924 research outputs found
Moderate deviations for the determinant of Wigner matrices
We establish a moderate deviations principle (MDP) for the log-determinant
of a Wigner matrix matching four moments with
either the GUE or GOE ensemble. Further we establish Cram\'er--type moderate
deviations and Berry-Esseen bounds for the log-determinant for the GUE and GOE
ensembles as well as for non-symmetric and non-Hermitian Gaussian random
matrices (Ginibre ensembles), respectively.Comment: 20 pages, one missing reference added; Limit Theorems in Probability,
Statistics and Number Theory, Springer Proceedings in Mathematics and
Statistics, 201
Asymptotics for the Wiener sausage among Poissonian obstacles
We consider the Wiener sausage among Poissonian obstacles. The obstacle is
called hard if Brownian motion entering the obstacle is immediately killed, and
is called soft if it is killed at certain rate. It is known that Brownian
motion conditioned to survive among obstacles is confined in a ball near its
starting point. We show the weak law of large numbers, large deviation
principle in special cases and the moment asymptotics for the volume of the
corresponding Wiener sausage. One of the consequence of our results is that the
trajectory of Brownian motion almost fills the confinement ball.Comment: 19 pages, Major revision made for publication in J. Stat. Phy
Self-intersection local time of planar Brownian motion based on a strong approximation by random walks
The main purpose of this work is to define planar self-intersection local
time by an alternative approach which is based on an almost sure pathwise
approximation of planar Brownian motion by simple, symmetric random walks. As a
result, Brownian self-intersection local time is obtained as an almost sure
limit of local averages of simple random walk self-intersection local times. An
important tool is a discrete version of the Tanaka--Rosen--Yor formula; the
continuous version of the formula is obtained as an almost sure limit of the
discrete version. The author hopes that this approach to self-intersection
local time is more transparent and elementary than other existing ones.Comment: 36 pages. A new part on renormalized self-intersection local time has
been added and several inaccuracies have been corrected. To appear in Journal
of Theoretical Probabilit
Duality and fluctuation relations for statistics of currents on cyclic graphs
We consider stochastic motion of a particle on a cyclic graph with
arbitrarily periodic time dependent kinetic rates. We demonstrate duality
relations for statistics of currents in this model and in its continuous
version of a diffusion in one dimension. Our duality relations are valid beyond
detailed balance constraints and lead to exact expressions that relate
statistics of currents induced by dual driving protocols. We also show that
previously known no-pumping theorems and some of the fluctuation relations,
when they are applied to cyclic graphs or to one dimensional diffusion, are
special consequences of our duality.Comment: 2 figure, 6 pages (In twocolumn). Accepted by JSTA
Large deviations for a damped telegraph process
In this paper we consider a slight generalization of the damped telegraph
process in Di Crescenzo and Martinucci (2010). We prove a large deviation
principle for this process and an asymptotic result for its level crossing
probabilities (as the level goes to infinity). Finally we compare our results
with the analogous well-known results for the standard telegraph process
Dynamic Phase Transitions in Cell Spreading
We monitored isotropic spreading of mouse embryonic fibroblasts on
fibronectin-coated substrates. Cell adhesion area versus time was measured via
total internal reflection fluorescence microscopy. Spreading proceeds in
well-defined phases. We found a power-law area growth with distinct exponents
a_i in three sequential phases, which we denote basal (a_1=0.4+-0.2), continous
(a_2=1.6+-0.9) and contractile (a_3=0.3+-0.2) spreading. High resolution
differential interference contrast microscopy was used to characterize local
membrane dynamics at the spreading front. Fourier power spectra of membrane
velocity reveal the sudden development of periodic membrane retractions at the
transition from continous to contractile spreading. We propose that the
classification of cell spreading into phases with distinct functional
characteristics and protein activity patterns serves as a paradigm for a
general program of a phase classification of cellular phenotype. Biological
variability is drastically reduced when only the corresponding phases are used
for comparison across species/different cell lines.Comment: 4 pages, 5 figure
A numerical approach to copolymers at selective interfaces
We consider a model of a random copolymer at a selective interface which
undergoes a localization/delocalization transition. In spite of the several
rigorous results available for this model, the theoretical characterization of
the phase transition has remained elusive and there is still no agreement about
several important issues, for example the behavior of the polymer near the
phase transition line. From a rigorous viewpoint non coinciding upper and lower
bounds on the critical line are known.
In this paper we combine numerical computations with rigorous arguments to
get to a better understanding of the phase diagram. Our main results include:
- Various numerical observations that suggest that the critical line lies
strictly in between the two bounds.
- A rigorous statistical test based on concentration inequalities and
super-additivity, for determining whether a given point of the phase diagram is
in the localized phase. This is applied in particular to show that, with a very
low level of error, the lower bound does not coincide with the critical line.
- An analysis of the precise asymptotic behavior of the partition function in
the delocalized phase, with particular attention to the effect of rare atypical
stretches in the disorder sequence and on whether or not in the delocalized
regime the polymer path has a Brownian scaling.
- A new proof of the lower bound on the critical line. This proof relies on a
characterization of the localized regime which is more appealing for
interpreting the numerical data.Comment: accepted for publication on J. Stat. Phy
Long time, large scale limit of the Wigner transform for a system of linear oscillators in one dimension
We consider the long time, large scale behavior of the Wigner transform
W_\eps(t,x,k) of the wave function corresponding to a discrete wave equation
on a 1-d integer lattice, with a weak multiplicative noise. This model has been
introduced in Basile, Bernardin, and Olla to describe a system of interacting
linear oscillators with a weak noise that conserves locally the kinetic energy
and the momentum. The kinetic limit for the Wigner transform has been shown in
Basile, Olla, and Spohn. In the present paper we prove that in the unpinned
case there exists such that for any the
weak limit of W_\eps(t/\eps^{3/2\gamma},x/\eps^{\gamma},k), as \eps\ll1,
satisfies a one dimensional fractional heat equation with . In the pinned case an analogous
result can be claimed for W_\eps(t/\eps^{2\gamma},x/\eps^{\gamma},k) but the
limit satisfies then the usual heat equation
Bose--Einstein Condensation in the Large Deviations Regime with Applications to Information System Models
We study the large deviations behavior of systems that admit a certain form
of a product distribution, which is frequently encountered both in Physics and
in various information system models. First, to fix ideas, we demonstrate a
simple calculation of the large deviations rate function for a single
constraint (event). Under certain conditions, the behavior of this function is
shown to exhibit an analogue of Bose--Einstein condensation (BEC). More
interestingly, we also study the large deviations rate function associated with
two constraints (and the extension to any number of constraints is conceptually
straightforward). The phase diagram of this rate function is shown to exhibit
as many as seven phases, and it suggests a two--dimensional generalization of
the notion of BEC (or more generally, a multi--dimensional BEC). While the
results are illustrated for a simple model, the underlying principles are
actually rather general. We also discuss several applications and implications
pertaining to information system models
Current fluctuations in stochastic systems with long-range memory
We propose a method to calculate the large deviations of current fluctuations
in a class of stochastic particle systems with history-dependent rates.
Long-range temporal correlations are seen to alter the speed of the large
deviation function in analogy with long-range spatial correlations in
equilibrium systems. We give some illuminating examples and discuss the
applicability of the Gallavotti-Cohen fluctuation theorem.Comment: 10 pages, 1 figure. v2: Minor alterations. v3: Very minor alterations
for consistency with published version appearing at
http://stacks.iop.org/1751-8121/42/34200
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