274 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

### Controlling quasiparticle excitations in a trapped Bose-Einstein condensate

We describe an approach to quantum control of the quasiparticle excitations
in a trapped Bose-Einstein condensate based on adiabatic and diabatic changes
in the trap anisotropy. We describe our approach in the context of Landau-Zener
transition at the avoided crossings in the quasiparticle excitation spectrum.
We show that there can be population oscillation between different modes at the
specific aspect ratios of the trapping potential at which the mode energies are
almost degenerate. These effects may have implications in the expansion of an
excited condensate as well as the dynamics of a moving condensate in an atomic
wave guide with a varying width

### Harmonic measure and SLE

In this paper we rigorously compute the average multifractal spectrum of
harmonic measure on the boundary of SLE clusters

### A covariance formula for topological events of smooth Gaussian fields

We derive a covariance formula for the class of `topological events' of smooth Gaussian fields on manifolds; these are events that depend only on the topology of the level sets of the field, for example (i) crossing events for level or excursion sets, (ii) events measurable with respect to the number of connected components of level or excursion sets of a given diffeomorphism class, and (iii) persistence events. As an application of the covariance formula, we derive strong mixing bounds for topological events, as well as lower concentration inequalities for additive topological functionals (e.g. the number of connected components) of the level sets that satisfy a law of large numbers. The covariance formula also gives an alternate justification of the Harris criterion, which conjecturally describes the boundary of the percolation university class for level sets of stationary Gaussian fields. Our work is inspired by a recent paper by Rivera and Vanneuville, in which a correlation inequality was derived for certain topological events on the plane, as well as by an old result of Piterbarg, in which a similar covariance formula was established for finite-dimensional Gaussian vectors

### Thermodynamics of a Bose-Einstein Condensate with Weak Disorder

We consider the thermodynamics of a homogeneous superfluid dilute Bose gas in
the presence of weak quenched disorder. Following the zero-temperature approach
of Huang and Meng, we diagonalize the Hamiltonian of a dilute Bose gas in an
external random delta-correlated potential by means of a Bogoliubov
transformation. We extend this approach to finite temperature by combining the
Popov and the many-body T-matrix approximations. This approach permits us to
include the quasi-particle interactions within this temperature range. We
derive the disorder-induced shifts of the Bose-Einstein critical temperature
and of the temperature for the onset of superfluidity by approaching the
transition points from below, i.e., from the superfluid phase. Our results lead
to a phase diagram consistent with that of the finite-temperature theory of
Lopatin and Vinokur which was based on the replica method, and in which the
transition points were approached from above.Comment: 11 pages, 5 figure

### Functional renormalization for quantum phase transitions with non-relativistic bosons

Functional renormalization yields a simple unified description of bosons at
zero temperature, in arbitrary space dimension $d$ and for $M$ complex fields.
We concentrate on nonrelativistic bosons and an action with a linear time
derivative. The ordered phase can be associated with a nonzero density of
(quasi) particles $n$. The behavior of observables and correlation functions in
the ordered phase depends crucially on the momentum $k_{ph}$, which is
characteristic for a given experiment. For the dilute regime $k_{ph}\gtrsim
n^{1/d}$ the quantum phase transition is simple, with the same ``mean field''
critical exponents for all $d$ and $M$. On the other hand, the dense regime
$k_{ph}\ll n^{1/d}$ reveals a rather rich spectrum of features, depending on
$d$ and $M$. In this regime one observes for $d\leq 3$ a crossover to a
relativistic action with second time derivatives. This admits order for $d>1$,
whereas $d=1$ shows a behavior similar to the low temperature phase of the
classical two-dimensional $O(2M)$-models.Comment: 31 pages, new reference

### Thermodynamics of the superfluid dilute Bose gas with disorder

We generalize the Beliaev-Popov diagrammatic technique for the problem of
interacting dilute Bose gas with weak disorder. Averaging over disorder is
implemented by the replica method. Low energy asymptotic form of the Green
function confirms that the low energy excitations of the superfluid dirty Boson
system are sound waves with velocity renormalized by the disorder and
additional dissipation due to the impurity scattering. We find the
thermodynamic potential and the superfluid density at any temperature below the
superfluid transition temperature and derive the phase diagram in temperature
vs. disorder plane.Comment: 4 page

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