1,057 research outputs found
Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function
It is known that the moments of the maximum value of a one-dimensional
conditional Brownian motion, the three-dimensional Bessel bridge with duration
1 started from the origin, are expressed using the Riemann zeta function. We
consider a system of two Bessel bridges, in which noncolliding condition is
imposed. We show that the moments of the maximum value is then expressed using
the double Dirichlet series, or using the integrals of products of the Jacobi
theta functions and its derivatives. Since the present system will be provided
as a diffusion scaling limit of a version of vicious walker model, the ensemble
of 2-watermelons with a wall, the dominant terms in long-time asymptotics of
moments of height of 2-watermelons are completely determined. For the height of
2-watermelons with a wall, the average value was recently studied by Fulmek by
a method of enumerative combinatorics.Comment: v2: LaTeX, 19 pages, 2 figures, minor corrections made for
publication in J. Stat. Phy
System of Complex Brownian Motions Associated with the O'Connell Process
The O'Connell process is a softened version (a geometric lifting with a
parameter ) of the noncolliding Brownian motion such that neighboring
particles can change the order of positions in one dimension within the
characteristic length . This process is not determinantal. Under a special
entrance law, however, Borodin and Corwin gave a Fredholm determinant
expression for the expectation of an observable, which is a softening of an
indicator of a particle position. We rewrite their integral kernel to a form
similar to the correlation kernels of determinantal processes and show, if the
number of particles is , the rank of the matrix of the Fredholm determinant
is . Then we give a representation for the quantity by using an -particle
system of complex Brownian motions (CBMs). The complex function, which gives
the determinantal expression to the weight of CBM paths, is not entire, but in
the combinatorial limit it becomes an entire function providing
conformal martingales and the CBM representation for the noncolliding Brownian
motion is recovered.Comment: v3: AMS_LaTeX, 25 pages, no figure, minor corrections made for
publication in J. Stat. Phy
Determinantal process starting from an orthogonal symmetry is a Pfaffian process
When the number of particles is finite, the noncolliding Brownian motion
(BM) and the noncolliding squared Bessel process with index
(BESQ) are determinantal processes for arbitrary fixed initial
configurations. In the present paper we prove that, if initial configurations
are distributed with orthogonal symmetry, they are Pfaffian processes in the
sense that any multitime correlation functions are expressed by Pfaffians. The
skew-symmetric matrix-valued correlation kernels of the Pfaffians
processes are explicitly obtained by the equivalence between the noncolliding
BM and an appropriate dilatation of a time reversal of the temporally
inhomogeneous version of noncolliding BM with finite duration in which all
particles start from the origin, , and by the equivalence between
the noncolliding BESQ and that of the noncolliding squared
generalized meander starting from .Comment: v2: AMS-LaTeX, 17 pages, no figure, corrections made for publication
in J.Stat.Phy
Colloquium: Physics of optical lattice clocks
Recently invented and demonstrated, optical lattice clocks hold great promise
for improving the precision of modern timekeeping. These clocks aim at the
10^-18 fractional accuracy, which translates into a clock that would neither
lose or gain a fraction of a second over an estimated age of the Universe. In
these clocks, millions of atoms are trapped and interrogated simultaneously,
dramatically improving clock stability. Here we discuss the principles of
operation of these clocks and, in particular, a novel concept of "magic"
trapping of atoms in optical lattices. We also highlight recently proposed
microwave lattice clocks and several applications that employ the optical
lattice clocks as a platform for precision measurements and quantum information
processing.Comment: 18 pages, 15 figure
Noncolliding Squared Bessel Processes
We consider a particle system of the squared Bessel processes with index conditioned never to collide with each other, in which if
the origin is assumed to be reflecting. When the number of particles is finite,
we prove for any fixed initial configuration that this noncolliding diffusion
process is determinantal in the sense that any multitime correlation function
is given by a determinant with a continuous kernel called the correlation
kernel. When the number of particles is infinite, we give sufficient conditions
for initial configurations so that the system is well defined. There the
process with an infinite number of particles is determinantal and the
correlation kernel is expressed using an entire function represented by the
Weierstrass canonical product, whose zeros on the positive part of the real
axis are given by the particle-positions in the initial configuration. From the
class of infinite-particle initial configurations satisfying our conditions, we
report one example in detail, which is a fixed configuration such that every
point of the square of positive zero of the Bessel function is
occupied by one particle. The process starting from this initial configuration
shows a relaxation phenomenon converging to the stationary process, which is
determinantal with the extended Bessel kernel, in the long-term limit.Comment: v3: LaTeX2e, 26 pages, no figure, corrections made for publication in
J. Stat. Phy
Lifetime measurement of the ^3P_2 metastable state of strontium atoms
We have measured the lifetime of the 5s5p ^3P_2 metastable state of strontium
atoms by magneto-optically trapping the decayed atoms to the ground state,
which allowed sensitive detection of the rare decay events. We found that the
blackbody radiation-induced decay was the dominant decay channel for the state
at T = 300 K. The lifetime was determined to be 500^{+280}_{-130} s in the
limit of zero temperature.Comment: 4 pages, 3 figures, submitted to Physical Review Letter
First Measurement of Muon Neutrino Charged Current Quasielastic (CCQE) Double Differential Cross Section
Using a high statistics sample of muon neutrino charged current quasielastic
(CCQE) events, we report the first measurement of the double differential cross
section as a function of muon energy and angle for this process. The result
features reduced model dependence and supplies the most complete information on
neutrino CCQE scattering to date. Measurements of the absolute cross section as
a function of neutrino energy and the single differential cross section as a
function of 4-momentum transfer squared are also provided, largely to
facilitate comparison with prior measurements. This data is of particular use
for understanding the axial-vector form factor of the nucleon as well as
improving the simulation of low energy neutrino interactions on nuclear
targets, which is of particular relevance for experiments searching for
neutrino oscillations.Comment: 6 pages, 6 figures, Proceedings of the 6th International Workshop on
Neutrino-Nucleus Interactions in the Few-GeV Region (NuInt09
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