964 research outputs found
On Bogomolny-Schmit conjecture
Bogomolny and Schmit proposed that the critical edge percolation on the
square lattice is a good model for the nodal domains of a random plane wave.
Based on this they made a conjecture about the number of nodal domains. Recent
computer experiments showed that the mean number of clusters per vertex and the
mean number of nodal domains per unit area are very close but different. Since
the original argument was mostly supported by numerics, it was believed that
the percolation model is wrong. In this paper we give some numerical evidence
in favour of the percolation model.Comment: 6 pages, 2 figures. To be published in Journal of Physics A:
Mathematical and Theoretica
Random conformal snowflakes
In many problems of classical analysis extremal configurations appear to
exhibit complicated fractal structure. This makes it much harder to describe
extremals and to attack such problems. Many of these problems are related to
the multifractal analysis of harmonic measure.
We argue that, searching for extremals in such problems, one should work with
random fractals rather than deterministic ones. We introduce a new class of
fractals random conformal snowflakes and investigate its properties developing
tools to estimate spectra and showing that extremals can be found in this
class. As an application we significantly improve known estimates from below on
the extremal behaviour of harmonic measure, showing how to constuct a rather
simple snowflake, which has a spectrum quite close to the conjectured extremal
value
A proof of factorization formula for critical percolation
We give mathematical proofs to a number of statements which appeared in the
series of papers by Kleban, Simmons and Ziff where they computed the
probabilities of several percolation crossing events.Comment: 13 pages, 1 figure. Version 2: introduction and some proofs expande
Two point function for critical points of a random plane wave
Random plane wave is conjectured to be a universal model for high-energy
eigenfunctions of the Laplace operator on generic compact Riemanian manifolds.
This is known to be true on average. In the present paper we discuss one of
important geometric observable: critical points. We first compute one-point
function for the critical point process, in particular we compute the expected
number of critical points inside any open set. After that we compute the
short-range asymptotic behaviour of the two-point function. This gives an
unexpected result that the second factorial moment of the number of critical
points in a small disc scales as the fourth power of the radius
Integral means spectrum of whole-plane SLE
We complete the mathematical analysis of the fine structure of harmonic
measure on SLE curves that was initiated by Beliaev and Smirnov, as described
by the averaged integral means spectrum. For the unbounded version of
whole-plane SLE as studied by Duplantier, Nguyen, Nguyen and Zinsmeister, and
Loutsenko and Yermolayeva, a phase transition has been shown to occur for high
enough moments from the bulk spectrum towards a novel spectrum related to the
point at infinity. For the bounded version of whole-plane SLE studied here, a
similar transition phenomenon, now associated with the SLE origin, is proved to
exist for low enough moments, but we show that it is superseded by the earlier
occurrence of the transition to the SLE tip spectrum.Comment: 14 pages, 1 figure; final versio
Field-induced decay dynamics in square-lattice antiferromagnet
Dynamical properties of the square-lattice Heisenberg antiferromagnet in
applied magnetic field are studied for arbitrary value S of the spin. Above the
threshold field for two-particle decays, the standard spin-wave theory yields
singular corrections to the excitation spectrum with logarithmic divergences
for certain momenta. We develop a self-consistent approximation applicable for
S >= 1, which avoids such singularities and provides regularized magnon decay
rates. Results for the dynamical structure factor obtained in this approach are
presented for S = 1 and S = 5/2.Comment: 12 pages, 11 figures, final versio
A Multiscale Guide to Brownian Motion
We revise the Levy's construction of Brownian motion as a simple though still
rigorous approach to operate with various Gaussian processes. A Brownian path
is explicitly constructed as a linear combination of wavelet-based "geometrical
features" at multiple length scales with random weights. Such a wavelet
representation gives a closed formula mapping of the unit interval onto the
functional space of Brownian paths. This formula elucidates many classical
results about Brownian motion (e.g., non-differentiability of its path),
providing intuitive feeling for non-mathematicians. The illustrative character
of the wavelet representation, along with the simple structure of the
underlying probability space, is different from the usual presentation of most
classical textbooks. Similar concepts are discussed for fractional Brownian
motion, Ornstein-Uhlenbeck process, Gaussian free field, and fractional
Gaussian fields. Wavelet representations and dyadic decompositions form the
basis of many highly efficient numerical methods to simulate Gaussian processes
and fields, including Brownian motion and other diffusive processes in
confining domains
Fermi-Bose mapping for one-dimensional Bose gases
One-dimensional Bose gases are considered, interacting either through the
hard-core potentials or through the contact delta potentials. Interest in these
gases gained momentum because of the recent experimental realization of
quasi-one-dimensional Bose gases in traps with tightly confined radial motion,
achieving the Tonks-Girardeau (TG) regime of strongly interacting atoms. For
such gases the Fermi-Bose mapping of wavefunctions is applicable. The aim of
the present communication is to give a brief survey of the problem and to
demonstrate the generality of this mapping by emphasizing that: (i) It is valid
for nonequilibrium wavefunctions, described by the time-dependent Schr\"odinger
equation, not merely for stationary wavefunctions. (ii) It gives the whole
spectrum of all excited states, not merely the ground state. (iii) It applies
to the Lieb-Liniger gas with the contact interaction, not merely to the TG gas
of impenetrable bosons.Comment: Brief review, Latex file, 15 page
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