26,864 research outputs found
Universality of free homogeneous sums in every dimension
We prove a general multidimensional invariance principle for a family of
U-statistics based on freely independent non-commutative random variables of
the type , where is the -th Chebyshev polynomial and is
a standard semicircular element on a fixed -probability space. As a
consequence, we deduce that homogeneous sums based on random variables of this
type are universal with respect to both semicircular and free Poisson
approximations.
Our results are stated in a general multidimensional setting and can be seen
as a genuine extension of some recent findings by Deya and Nourdin; our
techniques are based on the combination of the free Lindeberg method and the
Fourth moment Theorem
Interpolating between and
We study the dimensional continuation of the sphere free energy in conformal
field theories. In continuous dimension we define the quantity , where is the path integral of the Euclidean CFT on
the -dimensional round sphere. smoothly interpolates between
times the -anomaly coefficient in even , and
times the sphere free energy in odd . We calculate
in various examples of unitary CFT that can be continued to
non-integer dimensions, including free theories, double-trace deformations at
large , and perturbative fixed points in the expansion. For all
these examples is positive, and it decreases under RG flow. Using
perturbation theory in the coupling, we calculate in the
Wilson-Fisher fixed point of the vector model in to order
. We use this result to estimate the value of in the
3-dimensional Ising model, and find that it is only a few percent below of
the free conformally coupled scalar field. We use similar methods to estimate
the values for the Gross-Neveu model in and the model
in . Finally, we carry out the dimensional continuation of interacting
theories with 4 supercharges, for which we suggest that may be
calculated exactly using an appropriate version of localization on . Our
approach provides an interpolation between the -maximization in and
the -maximization in .Comment: 41 pages, 4 figures. v4: Eqs. (1.6), (4.13) and (5.37) corrected;
footnote 9 added discussing the Euler density counterter
One Loop Tests of Higher Spin AdS/CFT
Vasiliev's type A higher spin theories in AdS4 have been conjectured to be
dual to the U(N) or O(N) singlet sectors in 3-d conformal field theories with
N-component scalar fields. We compare the O(N^0) correction to the 3-sphere
free energy F in the CFTs with corresponding calculations in the higher spin
theories. This requires evaluating a regularized sum over one loop vacuum
energies of an infinite set of massless higher spin gauge fields in Euclidean
AdS4. For the Vasiliev theory including fields of all integer spin and a scalar
with Delta=1 boundary condition, we show that the regularized sum vanishes.
This is in perfect agreement with the vanishing of subleading corrections to F
in the U(N) singlet sector of the theory of N free complex scalar fields. For
the minimal Vasiliev theory including fields of only even spin, the regularized
sum remarkably equals the value of F for one free real scalar field. This
result may agree with the O(N) singlet sector of the theory of N real scalar
fields, provided the coupling constant in the Vasiliev theory is identified as
G_N ~ 1/(N-1). Similarly, consideration of the USp(N) singlet sector for N
complex scalar fields, which we conjecture to be dual to the husp(2;0|4)
Vasiliev theory, requires G_N ~ 1/(N+1). We also test the higher spin AdS3/CFT2
conjectures by calculating the regularized sum over one loop vacuum energies of
higher spin fields in AdS3. We match the result with the O(N^0) term in the
central charge of the W_N minimal models; this requires a certain truncation of
the CFT operator spectrum so that the bulk theory contains two real scalar
fields with the same boundary conditions.Comment: 20 pages. v3: minor corrections, version published in JHE
Jet Veto Clustering Logarithms Beyond Leading Order
Many experimental analyses separate events into exclusive jet bins, using a
jet algorithm to cluster the final state and then veto on jets. Jet clustering
induces logarithmic dependence on the jet radius R in the cross section for
exclusive jet bins, a dependence that is poorly controlled due to the
non-global nature of the clustering. At jet radii of experimental interest, the
leading order (LO) clustering effects are numerically significant, but the
higher order effects are currently unknown. We rectify this situation by
calculating the most important part of the next-to-leading order (NLO)
clustering logarithms of R for any 0-jet process, which enter as
corrections to the cross section. The calculation blends
