274 research outputs found

    On Bogomolny-Schmit conjecture

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    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

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    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

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    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

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    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

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    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

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    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

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    Functional renormalization yields a simple unified description of bosons at zero temperature, in arbitrary space dimension dd and for MM 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 nn. The behavior of observables and correlation functions in the ordered phase depends crucially on the momentum kphk_{ph}, which is characteristic for a given experiment. For the dilute regime kph‚Č≥n1/dk_{ph}\gtrsim n^{1/d} the quantum phase transition is simple, with the same ``mean field'' critical exponents for all dd and MM. On the other hand, the dense regime kph‚Č™n1/dk_{ph}\ll n^{1/d} reveals a rather rich spectrum of features, depending on dd and MM. In this regime one observes for d‚ȧ3d\leq 3 a crossover to a relativistic action with second time derivatives. This admits order for d>1d>1, whereas d=1d=1 shows a behavior similar to the low temperature phase of the classical two-dimensional O(2M)O(2M)-models.Comment: 31 pages, new reference

    Thermodynamics of the superfluid dilute Bose gas with disorder

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    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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