10,947 research outputs found
On the Symmetry Foundation of Double Soft Theorems
Double-soft theorems, like its single-soft counterparts, arises from the
underlying symmetry principles that constrain the interactions of massless
particles. While single soft theorems can be derived in a non-perturbative
fashion by employing current algebras, recent attempts of extending such an
approach to known double soft theorems has been met with difficulties. In this
work, we have traced the difficulty to two inequivalent expansion schemes,
depending on whether the soft limit is taken asymmetrically or symmetrically,
which we denote as type A and B respectively. We show that soft-behaviour for
type A scheme can simply be derived from single soft theorems, and are thus
non-preturbatively protected. For type B, the information of the four-point
vertex is required to determine the corresponding soft theorems, and thus are
in general not protected. This argument can be readily extended to general
multi-soft theorems. We also ask whether unitarity can be emergent from
locality together with the two kinds of soft theorems, which has not been fully
investigated before.Comment: 45 pages, 7 figure
Anomalous scaling due to correlations: Limit theorems and self-similar processes
We derive theorems which outline explicit mechanisms by which anomalous
scaling for the probability density function of the sum of many correlated
random variables asymptotically prevails. The results characterize general
anomalous scaling forms, justify their universal character, and specify
universality domains in the spaces of joint probability density functions of
the summand variables. These density functions are assumed to be invariant
under arbitrary permutations of their arguments. Examples from the theory of
critical phenomena are discussed. The novel notion of stability implied by the
limit theorems also allows us to define sequences of random variables whose sum
satisfies anomalous scaling for any finite number of summands. If regarded as
developing in time, the stochastic processes described by these variables are
non-Markovian generalizations of Gaussian processes with uncorrelated
increments, and provide, e.g., explicit realizations of a recently proposed
model of index evolution in finance.Comment: Through text revision. 15 pages, 3 figure
On bi-free De Finetti theorems
We investigate possible generalizations of the de Finetti theorem to bi-free
probability. We first introduce a twisted action of the quantum permutation
groups corresponding to the combinatorics of bi-freeness. We then study
properties of families of pairs of variables which are invariant under this
action, both in the bi-noncommutative setting and in the usual noncommutative
setting. We do not have a completely satisfying analogue of the de Finetti
theorem, but we have partial results leading the way. We end with suggestions
concerning the symmetries of a potential notion of n-freeness.Comment: 16 pages. Major rewriting. In the first version the main theorem was
stated through an embedding into a B-B-noncommutative probability space
making it much weaker than what the proof really contains. It has therefore
been split into two independent statements clarifying how far we are able to
extend the de Finetti theorem to the bi-free settin
Hidden Conformal Symmetry in Tree-Level Graviton Scattering
We argue that the scattering of gravitons in ordinary Einstein gravity
possesses a hidden conformal symmetry at tree level in any number of
dimensions. The presence of this conformal symmetry is indicated by the dilaton
soft theorem in string theory, and it is reminiscent of the conformal
invariance of gluon tree-level amplitudes in four dimensions. To motivate the
underlying prescription, we demonstrate that formulating the conformal symmetry
of gluon amplitudes in terms of momenta and polarization vectors requires
manifest reversal and cyclic symmetry. Similarly, our formulation of the
conformal symmetry of graviton amplitudes relies on a manifestly permutation
symmetric form of the amplitude function.Comment: 35 pages, 3 figure
De Finetti theorems for a Boolean analogue of easy quantum groups
We show an organized form of quantum de Finetti theorem for Boolean
independence. We define a Boolean analogue of easy quantum groups for the
categories of interval partitions, which is a family of sequences of quantum
semigroups.
We construct the Haar states on those quantum semigroups. The proof of our de
Finetti theorem is based on the analysis of the Haar states.
[Modified]Definition of the Boolean quantum semigroups on categories of
interval partitions
[Delete]Classification of categories of interval partitions
[Add]Proof of the positiveness of the Haar functionals (in particular they
are Haar states)Comment: 26 page
De Finetti theorems for easy quantum groups
We study sequences of noncommutative random variables which are invariant
under "quantum transformations" coming from an orthogonal quantum group
satisfying the "easiness" condition axiomatized in our previous paper. For 10
easy quantum groups, we obtain de Finetti type theorems characterizing the
joint distribution of any infinite quantum invariant sequence. In particular,
we give a new and unified proof of the classical results of de Finetti and
Freedman for the easy groups S_n, O_n, which is based on the combinatorial
theory of cumulants. We also recover the free de Finetti theorem of K\"ostler
and Speicher, and the characterization of operator-valued free semicircular
families due to Curran. We consider also finite sequences, and prove an
approximation result in the spirit of Diaconis and Freedman.Comment: Published in at http://dx.doi.org/10.1214/10-AOP619 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On permutations of lacunary series
It is a well known fact that for periodic measurable and rapidly
increasing the sequence behaves like a
sequence of independent, identically distributed random variables. For example,
if is a periodic Lipschitz function, then satisfies
the central limit theorem, the law of the iterated logarithm and several
further limit theorems for i.i.d.\ random variables. Since an i.i.d.\ sequence
remains i.i.d.\ after any permutation of its terms, it is natural to expect
that the asymptotic properties of lacunary series are also
permutation-invariant. Recently, however, Fukuyama (2009) showed that a
rearrangement of the sequence can change substantially its
asymptotic behavior, a very surprising result. The purpose of the present paper
is to investigate this interesting phenomenon in detail and to give necessary
and sufficient criteria for the permutation-invariance of the CLT and LIL for
- …