430 research outputs found
Relaxation time of -reversal chains and other chromosome shuffles
We prove tight bounds on the relaxation time of the so-called -reversal
chain, which was introduced by R. Durrett as a stochastic model for the
evolution of chromosome chains. The process is described as follows. We have
distinct letters on the vertices of the -cycle ( mod
); at each step, a connected subset of the graph is chosen uniformly at
random among all those of length at most , and the current permutation is
shuffled by reversing the order of the letters over that subset. We show that
the relaxation time , defined as the inverse of the spectral gap of
the associated Markov generator, satisfies . Our results can be interpreted as strong evidence for a
conjecture of R. Durrett predicting a similar behavior for the mixing time of
the chain.Comment: Published at http://dx.doi.org/10.1214/105051606000000295 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Relaxation times of kinetically constrained spin models with glassy dynamics
We analyze the density and size dependence of the relaxation time for
kinetically constrained spin systems. These have been proposed as models for
strong or fragile glasses and for systems undergoing jamming transitions. For
the one (FA1f) or two (FA2f) spin facilitated Fredrickson-Andersen model at any
density and for the Knight model below the critical density at which
the glass transition occurs, we show that the persistence and the spin-spin
time auto-correlation functions decay exponentially. This excludes the
stretched exponential relaxation which was derived by numerical simulations.
For FA2f in , we also prove a super-Arrhenius scaling of the form
. For FA1f in = we
rigorously prove the power law scalings recently derived in \cite{JMS} while in
we obtain upper and lower bounds consistent with findings therein.
Our results are based on a novel multi-scale approach which allows to analyze
in presence of kinetic constraints and to connect time-scales and
dynamical heterogeneities. The techniques are flexible enough to allow a
variety of constraints and can also be applied to conservative stochastic
lattice gases in presence of kinetic constraints.Comment: 4 page
Cutoff for the East process
The East process is a 1D kinetically constrained interacting particle system,
introduced in the physics literature in the early 90's to model liquid-glass
transitions. Spectral gap estimates of Aldous and Diaconis in 2002 imply that
its mixing time on sites has order . We complement that result and show
cutoff with an -window.
The main ingredient is an analysis of the front of the process (its rightmost
zero in the setup where zeros facilitate updates to their right). One expects
the front to advance as a biased random walk, whose normal fluctuations would
imply cutoff with an -window. The law of the process behind the
front plays a crucial role: Blondel showed that it converges to an invariant
measure , on which very little is known. Here we obtain quantitative
bounds on the speed of convergence to , finding that it is exponentially
fast. We then derive that the increments of the front behave as a stationary
mixing sequence of random variables, and a Stein-method based argument of
Bolthausen ('82) implies a CLT for the location of the front, yielding the
cutoff result.
Finally, we supplement these results by a study of analogous kinetically
constrained models on trees, again establishing cutoff, yet this time with an
-window.Comment: 33 pages, 2 figure
Activity phase transition for constrained dynamics
We consider two cases of kinetically constrained models, namely East and
FA-1f models. The object of interest of our work is the activity A(t) defined
as the total number of configuration changes in the interval [0,t] for the
dynamics on a finite domain. It has been shown in [GJLPDW1,GJLPDW2] that the
large deviations of the activity exhibit a non-equilibirum phase transition in
the thermodynamic limit and that reducing the activity is more likely than
increasing it due to a blocking mechanism induced by the constraints. In this
paper, we study the finite size effects around this first order phase
transition and analyze the phase coexistence between the active and inactive
dynamical phases in dimension 1. In higher dimensions, we show that the finite
size effects are also determined by the dimension and the choice of boundary
conditions.Comment: 38 pages, 3 figure
Exclusion processes with degenerate rates: convergence to equilibrium and tagged particle
Stochastic lattice gases with degenerate rates, namely conservative particle
