pinecone/core-2020-05-10-deduplication huggingface dataset:
A volumetric Penrose inequality for conformally flat manifolds

We consider asymptotically flat Riemannian manifolds with nonnegative scalar
curvature that are conformal to $\R^{n}\setminus \Omega, n\ge 3$, and so that
their boundary is a minimal hypersurface. (Here, $\Omega\subset \R^{n}$ is open
bounded with smooth mean-convex boundary.) We prove that the ADM mass of any
such manifold is bounded below by $(V/\beta_{n})^{(n-2)/n}$, where $V$ is the
Euclidean volume of $\Omega$ and $\beta_{n}$ is the volume of the Euclidean
unit $n$-ball. This gives a partial proof to a conjecture of Bray and Iga
\cite{brayiga}. Surprisingly, we do not require the boundary to be outermost.Comment: 7 page
Asymptotes in SU(2) Recoupling Theory: Wigner Matrices, $3j$ Symbols,
  and Character Localization

In this paper we employ a novel technique combining the Euler Maclaurin
formula with the saddle point approximation method to obtain the asymptotic
behavior (in the limit of large representation index $J$) of generic Wigner
matrix elements $D^{J}_{MM'}(g)$. We use this result to derive asymptotic
formulae for the character $\chi^J(g)$ of an SU(2) group element and for
Wigner's $3j$ symbol. Surprisingly, given that we perform five successive
layers of approximations, the asymptotic formula we obtain for $\chi^J(g)$ is
in fact exact. This result provides a non trivial example of a
Duistermaat-Heckman like localization property for discrete sums.Comment: 36 pages, 3 figure
Linear perturbations for the vacuum axisymmetric Einstein equations

In axial symmetry, there is a gauge for Einstein equations such that the
total mass of the spacetime can be written as a conserved, positive definite,
integral on the spacelike slices. This property is expected to play an
important role in the global evolution. In this gauge the equations reduce to a
coupled hyperbolic-elliptic system which is formally singular at the axis. Due
to the rather peculiar properties of the system, the local in time existence
has proved to resist analysis by standard methods. To analyze the principal
part of the equations, which may represent the main source of the difficulties,
we study linear perturbation around the flat Minkowski solution in this gauge.
In this article we solve this linearized system explicitly in terms of integral
transformations in a remarkable simple form. This representation is well suited
to obtain useful estimates to apply in the non-linear case.Comment: 13 pages. We suppressed the statements about decay at infinity. The
  proofs of these statements were incomplete. The complete proofs will require
  extensive technical analysis. We will studied this in a subsequent work. We
  also have rewritten the introduction and slighted changed the titl
The Cauchy problem on a characteristic cone for the Einstein equations
  in arbitrary dimensions

We derive explicit formulae for a set of constraints for the Einstein
equations on a null hypersurface, in arbitrary dimensions. We solve these
constraints and show that they provide necessary and sufficient conditions so
that a spacetime solution of the Cauchy problem on a characteristic cone for
the hyperbolic system of the reduced Einstein equations in wave-map gauge also
satisfies the full Einstein equations. We prove a geometric uniqueness theorem
for this Cauchy problem in the vacuum case.Comment: 83 pages, 1 figur
Further restrictions on the topology of stationary black holes in five
  dimensions

We place further restriction on the possible topology of stationary
asymptotically flat vacuum black holes in 5 spacetime dimensions. We prove that
the horizon manifold can be either a connected sum of Lens spaces and "handles"
$S^1 \times S^2$, or the quotient of $S^3$ by certain finite groups of
isometries (with no "handles"). The resulting horizon topologies include Prism
manifolds and quotients of the Poincare homology sphere. We also show that the
topology of the domain of outer communication is a cartesian product of the
time direction with a finite connected sum of $\mathbb R^4,S^2 \times S^2$'s
and $CP^2$'s, minus the black hole itself. We do not assume the existence of
any Killing vector beside the asymptotically timelike one required by
definition for stationarity.Comment: LaTex, 22 pages, 9 figure
Fermi coordinates, simultaneity, and expanding space in Robertson-Walker
  cosmologies

