517 research outputs found
Non-Gaussian Minkowski functionals & extrema counts in redshift space
In the context of upcoming large-scale structure surveys such as Euclid, it
is of prime importance to quantify the effect of peculiar velocities on
geometric probes. Hence the formalism to compute in redshift space the
geometrical and topological one-point statistics of mildly non-Gaussian 2D and
3D cosmic fields is developed. Leveraging the partial isotropy of the target
statistics, the Gram-Charlier expansion of the joint probability distribution
of the field and its derivatives is reformulated in terms of the corresponding
anisotropic variables. In particular, the cosmic non-linear evolution of the
Minkowski functionals, together with the statistics of extrema are investigated
in turn for 3D catalogues and 2D slabs. The amplitude of the non-Gaussian
redshift distortion correction is estimated for these geometric probes. In 3D,
gravitational perturbation theory is implemented in redshift space to predict
the cosmic evolution of all relevant Gram-Charlier coefficients. Applications
to the estimation of the cosmic parameters sigma(z) and beta=f/b1 from upcoming
surveys is discussed. Such statistics are of interest for anisotropic fields
beyond cosmology.Comment: 35 pages, 15 figures, matches version published in MNRAS with a typo
corrected in eq A1
The invariant joint distribution of a stationary random field and its derivatives: Euler characteristic and critical point counts in 2 and 3D
The full moments expansion of the joint probability distribution of an
isotropic random field, its gradient and invariants of the Hessian is presented
in 2 and 3D. It allows for explicit expression for the Euler characteristic in
ND and computation of extrema counts as functions of the excursion set
threshold and the spectral parameter, as illustrated on model examples.Comment: 4 pages, 2 figures. Corrected expansion coefficients for orders n>=5.
Relation between Gram-Charlier and Edgeworth expansions is clarified
On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures
This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev
inequalities for a class of Boltzmann-Gibbs measures with singular interaction.
Such measures allow to model one-dimensional particles with confinement and
singular pair interaction. The functional inequalities come from convexity. We
prove and characterize optimality in the case of quadratic confinement via a
factorization of the measure. This optimality phenomenon holds for all beta
Hermite ensembles including the Gaussian unitary ensemble, a famous exactly
solvable model of random matrix theory. We further explore exact solvability by
reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting
the Hermite-Lassalle orthogonal polynomials as a complete set of
eigenfunctions. We also discuss the consequence of the log-Sobolev inequality
in terms of concentration of measure for Lipschitz functions such as maxima and
linear statistics.Comment: Minor improvements. To appear in Geometric Aspects of Functional
Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics
225
Moment transport equations for the primordial curvature perturbation
In a recent publication, we proposed that inflationary perturbation theory
can be reformulated in terms of a probability transport equation, whose moments
determine the correlation properties of the primordial curvature perturbation.
In this paper we generalize this formulation to an arbitrary number of fields.
We deduce ordinary differential equations for the evolution of the moments of
zeta on superhorizon scales, which can be used to obtain an evolution equation
for the dimensionless bispectrum, fNL. Our equations are covariant in field
space and allow identification of the source terms responsible for evolution of
fNL. In a model with M scalar fields, the number of numerical integrations
required to obtain solutions of these equations scales like O(M^3). The
performance of the moment transport algorithm means that numerical calculations
with M >> 1 fields are straightforward. We illustrate this performance with a
numerical calculation of fNL in Nflation models containing M ~ 10^2 fields,
finding agreement with existing analytic calculations. We comment briefly on
extensions of the method beyond the slow-roll approximation, or to calculate
higher order parameters such as gNL.Comment: 23 pages, plus appendices and references; 4 figures. v2: incorrect
statements regarding numerical delta N removed from Sec. 4.3. Minor
modifications elsewher
-Gaussian processes: non-commutative and classical aspects
We examine, for , -Gaussian processes, i.e. families of operators
(non-commutative random variables) -- where the fulfill
the -commutation relations a_sa_t^*-qa_t^*a_s=c(s,t)\cdot \id for some
covariance function -- equipped with the vacuum expectation
state. We show that there is a -analogue of the Gaussian functor of second
quantization behind these processes and that this structure can be used to
translate questions on -Gaussian processes into corresponding (and much
simpler) questions in the underlying Hilbert space. In particular, we use this
idea to show that a large class of -Gaussian processes possess a
non-commutative kind of Markov property, which ensures that there exist
classical versions of these non-commutative processes. This answers an old
question of Frisch and Bourret \cite{FB}.Comment: AMS-TeX 2.
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