systems where the exchange rates vanish for some configurations, have been
introduced as simplified models for glassy dynamics. We introduce two
particular models and consider them in a finite volume of size in
contact with particle reservoirs at the boundary. We prove that, as for
non--degenerate rates, the inverse of the spectral gap and the logarithmic
Sobolev constant grow as . It is also shown how one can obtain, via a
scaling limit from the logarithmic Sobolev inequality, the exponential decay of
a macroscopic entropy associated to a degenerate parabolic differential
equation (porous media equation). We analyze finally the tagged particle
displacement for the stationary process in infinite volume. In dimension larger
than two we prove that, in the diffusive scaling limit, it converges to a
Brownian motion with non--degenerate diffusion coefficient.Comment: 25 pages, 3 figure
Theophylline as a precision therapy in a young girl with PIK3R1 immunodeficiency
Based on its phosphatidylinositol 3-kinase-delta (PI3Kd) inhibitory properties, theophylline was administered to a young girl with activated PI3Kd syndrome (APDS). We report reduced frequency of infections, decreased lymphoproliferation, and noticeable changes in immunophenotype, encouraging further trials with theophylline in children with APDS
Non-equilibrium dynamics of spin facilitated glass models
We consider the dynamics of spin facilitated models of glasses in the
non-equilibrium aging regime following a sudden quench from high to low
temperatures. We briefly review known results obtained for the broad class of
kinetically constrained models, and then present new results for the behaviour
of the one-spin facilitated Fredrickson-Andersen and East models in various
spatial dimensions. The time evolution of one-time quantities, such as the
energy density, and the detailed properties of two-time correlation and
response functions are studied using a combination of theoretical approaches,
including exact mappings of master operators and reductions to integrable
quantum spin chains, field theory and renormalization group, and independent
interval and timescale separation methods. The resulting analytical predictions
are confirmed by means of detailed numerical simulations. The models we
consider are characterized by trivial static properties, with no finite
temperature singularities, but they nevertheless display a surprising variety
of dynamic behaviour during aging, which can be directly related to the
existence and growth in time of dynamic lengthscales. Well-behaved
fluctuation-dissipation ratios can be defined for these models, and we study
their properties in detail. We confirm in particular the existence of negative
fluctuation-dissipation ratios for a large number of observables. Our results
suggest that well-defined violations of fluctuation-dissipation relations, of a
purely dynamic origin and unrelated to the thermodynamic concept of effective
temperatures, could in general be present in non-equilibrium glassy materials.Comment: 72 pages, invited contribution to special issue of JSTAT on
"Principles of Dynamics of Nonequilibrium Systems" (Programme at Newton
Institute Cambridge). v2: New data added to Figs. 11, 23, 24, new Fig. 26 on
East model in d=3, minor improvements to tex
The role of immune PSA complex (iXip) in the prediction of prostate cancer
Purpose: To analyse the performance of iXip in the prediction of prostate cancer (PCa) and high-grade PCa. Methods: A consecutive series of men undergoing MRI/FUSION prostate biopsies were enrolled in one centre. Indications for prostate biopsy included abnormal prostate-specific antigen (PSA) levels (PSA>4 ng/ml) and/or abnormal digital rectal examination (DRE) and/or abnormal MRI. All patients underwent the evaluation of serum PSA-IgM concentration and the iXip ratio was calculated. Accuracy iXip for the prediction of PCa was evaluated using multivariable binary regression analysis and receiver operator characteristics (ROC) curves. Results: Overall 160 patients with a median age of 65 (62/73) years were enrolled. Overall, 42% patients were diagnosed with PCa and 75% of them had high-grade cancer (Epstein ≥ 3). Patients with PCa were older and presented higher PSA levels, higher PIRADS scores and lower prostate volumes (PVs). On ROC analysis iXip presented an area under the curve (AUC) of 0.57 in the prediction of PCa and of 0.54 for the prediction of high-grade PCa. Conclusions: In our experience, immune PSA complexes are not predictors of PCa. iXip analysis should not be included in the diagnostic pathway of patients at increased risk of PCa
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