Explicit Fermi coordinates are given for geodesic observers comoving with the
Hubble flow in expanding Robertson-Walker spacetimes, along with exact
expressions for the metric tensors in Fermi coordinates. For the case of non
inflationary cosmologies, it is shown that Fermi coordinate charts are global,
and space-time is foliated by space slices of constant Fermi (proper) time that
have finite extent. A universal upper bound for the proper radius of any leaf
of the foliation, i.e., for the proper radius of the spatial universe at any
fixed time of the geodesic observer, is given. A general expression is derived
for the geometrically defined Fermi relative velocity of a test particle (e.g.
a galaxy) comoving with the Hubble flow away from the observer. Least upper
bounds of superluminal recessional Fermi velocities are given for spacetimes
whose scale factors follow power laws, including matter-dominated and
radiation-dominated cosmologies. Exact expressions for the proper radius of any
leaf of the foliation for this same class of spacetimes are given. It is shown
that the radii increase linearly with proper time of the observer moving with
the Hubble flow. These results are applied to particular cosmological models.Comment: This revised version corrects minor typo
On Breakdown Criteria for Nonvacuum Einstein Equations

The recent "breakdown criterion" result of S. Klainerman and I. Rodnianski
stated roughly that an Einstein-vacuum spacetime, given as a CMC foliation, can
be further extended in time if the second fundamental form and the derivative
of the lapse of the foliation are uniformly bounded. This theorem and its proof
were extended to Einstein-scalar and Einstein-Maxwell spacetimes in the
author's Ph.D. thesis. In this paper, we state the main results of the thesis,
and we summarize and discuss their proofs. In particular, we will discuss the
various issues resulting from nontrivial Ricci curvature and the coupling
between the Einstein and the field equations.Comment: 62 pages This version: corrected minor typos, expanded Section 6
  (geometry of null cones
Divergences in quantum field theory on the noncommutative
  two-dimensional Minkowski space with Grosse-Wulkenhaar potential

Quantum field theory on the noncommutative two-dimensional Minkowski space
with Grosse-Wulkenhaar potential is discussed in two ways: In terms of a
continuous set of generalised eigenfunctions of the wave operator, and directly
in position space. In both settings, we find a new type of divergence in planar
graphs. It is present at and above the self-dual point. This new kind of
divergence might make the construction of a Minkowski space version of the
Grosse-Wulkenhaar model impossible.Comment: 26 pages, published versio
Areas and volumes for null cones

Motivated by recent work of Choquet-Bruhat, Chrusciel, and Martin-Garcia, we
prove monotonicity properties and comparison results for the area of slices of
the null cone of a point in a Lorentzian manifold. We also prove volume
comparison results for subsets of the null cone analogous to the Bishop-Gromov
relative volume monotonicity theorem and Guenther's volume comparison theorem.
We briefly discuss how these estimates may be used to control the null second
fundamental form of slices of the null cone in Ricci-flat Lorentzian
four-manifolds with null curvature bounded above.Comment: 16 pages, no figures. Typos fixed. One garbled proof corrected.
  Published versio
Ground States in the Spin Boson Model

We prove that the Hamiltonian of the model describing a spin which is
linearly coupled to a field of relativistic and massless bosons, also known as
the spin-boson model, admits a ground state for small values of the coupling
constant lambda. We show that the ground state energy is an analytic function
of lambda and that the corresponding ground state can also be chosen to be an
analytic function of lambda. No infrared regularization is imposed. Our proof
is based on a modified version of the BFS operator theoretic renormalization
analysis. Moreover, using a positivity argument we prove that the ground state
of the spin-boson model is unique. We show that the expansion coefficients of
the ground state and the ground state energy can be calculated using regular
analytic perturbation theory
Quantization of edge currents along magnetic barriers and magnetic
  guides

We investigate the edge conductance of particles submitted to an Iwatsuka
magnetic field, playing the role of a purely magnetic barrier. We also consider
magnetic guides generated by generalized Iwatsuka potentials. In both cases we
prove quantization of the edge conductance. Next, we consider magnetic
perturbations of such magnetic barriers or guides, and prove stability of the
quantized value of the edge conductance. Further, we establish a sum rule for
edge conductances. Regularization within the context of disordered systems is
discussed as well.Comment: 25 page
A class of dust-like self-similar solutions of the massless
  Einstein-Vlasov system

