113 research outputs found
A Universal Magnification Theorem III. Caustics Beyond Codimension Five
In the final paper of this series, we extend our results on magnification
invariants to the infinite family of A, D, E caustic singularities. We prove
that for families of general mappings between planes exhibiting any caustic
singularity of the A, D, E family, and for a point in the target space lying
anywhere in the region giving rise to the maximum number of lensed images (real
pre-images), the total signed magnification of the lensed images will always
sum to zero. The proof is algebraic in nature and relies on the Euler trace
formula.Comment: 8 page
Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
We compute exact asymptotic results for the probability of the occurrence of
large deviations of the largest (smallest) eigenvalue of random matrices
belonging to the Gaussian orthogonal, unitary and symplectic ensembles. In
particular, we show that the probability that all the eigenvalues of an (NxN)
random matrix are positive (negative) decreases for large N as ~\exp[-\beta
\theta(0) N^2] where the Dyson index \beta characterizes the ensemble and the
exponent \theta(0)=(\ln 3)/4=0.274653... is universal. We compute the
probability that the eigenvalues lie in the interval [\zeta_1,\zeta_2] which
allows us to calculate the joint probability distribution of the minimum and
the maximum eigenvalue. As a byproduct, we also obtain exactly the average
density of states in Gaussian ensembles whose eigenvalues are restricted to lie
in the interval [\zeta_1,\zeta_2], thus generalizing the celebrated Wigner
semi-circle law to these restricted ensembles. It is found that the density of
states generically exhibits an inverse square-root singularity at the location
of the barriers. These results are confirmed by numerical simulations.Comment: 17 pages Revtex, 5 .eps figures include
The Statistics of the Number of Minima in a Random Energy Landscape
We consider random energy landscapes constructed from d-dimensional lattices
or trees. The distribution of the number of local minima in such landscapes
follows a large deviation principle and we derive the associated law exactly
for dimension 1. Also of interest is the probability of the maximum possible
number of minima; this probability scales exponentially with the number of
sites. We calculate analytically the corresponding exponent for the Cayley tree
and the two-leg ladder; for 2 to 5 dimensional hypercubic lattices, we compute
the exponent numerically and compare to the Cayley tree case.Comment: 18 pages, 8 figures, added background on landscapes and reference
Lensing by Kerr Black Holes. II: Analytical Study of Quasi-Equatorial Lensing Observables
In this second paper, we develop an analytical theory of quasi-equatorial
lensing by Kerr black holes. In this setting we solve perturbatively our
general lens equation with displacement given in Paper I, going beyond
weak-deflection Kerr lensing to third order in our expansion parameter epsilon,
which is the ratio of the angular gravitational radius to the angular Einstein
radius. We obtain new formulas and results for the bending angle, image
positions, image magnifications, total unsigned magnification, and centroid,
all to third order in epsilon and including the displacement. New results on
the time delay between images are also given to second order in epsilon, again
including displacement. For all lensing observables we show that the
displacement begins to appear only at second order in epsilon. When there is no
spin, we obtain new results on the lensing observables for Schwarzschild
lensing with displacement.Comment: 23 pages; final published versio
Exploring a string-like landscape
We explore inflationary trajectories within randomly-generated
two-dimensional potentials, considered as a toy model of the string landscape.
Both the background and perturbation equations are solved numerically, the
latter using the two-field formalism of Peterson and Tegmark which fully
incorporates the effect of isocurvature perturbations. Sufficient inflation is
a rare event, occurring for only roughly one in potentials. For models
generating sufficient inflation, we find that the majority of runs satisfy
current constraints from WMAP. The scalar spectral index is less than 1 in all
runs. The tensor-to-scalar ratio is below the current limit, while typically
large enough to be detected by next-generation CMB experiments and perhaps also
by Planck. In many cases the inflationary consistency equation is broken by the
effect of isocurvature modes.Comment: 24 pages with 8 figures incorporated, matches version accepted by
JCA
Parameterized Inapproximability of Target Set Selection and Generalizations
In this paper, we consider the Target Set Selection problem: given a graph
and a threshold value for any vertex of the graph, find a minimum
size vertex-subset to "activate" s.t. all the vertices of the graph are
activated at the end of the propagation process. A vertex is activated
during the propagation process if at least of its neighbors are
activated. This problem models several practical issues like faults in
distributed networks or word-to-mouth recommendations in social networks. We
show that for any functions and this problem cannot be approximated
within a factor of in time, unless FPT = W[P],
even for restricted thresholds (namely constant and majority thresholds). We
also study the cardinality constraint maximization and minimization versions of
the problem for which we prove similar hardness results
Supersymmetric Vacua in Random Supergravity
We determine the spectrum of scalar masses in a supersymmetric vacuum of a
general N=1 supergravity theory, with the Kahler potential and superpotential
taken to be random functions of N complex scalar fields. We derive a random
matrix model for the Hessian matrix and compute the eigenvalue spectrum.
Tachyons consistent with the Breitenlohner-Freedman bound are generically
present, and although these tachyons cannot destabilize the supersymmetric
vacuum, they do influence the likelihood of the existence of an `uplift' to a
metastable vacuum with positive cosmological constant. We show that the
probability that a supersymmetric AdS vacuum has no tachyons is formally
equivalent to the probability of a large fluctuation of the smallest eigenvalue
of a certain real Wishart matrix. For normally-distributed matrix entries and
any N, this probability is given exactly by P = exp(-2N^2|W|^2/m_{susy}^2),
with W denoting the superpotential and m_{susy} the supersymmetric mass scale;
for more general distributions of the entries, our result is accurate when N >>
1. We conclude that for |W| \gtrsim m_{susy}/N, tachyonic instabilities are
ubiquitous in configurations obtained by uplifting supersymmetric vacua.Comment: 26 pages, 6 figure
A Stringy Mechanism for A Small Cosmological Constant
Based on the probability distributions of products of random variables, we
propose a simple stringy mechanism that prefers the meta-stable vacua with a
small cosmological constant. We state some relevant properties of the
probability distributions of functions of random variables. We then illustrate
the mechanism within the flux compactification models in Type IIB string
theory. As a result of the stringy dynamics, we argue that the generic
probability distribution for the meta-stable vacua typically peaks with a
divergent behavior at the zero value of the cosmological constant. However, its
suppression in the single modulus model studied here is modest.Comment: 36 pages, 8 figure
Large Deviations of the Maximum Eigenvalue in Wishart Random Matrices
We compute analytically the probability of large fluctuations to the left of
the mean of the largest eigenvalue in the Wishart (Laguerre) ensemble of
positive definite random matrices. We show that the probability that all the
eigenvalues of a (N x N) Wishart matrix W=X^T X (where X is a rectangular M x N
matrix with independent Gaussian entries) are smaller than the mean value
=N/c decreases for large N as , where \beta=1,2 correspond respectively to
real and complex Wishart matrices, c=N/M < 1 and \Phi_{-}(x;c) is a large
deviation function that we compute explicitly. The result for the Anti-Wishart
case (M < N) simply follows by exchanging M and N. We also analytically
determine the average spectral density of an ensemble of constrained Wishart
matrices whose eigenvalues are forced to be smaller than a fixed barrier. The
numerical simulations are in excellent agreement with the analytical
predictions.Comment: Published version. References and appendix adde
- âŠ