1,608 research outputs found
Density of States of Quantum Spin Systems from Isotropic Entanglement
We propose a method which we call "Isotropic Entanglement" (IE), that
predicts the eigenvalue distribution of quantum many body (spin) systems (QMBS)
with generic interactions. We interpolate between two known approximations by
matching fourth moments. Though, such problems can be QMA-complete, our
examples show that IE provides an accurate picture of the spectra well beyond
what one expects from the first four moments alone. We further show that the
interpolation is universal, i.e., independent of the choice of local terms.Comment: 4+ pages, content is as in the published versio
Systematics of Aligned Axions
We describe a novel technique that renders theories of axions tractable,
and more generally can be used to efficiently analyze a large class of periodic
potentials of arbitrary dimension. Such potentials are complex energy
landscapes with a number of local minima that scales as , and so for
large appear to be analytically and numerically intractable. Our method is
based on uncovering a set of approximate symmetries that exist in addition to
the periods. These approximate symmetries, which are exponentially close to
exact, allow us to locate the minima very efficiently and accurately and to
analyze other characteristics of the potential. We apply our framework to
evaluate the diameters of flat regions suitable for slow-roll inflation, which
unifies, corrects and extends several forms of "axion alignment" previously
observed in the literature. We find that in a broad class of random theories,
the potential is smooth over diameters enhanced by compared to the
typical scale of the potential. A Mathematica implementation of our framework
is available online.Comment: 68 pages, 17 figure
Approximate null distribution of the largest root in multivariate analysis
The greatest root distribution occurs everywhere in classical multivariate
analysis, but even under the null hypothesis the exact distribution has
required extensive tables or special purpose software. We describe a simple
approximation, based on the Tracy--Widom distribution, that in many cases can
be used instead of tables or software, at least for initial screening. The
quality of approximation is studied, and its use illustrated in a variety of
setttings.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS220 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Superstatistical generalisations of Wishart-Laguerre ensembles of random matrices
Using Beck and Cohen's superstatistics, we introduce in a systematic way a family of generalized WishartâLaguerre ensembles of random matrices with Dyson index β = 1, 2 and 4. The entries of the data matrix are Gaussian random variables whose variances Ρ fluctuate from one sample to another according to a certain probability density f(Ρ) and a single deformation parameter Îł. Three superstatistical classes for f(Ρ) are usually considered: Ď2-, inverse Ď2- and log-normal distributions. While the first class, already considered by two of the authors, leads to a power-law decay of the spectral density, we here introduce and solve exactly a superposition of WishartâLaguerre ensembles with inverse Ď2-distribution. The corresponding macroscopic spectral density is given by a Îł-deformation of the semi-circle and MarÄenkoâPastur laws, on a non-compact support with exponential tails. After discussing in detail the validity of Wigner's surmise in the WishartâLaguerre class, we introduce a generalized Îł-dependent surmise with stretched-exponential tails, which well approximates the individual level spacing distribution in the bulk. The analytical results are in excellent agreement with numerical simulations. To illustrate our findings we compare the Ď2- and inverse Ď2-classes to empirical data from financial covariance matrices
Planckian Axions in String Theory
We argue that super-Planckian diameters of axion fundamental domains can
naturally arise in Calabi-Yau compactifications of string theory. In a theory
with axions , the fundamental domain is a polytope defined by the
periodicities of the axions, via constraints of the form . We compute the diameter of the fundamental domain in terms of
the eigenvalues of the metric on field space, and also,
crucially, the largest eigenvalue of . At large ,
approaches a Wishart matrix, due to universality, and we show that
the diameter is at least , exceeding the naive Pythagorean range by a
factor . This result is robust in the presence of constraints,
while for the diameter is further enhanced by eigenvector delocalization
to . We directly verify our results in explicit Calabi-Yau
compactifications of type IIB string theory. In the classic example with
where parametrically controlled moduli stabilization was
demonstrated by Denef et al. in [1], the largest metric eigenvalue obeys . The random matrix analysis then predicts, and we
exhibit, axion diameters for the precise vacuum parameters found in
[1]. Our results provide a framework for achieving large-field axion inflation
in well-understood flux vacua.Comment: 42 pages, 4 figure
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