In this paper the existence of a class of self-similar solutions of the
Einstein-Vlasov system is proved. The initial data for these solutions are not
smooth, with their particle density being supported in a submanifold of
codimension one. They can be thought of as intermediate between smooth
solutions of the Einstein-Vlasov system and dust. The motivation for studying
them is to obtain insights into possible violation of weak cosmic censorship by
solutions of the Einstein-Vlasov system. By assuming a suitable form of the
unknowns it is shown that the existence question can be reduced to that of the
existence of a certain type of solution of a four-dimensional system of
ordinary differential equations depending on two parameters. This solution
starts at a particular point $P_0$ and converges to a stationary solution $P_1$
as the independent variable tends to infinity. The existence proof is based on
a shooting argument and involves relating the dynamics of solutions of the
four-dimensional system to that of solutions of certain two- and
three-dimensional systems obtained from it by limiting processes.Comment: 47 page
Thermal effects in gravitational Hartree systems

We consider the non-relativistic Hartree model in the gravitational case,
i.e. with attractive Coulomb-Newton interaction. For a given mass, we construct
stationary states with non-zero temperature by minimizing the corresponding
free energy functional. It is proved that minimizers exist if and only if the
temperature of the system is below a certain threshold(possibly infinite),
which itself depends on the specific choice of the entropy functional. We also
investigate whether the corresponding minimizers are mixed or pure quantum
states and characterize a positive critical temperature above which mixed
states appear
Critical points of Wang-Yau quasi-local energy

In this paper, we prove the following theorem regarding the Wang-Yau
quasi-local energy of a spacelike two-surface in a spacetime: Let $\Sigma$ be a
boundary component of some compact, time-symmetric, spacelike hypersurface
$\Omega$ in a time-oriented spacetime $N$ satisfying the dominant energy
condition. Suppose the induced metric on $\Sigma$ has positive Gaussian
curvature and all boundary components of $\Omega$ have positive mean curvature.
Suppose $H \le H_0$ where $H$ is the mean curvature of $\Sigma$ in $\Omega$ and
$H_0$ is the mean curvature of $\Sigma$ when isometrically embedded in $R^3$.
If $\Omega$ is not isometric to a domain in $R^3$, then 1. the Brown-York mass
of $\Sigma$ in $\Omega$ is a strict local minimum of the Wang-Yau quasi-local
energy of $\Sigma$, 2. on a small perturbation $\tilde{\Sigma}$ of $\Sigma$ in
$N$, there exists a critical point of the Wang-Yau quasi-local energy of
$\tilde{\Sigma}$.Comment: substantially revised, main theorem replaced, Section 3 adde
The Ground State Energy of The Massless Spin-Boson Model

We provide an explicit combinatorial expansion for the ground state energy of
the massless spin-Boson model as a power series in the coupling parameter. Our
method uses the technique of cluster expansion in constructive quantum field
theory and takes as a starting point the functional integral representation and
its reduction to an Ising model on the real line with long range interactions.
We prove the analyticity of our expansion and provide an explicit lower bound
on the radius of convergence. We do not need multiscale nor renormalization
group analysis. A connection to the loop-erased random walk is indicated.Comment: 32 pages, 4 figures, a remark and references added, typos corrected,
  minor computational errors correcte
Quantum Diffusion and Delocalization for Band Matrices with General
  Distribution

We consider Hermitian and symmetric random band matrices $H$ in $d \geq 1$
dimensions. The matrix elements $H_{xy}$, indexed by $x,y \in \Lambda \subset
\Z^d$, are independent and their variances satisfy $\sigma_{xy}^2:=\E
\abs{H_{xy}}^2 = W^{-d} f((x - y)/W)$ for some probability density $f$. We
assume that the law of each matrix element $H_{xy}$ is symmetric and exhibits
subexponential decay. We prove that the time evolution of a quantum particle
subject to the Hamiltonian $H$ is diffusive on time scales $t\ll W^{d/3}$. We
also show that the localization length of the eigenvectors of $H$ is larger
than a factor $W^{d/6}$ times the band width $W$. All results are uniform in
the size $\abs{\Lambda}$ of the matrix. This extends our recent result
\cite{erdosknowles} to general band matrices. As another consequence of our
proof we show that, for a larger class of random matrices satisfying
$\sum_x\sigma_{xy}^2=1$ for all $y$, the largest eigenvalue of $H$ is bounded
with high probability by $2 + M^{-2/3 + \epsilon}$ for any $\epsilon > 0$,
where $M \deq 1 / (\max_{x,y} \sigma_{xy}^2)$.Comment: Corrected typos and some inaccuracies in appendix
Spacelike localization of long-range fields in a model of asymptotic
  electrodynamics

