540 research outputs found
Permanental processes from products of complex and quaternionic induced Ginibre ensembles
We consider products of independent random matrices taken from the induced
Ginibre ensemble with complex or quaternion elements. The joint densities for
the complex eigenvalues of the product matrix can be written down exactly for a
product of any fixed number of matrices and any finite matrix size. We show
that the squared absolute values of the eigenvalues form a permanental process,
generalising the results of Kostlan and Rider for single matrices to products
of complex and quaternionic matrices. Based on these findings, we can first
write down exact results and asymptotic expansions for the so-called hole
probabilities, that a disc centered at the origin is void of eigenvalues.
Second, we compute the asymptotic expansion for the opposite problem, that a
large fraction of complex eigenvalues occupies a disc of fixed radius centered
at the origin; this is known as the overcrowding problem. While the expressions
for finite matrix size depend on the parameters of the induced ensembles, the
asymptotic results agree to leading order with previous results for products of
square Ginibre matrices.Comment: 47 pages, v2: typos corrected, 1 reference added, published versio
Systematic Derivation of Amplitude Equations and Normal Forms for Dynamical Systems
We present a systematic approach to deriving normal forms and related
amplitude equations for flows and discrete dynamics on the center manifold of a
dynamical system at local bifurcations and unfoldings of these. We derive a
general, explicit recurrence relation that completely determines the amplitude
equation and the associated transformation from amplitudes to physical space.
At any order, the relation provides explicit expressions for all the
nonvanishing coefficients of the amplitude equation together with
straightforward linear equations for the coefficients of the transformation.
The recurrence relation therefore provides all the machinery needed to solve a
given physical problem in physical terms through an amplitude equation. The new
result applies to any local bifurcation of a flow or map for which all the
critical eigenvalues are semisimple i.e. have Riesz index unity). The method is
an efficient and rigorous alternative to more intuitive approaches in terms of
multiple time scales. We illustrate the use of the method by deriving amplitude
equations and associated transformations for the most common simple
bifurcations in flows and iterated maps. The results are expressed in tables in
a form that can be immediately applied to specific problems.Comment: 40 pages, 1 figure, 4 tables. Submitted to Chaos. Please address any
correspondence by email to [email protected]
A note on the Lee-Yang singularity coupled to 2d quantum gravity
We show how to obtain the critical exponent of magnetization in the Lee-Yang
edge singularity model coupled to two-dimensional quantum gravity
Derivation of the Statistical Distribution of the Mass Peak Centroids of Mass Spectrometers Employing Analog-to-Digital Converters and Electron Multipliers
The statistical distribution of mass peak centroids recorded on mass spectrometers employing analog-to-digital converters (ADCs) and electron multipliers is derived from the first principles of the data generation process. The resulting Gaussian model is discussed and is validated with experimental data and with Monte Carlo simulations
Random-lattice models and simulation algorithms for the phase equilibria in two-dimensional condensed systems of particles with coupled internal and translational degrees of freedom
Series of Concentration-Induced Phase Transitions in Cholesterol/Phosphatidylcholine Mixtures
In lipid membranes, temperature-induced transition from gel-to-fluid phase increases the lateral diffusion of the lipid molecules by three orders of magnitude. In cell membranes, a similar phase change may trigger the communication between the membrane components. Here concentration-induced phase transition properties of our recently developed statistical mechanical model of cholesterol/phospholipid mixtures are investigated. A slight (<1%) decrease in the model parameter values, controlling the lateral interaction energies, reveals the existence of a series of first- or second-order phase transitions. By weakening the lateral interactions first, the proportion of the ordered (i.e., superlattice) phase (Areg) is slightly and continuously decreasing at every cholesterol mole fraction. Then sudden decreases in Areg appear at the 0.18–0.26 range of cholesterol mole fractions. We point out that the sudden changes in Areg represent first- or second-order concentration-induced phase transitions from fluid to superlattice and from superlattice to fluid phase. Sudden changes like these were detected in our previous experiments at 0.2, 0.222, and 0.25 sterol mole fractions in ergosterol/DMPC mixtures. By further decreasing the lateral interactions, the fluid phase will dominate throughout the 0.18–0.26 interval, whereas outside this interval sudden increases in Areg may appear. Lipid composition-induced phase transitions as specified here should have far more important biological implications than temperature- or pressure-induced phase transitions. This is the case because temperature and pressure in cell membranes are largely invariant under physiological conditions
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