663 research outputs found
Open Quantum Symmetric Simple Exclusion Process
We introduce and solve a model of fermions hopping between neighbouring sites
on a line with random Brownian amplitudes and open boundary conditions driving
the system out of equilibrium. The average dynamics reduces to that of the
symmetric simple exclusion process. However, the full distribution encodes for
a richer behaviour entailing fluctuating quantum coherences which survive in
the steady limit. We determine exactly the system state steady distribution. We
show that these out of equilibrium quantum fluctuations fulfil a large
deviation principle and we present a method to recursively compute exactly the
large deviation function. On the way, our approach gives a solution of the
classical symmetric simple exclusion process using fermion technology. Our
results open the route towards the extension of the macroscopic fluctuation
theory to many body quantum systems.Comment: 5 pages + SM, 2 figure
Estimating the Null and the Proportion of non-Null effects in Large-scale Multiple Comparisons
An important issue raised by Efron in the context of large-scale multiple
comparisons is that in many applications the usual assumption that the null
distribution is known is incorrect, and seemingly negligible differences in the
null may result in large differences in subsequent studies. This suggests that
a careful study of estimation of the null is indispensable.
In this paper, we consider the problem of estimating a null normal
distribution, and a closely related problem, estimation of the proportion of
non-null effects. We develop an approach based on the empirical characteristic
function and Fourier analysis. The estimators are shown to be uniformly
consistent over a wide class of parameters. Numerical performance of the
estimators is investigated using both simulated and real data. In particular,
we apply our procedure to the analysis of breast cancer and HIV microarray data
sets. The estimators perform favorably in comparison to existing methods.Comment: 42 pages, 6 figure
Optimal rates of convergence for estimating the null density and proportion of nonnull effects in large-scale multiple testing
An important estimation problem that is closely related to large-scale
multiple testing is that of estimating the null density and the proportion of
nonnull effects. A few estimators have been introduced in the literature;
however, several important problems, including the evaluation of the minimax
rate of convergence and the construction of rate-optimal estimators, remain
open. In this paper, we consider optimal estimation of the null density and the
proportion of nonnull effects. Both minimax lower and upper bounds are derived.
The lower bound is established by a two-point testing argument, where at the
core is the novel construction of two least favorable marginal densities
and . The density is heavy tailed both in the spatial and frequency
domains and is a perturbation of such that the characteristic
functions associated with and match each other in low frequencies.
The minimax upper bound is obtained by constructing estimators which rely on
the empirical characteristic function and Fourier analysis. The estimator is
shown to be minimax rate optimal. Compared to existing methods in the
literature, the proposed procedure not only provides more precise estimates of
the null density and the proportion of the nonnull effects, but also yields
more accurate results when used inside some multiple testing procedures which
aim at controlling the False Discovery Rate (FDR). The procedure is easy to
implement and numerical results are given.Comment: Published in at http://dx.doi.org/10.1214/09-AOS696 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Equilibration of quantum cat states
We study the equilibration properties of isolated ergodic quantum systems
initially prepared in a cat state, i.e a macroscopic quantum superposition of
states. Our main result consists in showing that, even though decoherence is at
work in the mean, there exists a remnant of the initial quantum coherences
visible in the strength of the fluctuations of the steady state. We back-up our
analysis with numerical results obtained on the XXX spin chain with a random
field along the z-axis in the ergodic regime and find good qualitative and
quantitative agreement with the theory. We also present and discuss a framework
where equilibrium quantities can be computed from general statistical ensembles
without relying on microscopic details about the initial state, akin to the
eigenstate thermalization hypothesis.Comment: 18 pages, 3 figure
Estimation and confidence sets for sparse normal mixtures
For high dimensional statistical models, researchers have begun to focus on
situations which can be described as having relatively few moderately large
coefficients. Such situations lead to some very subtle statistical problems. In
particular, Ingster and Donoho and Jin have considered a sparse normal means
testing problem, in which they described the precise demarcation or detection
boundary. Meinshausen and Rice have shown that it is even possible to estimate
consistently the fraction of nonzero coordinates on a subset of the detectable
region, but leave unanswered the question of exactly in which parts of the
detectable region consistent estimation is possible. In the present paper we
develop a new approach for estimating the fraction of nonzero means for
problems where the nonzero means are moderately large. We show that the
detection region described by Ingster and Donoho and Jin turns out to be the
region where it is possible to consistently estimate the expected fraction of
nonzero coordinates. This theory is developed further and minimax rates of
convergence are derived. A procedure is constructed which attains the optimal
rate of convergence in this setting. Furthermore, the procedure also provides
an honest lower bound for confidence intervals while minimizing the expected
length of such an interval. Simulations are used to enable comparison with the
work of Meinshausen and Rice, where a procedure is given but where rates of
convergence have not been discussed. Extensions to more general Gaussian
mixture models are also given.Comment: Published in at http://dx.doi.org/10.1214/009053607000000334 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Solution to the Quantum Symmetric Simple Exclusion Process : the Continuous Case
The Quantum Symmetric Simple Exclusion Process (Q-SSEP) is a model for
quantum stochastic dynamics of fermions hopping along the edges of a graph with
Brownian noisy amplitudes and driven out-of-equilibrium by injection-extraction
processes at a few vertices. We present a solution for the invariant
probability measure of the one dimensional Q-SSEP in the infinite size limit by
constructing the steady correlation functions of the system density matrix and
quantum expectation values. These correlation functions code for a rich
structure of fluctuating quantum correlations and coherences. Although our
construction does not rely on the standard techniques from the theory of
integrable systems, it is based on a remarkable interplay between the
permutation groups and polynomials. We incidentally point out a possible
combinatorial interpretation of the Q-SSEP correlation functions via a
surprising connexion with geometric combinatorics and the associahedron
polytopes.Comment: 46 pages, 3 figure
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