subtraction methods for NLO calculations with factorization properties of QCD
and soft-collinear effective theory (SCET). We compare the size of the known LO
and new NLO clustering logarithms and find that the impact of the NLO terms on
the 0-jet cross section in Higgs production is small. This brings clustering
effects under better control and may be used to improve uncertainty estimates
on cross sections with a jet veto.Comment: 39 pages, 5 figures. v2: journal version. v3: added missing term in
calculation, conclusions unchange
CUB models: a preliminary fuzzy approach to heterogeneity
In line with the increasing attention paid to deal with uncertainty in
ordinal data models, we propose to combine Fuzzy models with \cub models within
questionnaire analysis. In particular, the focus will be on \cub models'
uncertainty parameter and its interpretation as a preliminary measure of
heterogeneity, by introducing membership, non-membership and uncertainty
functions in the more general framework of Intuitionistic Fuzzy Sets. Our
proposal is discussed on the basis of the Evaluation of Orientation Services
survey collected at University of Naples Federico II.Comment: 10 pages, invited contribution at SIS2016 (Salerno, Italy), in
SIS2016 proceeding
Higher Spin AdS/CFT at One Loop
Following arXiv:1308.2337, we carry out one loop tests of higher spin
AdS/CFT correspondences for . The Vasiliev theories in
AdS, which contain each integer spin once, are related to the
singlet sector of the -dimensional CFT of free complex scalar fields;
the minimal theories containing each even spin once -- to the singlet
sector of the CFT of free real scalar fields. Using analytic continuation
of higher spin zeta functions, which naturally regulate the spin sums, we
calculate one loop vacuum energies in Euclidean AdS. In even we
compare the result with the correction to the -coefficient of the
Weyl anomaly; in odd -- with the correction to the free energy
on the -dimensional sphere. For the theories of integer spins, the
correction vanishes in agreement with the CFT of free complex scalars. For
the minimal theories, the correction always equals the contribution of one real
conformal scalar field in dimensions. As explained in arXiv:1308.2337, this
result may agree with the singlet sector of the theory of real
scalar fields, provided the coupling constant in the higher spin theory is
identified as . Our calculations in even are closely
related to finding the regularized -anomalies of conformal higher spin
theories. In each even we identify two such theories with vanishing
-anomaly: a theory of all integer spins, and a theory of all even spins
coupled to a complex conformal scalar. We also discuss an interacting UV fixed
point in obtained from the free scalar theory via an irrelevant
double-trace quartic interaction. This interacting large theory is dual to
the Vasiliev theory in AdS where the bulk scalar is quantized with the
alternate boundary condition.Comment: 35 pages. v2: minor improvement
Theory of Optical Nonlocality in Polar Dielectrics
Sub-wavelength confinement of mid-infrared light can be achieved exploiting
the metal-like optical response of polar dielectric crystals in their
Reststrahlen spectral region, where they support evanescent modes termed
surface phonon polaritons. In the past few years the investigation of phonon
polaritons localised in nanoresonators and layered heterostructures has enjoyed
remarkable success, highlighting them as a promising platform for mid-infrared
nanophotonic applications. Here we prove that the standard local dielectric
description of phonon polaritons in nanometric objects fails due to the
nonlocal nature of the phonon response and we develop the corresponding
nonlocal theory. Application of our general theory to both dielectric
nanospheres and thin films demonstrates that polar dielectrics exhibit a rich
nonlocal phenomenology, qualitatively different from the one of plasmonic
systems, due to the negative dispersion of phononic optical modes.Comment: 13 pages, 6 figure
Theory of four-wave-mixing in phonon polaritons
Third order anharmonic scattering in light-matter systems can drive a wide
variety of practical and physically interesting processes from lasing to
polariton condensation. Motivated by recent experimental results in the
nonlinear optics of localised phonon polaritons, in this Letter we develop a
quantum theory capable of describing four-wave mixing in arbitrarily
inhomogeneous photonic environments. Using it we investigate Kerr
self-interaction and parametric scattering of surface and localised phonon
polaritons, showing both processes to be within experimental reach.Comment: 20 pages, 3 figure
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