A previously proposed algebra of asymptotic fields in quantum electrodynamics
is formulated as a net of algebras localized in regions which in general have
unbounded spacelike extension. Electromagnetic fields may be localized in
`symmetrical spacelike cones', but there are strong indications this is not
possible in the present model for charged fields, which have tails extending in
all space directions. Nevertheless, products of appropriately `dressed' fermion
fields (with compensating charges) yield bi-localized observables.Comment: 29 pages, accepted for publication in Annales Henri Poincar\'
On Quantum Markov Chains on Cayley tree II: Phase transitions for the
  associated chain with XY-model on the Cayley tree of order three

In the present paper we study forward Quantum Markov Chains (QMC) defined on
a Cayley tree. Using the tree structure of graphs, we give a construction of
quantum Markov chains on a Cayley tree. By means of such constructions we prove
the existence of a phase transition for the XY-model on a Cayley tree of order
three in QMC scheme. By the phase transition we mean the existence of two now
quasi equivalent QMC for the given family of interaction operators
$\{K_{<x,y>}\}$.Comment: 34 pages, 1 figur
Resolvent estimates for normally hyperbolic trapped sets

We give pole free strips and estimates for resolvents of semiclassical
operators which, on the level of the classical flow, have normally hyperbolic
smooth trapped sets of codimension two in phase space. Such trapped sets are
structurally stable and our motivation comes partly from considering the wave
equation for Kerr black holes and their perturbations, whose trapped sets have
precisely this structure. We give applications including local smoothing
effects with epsilon derivative loss for the Schr\"odinger propagator as well
as local energy decay results for the wave equation.Comment: Further changes to erratum correcting small problems with Section 3.5
  and Lemma 4.1; this now also corrects hypotheses, explicitly requiring
  trapped set to be symplectic. Erratum follows references in this versio
Kochen-Specker Sets and Generalized Orthoarguesian Equations

Every set (finite or infinite) of quantum vectors (states) satisfies
generalized orthoarguesian equations ($n$OA). We consider two 3-dim
Kochen-Specker (KS) sets of vectors and show how each of them should be
represented by means of a Hasse diagram---a lattice, an algebra of subspaces of
a Hilbert space--that contains rays and planes determined by the vectors so as
to satisfy $n$OA. That also shows why they cannot be represented by a special
kind of Hasse diagram called a Greechie diagram, as has been erroneously done
in the literature. One of the KS sets (Peres') is an example of a lattice in
which 6OA pass and 7OA fails, and that closes an open question of whether the
7oa class of lattices properly contains the 6oa class. This result is important
because it provides additional evidence that our previously given proof of noa
=< (n+1)oa can be extended to proper inclusion noa < (n+1)oa and that nOA form
an infinite sequence of successively stronger equations.Comment: 16 pages and 5 figure
Approximate KMS states for scalar and spinor fields in
  Friedmann-Robertson-Walker spacetimes

We construct and discuss Hadamard states for both scalar and Dirac spinor
fields in a large class of spatially flat Friedmann-Robertson-Walker spacetimes
characterised by an initial phase either of exponential or of power-law
expansion. The states we obtain can be interpreted as being in thermal
equilibrium at the time when the scale factor a has a specific value a=a_0. In
the case a_0=0, these states fulfil a strict KMS condition on the boundary of
the spacetime, which is either a cosmological horizon, or a Big Bang
hypersurface. Furthermore, in the conformally invariant case, they are
conformal KMS states on the full spacetime. However, they provide a natural
notion of an approximate KMS state also in the remaining cases, especially for
massive fields. On the technical side, our results are based on a
bulk-to-boundary reconstruction technique already successfully applied in the
scalar case and here proven to be suitable also for spinor fields. The
potential applications of the states we find range over a broad spectrum, but
they appear to be suited to discuss in particular thermal phenomena such as the
cosmic neutrino background or the quantum state of dark matter.Comment: 42 page
Anderson localization for a class of models with a sign-indefinite
  single-site potential via fractional moment method

