Relaxation time of LL-reversal chains and other chromosome shuffles

We prove tight bounds on the relaxation time of the so-called $L$-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 $n$ distinct letters on the vertices of the ${n}$-cycle (${{\mathbb{Z}}}$ mod $n$); at each step, a connected subset of the graph is chosen uniformly at random among all those of length at most $L$, and the current permutation is shuffled by reversing the order of the letters over that subset. We show that the relaxation time $\tau (n,L)$, defined as the inverse of the spectral gap of the associated Markov generator, satisfies $\tau (n,L)=O(n\vee \frac{n^3}{L^3})$. 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 $\tau$ 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 $\rho<1$ 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 $d\geq 2$, we also prove a super-Arrhenius scaling of the form $\exp(1/(1-\rho))\leq \tau\leq\exp(1/(1-\rho)^2)$. For FA1f in $d$=$1,2$ we rigorously prove the power law scalings recently derived in \cite{JMS} while in $d\geq 3$ we obtain upper and lower bounds consistent with findings therein. Our results are based on a novel multi-scale approach which allows to analyze $\tau$ 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 $L$ sites has order $L$. We complement that result and show cutoff with an $O(\sqrt{L})$-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 $O(\sqrt{L})$-window. The law of the process behind the front plays a crucial role: Blondel showed that it converges to an invariant measure $\nu$, on which very little is known. Here we obtain quantitative bounds on the speed of convergence to $\nu$, 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 $O(1)$-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 $\ell$ 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 $\ell^2$. 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&gt;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