183 research outputs found
Convergence of three-dimensional loop-erased random walk in the natural parametrization
In this work, we consider loop-erased random walk (LERW) and its scaling
limit in three dimensions, and prove that 3D LERW parametrized by renormalized
length converges to its scaling limit parametrized by some suitable measure
with respect to the uniform convergence topology in the lattice size scaling
limit. Our result improves the previous work of Gady Kozma (Acta Math.
199(1):29-152), which shows that the rescaled trace of 3D LERW converges weakly
to a random compact set with respect to the Hausdorff distance.Comment: 74 pages, 3 figure
Parity Violation of Gravitons in the CMB Bispectrum
We investigate the cosmic microwave background (CMB) bispectra of the
intensity (temperature) and polarization modes induced by the graviton
non-Gaussianities, which arise from the parity-conserving and parity-violating
Weyl cubic terms with time-dependent coupling. By considering the
time-dependent coupling, we find that even in the exact de Sitter space time,
the parity violation still appears in the three-point function of the
primordial gravitational waves and could become large. Through the estimation
of the CMB bispectra, we demonstrate that the signals generated from the
parity-conserving and parity-violating terms appear in completely different
configurations of multipoles. For example, the parity-conserving
non-Gaussianity induces the nonzero CMB temperature bispectrum in the
configuration with and, while due to the
parity-violating non-Gaussianity, the CMB temperature bispectrum also appears
for . This signal is just good evidence of the
parity violation in the non-Gaussianity of primordial gravitational waves. We
find that the shape of this non-Gaussianity is similar to the so-called
equilateral one and the amplitudes of these spectra at large scale are roughly
estimated as , where is an energy scale that
sets the magnitude of the Weyl cubic terms (higher derivative corrections) and
is a tensor-to-scalar ratio. Taking the limit for the nonlinearity
parameter of the equilateral type as , we can obtain
a bound as , assuming .Comment: 23 pages, 3 figures. Accepted for publication in PTP. Version 3
includes errata in Fig.
Volume and heat kernel fluctuations for the three-dimensional uniform spanning tree
Let be the uniform spanning tree on . We show
the occurrence of log-logarithmic fluctuations around the leading order for the
volume of intrinsic balls in . As an application, we obtain
similar fluctuations for the quenched heat kernel of the simple random walk on
.Comment: 39 pages, 20 figure
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