A technically convenient signature of Anderson localization is exponential
decay of the fractional moments of the Green function within appropriate energy
ranges. We consider a random Hamiltonian on a lattice whose randomness is
generated by the sign-indefinite single-site potential, which is however
sign-definite at the boundary of its support. For this class of Anderson
operators we establish a finite-volume criterion which implies that above
mentioned the fractional moment decay property holds. This constructive
criterion is satisfied at typical perturbative regimes, e. g. at spectral
boundaries which satisfy 'Lifshitz tail estimates' on the density of states and
for sufficiently strong disorder. We also show how the fractional moment method
facilitates the proof of exponential (spectral) localization for such random
potentials.Comment: 29 pages, 1 figure, to appear in AH
From constructive field theory to fractional stochastic calculus. (II)
  Constructive proof of convergence for the L\'evy area of fractional Brownian
  motion with Hurst index $\alpha\in(1/8,1/4)$

{Let $B=(B_1(t),...,B_d(t))$ be a $d$-dimensional fractional Brownian motion
with Hurst index $\alpha<1/4$, or more generally a Gaussian process whose paths
have the same local regularity. Defining properly iterated integrals of $B$ is
a difficult task because of the low H\"older regularity index of its paths. Yet
rough path theory shows it is the key to the construction of a stochastic
calculus with respect to $B$, or to solving differential equations driven by
$B$.
  We intend to show in a series of papers how to desingularize iterated
integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure
defined by a limit in law procedure. Convergence is proved by using "standard"
tools of constructive field theory, in particular cluster expansions and
renormalization. These powerful tools allow optimal estimates, and call for an
extension of Gaussian tools such as for instance the Malliavin calculus.
  After a first introductory paper \cite{MagUnt1}, this one concentrates on the
details of the constructive proof of convergence for second-order iterated
integrals, also known as L\'evy area
Random skew plane partitions with a piecewise periodic back wall

Random skew plane partitions of large size distributed according to an
appropriately scaled Schur process develop limit shapes. In the present work we
consider the limit of large random skew plane partitions where the inner
boundary approaches a piecewise linear curve with non-lattice slopes,
describing the limit shape and the local fluctuations in various regions. This
analysis is fairly similar to that in [OR2], but we do find some new behavior.
For instance, the boundary of the limit shape is now a single smooth (not
algebraic) curve, whereas the boundary in [OR2] is singular. We also observe
the bead process introduced in [B] appearing in the asymptotics at the top of
the limit shape.Comment: 24 pages. This version to appear in Annales Henri Poincar
Asymptotic simplicity and static data

The present article considers time symmetric initial data sets for the vacuum
Einstein field equations which in a neighbourhood of infinity have the same
massless part as that of some static initial data set. It is shown that the
solutions to the regular finite initial value problem at spatial infinity for
this class of initial data sets extend smoothly through the critical sets where
null infinity touches spatial infinity if and only if the initial data sets
coincide with static data in a neighbourhood of infinity. This result
highlights the special role played by static data among the class of initial
data sets for the Einstein field equations whose development gives rise to a
spacetime with a smooth conformal compactification at null infinity.Comment: 25 page
Dynamics of nodal points and the nodal count on a family of quantum
  graphs

We investigate the properties of the zeros of the eigenfunctions on quantum
graphs (metric graphs with a Schr\"odinger-type differential operator). Using
tools such as scattering approach and eigenvalue interlacing inequalities we
derive several formulas relating the number of the zeros of the n-th
eigenfunction to the spectrum of the graph and of some of its subgraphs. In a
special case of the so-called dihedral graph we prove an explicit formula that
only uses the lengths of the edges, entirely bypassing the information about
the graph's eigenvalues. The results are explained from the point of view of
the dynamics of zeros of the solutions to the scattering problem.Comment: 34 pages, 12 figure