changes in rank and
are smoothly increasing over time, while return fluctuations are driven by
volatile relative price changes. Our results imply that in a closed,
dividend-free market in which the relative price of bottom-ranked assets is
approximately constant, a portfolio of those bottom-ranked assets will
outperform the market portfolio over time. We show that bottom-ranked relative
commodity futures prices have increased only slightly, and confirm the
existence of substantial excess returns predicted by our theory. If these
excess returns did not exist, then top-ranked relative prices would have had to
be much higher in 2018 than those actually observed -- this would imply a
radically different commodity price distribution.Comment: 47 pages, 10 figures, 5 tables. arXiv admin note: substantial text
overlap with arXiv:1810.1284
Exciton dynamics in emergent Rydberg lattices
The dynamics of excitons in a one-dimensional ensemble with partial spatial
order are studied. During optical excitation, cold Rydberg atoms spontaneously
organize into regular spatial arrangements due to their mutual interactions.
This emergent lattice is used as the starting point to study resonant energy
transfer triggered by driving a to transition using a
microwave field. The dynamics are probed by detecting the survival probability
of atoms in the Rydberg state. Experimental data qualitatively agree with
our theoretical predictions including the mapping onto XXZ spin model in the
strong-driving limit. Our results suggest that emergent Rydberg lattices
provide an ideal platform to study coherent energy transfer in structured media
without the need for externally imposed potentials
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