1,587 research outputs found
The energy of a deterministic Loewner chain: Reversibility and interpretation via SLE
We study some features of the energy of a deterministic chordal Loewner
chain, which is defined as the Dirichlet energy of its driving function. In
particular, using an interpretation of this energy as a large deviation rate
function for SLE as tends to 0 and the known reversibility of
the SLE curves for small , we show that the energy of a
deterministic curve from one boundary point A of a simply connected domain D to
another boundary point B, is equal to the energy of its time-reversal ie. of
the same curve but viewed as going from B to A in D.Comment: 28 pages, 5 figures, minor changes in Sec. 2.2., to appear in J.
Europ. Math. So
A note on Loewner energy, conformal restriction and Werner's measure on self-avoiding loops
In this note, we establish an expression of the Loewner energy of a Jordan
curve on the Riemann sphere in terms of Werner's measure on simple loops of
SLE type. The proof is based on a formula for the change of the Loewner
energy under a conformal map that is reminiscent of the restriction properties
derived for SLE processes.Comment: 11 pages, 3 figure
A new inflationary Universe scenario with inhomogeneous quantum vacuum
We investigate the quantum vacuum and find the fluctuations can lead to the
inhomogeneous quantum vacuum. We find that the vacuum fluctuations can
significantly influence the cosmological inhomogeneity, which is different from
what previously expected. By introducing the modified Green's function, we
reach a new inflationary scenario which can explain why the Universe is still
expanding without slowing down. We also calculate the tunneling amplitude of
the Universe based on the inhomogeneous vacuum. We find that the inhomogeneity
can lead to the penetration of the universe over the potential barrier faster
than previously thought.Comment: 16 pages, 9 figure
Interplay between Loewner and Dirichlet energies via conformal welding and flow-lines
The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner
driving term. It is finite if and only if the curve is a Weil-Petersson
quasicircle. In this paper, we describe cutting and welding operations on
finite Dirichlet energy functions defined in the plane, allowing expression of
the Loewner energy in terms of Dirichlet energy dissipation. We show that the
Loewner energy of a unit vector field flow-line is equal to the Dirichlet
energy of the harmonically extended winding. We also give an identity involving
a complex-valued function of finite Dirichlet energy that expresses the welding
and flow-line identities simultaneously. As applications, we prove that
arclength isometric welding of two domains is sub-additive in the energy, and
that the energy of equipotentials in a simply connected domain is monotone. Our
main identities can be viewed as action functional analogs of both the welding
and flow-line couplings of Schramm-Loewner evolution curves with the Gaussian
free field.Comment: 28 pages, 3 figures. Minor revision according to referees' repor
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