Metastability and low lying spectra in reversible Markov chains

We study a large class of reversible Markov chains with discrete state space and transition matrix $P_N$. We define the notion of a set of {\it metastable points} as a subset of the state space \G_N such that (i) this set is reached from any point x\in \G_N without return to x with probability at least $b_N$, while (ii) for any two point x,y in the metastable set, the probability $T^{-1}_{x,y}$ to reach y from x without return to x is smaller than $a_N^{-1}\ll b_N$. Under some additional non-degeneracy assumption, we show that in such a situation: \item{(i)} To each metastable point corresponds a metastable state, whose mean exit time can be computed precisely. \item{(ii)} To each metastable point corresponds one simple eigenvalue of $1-P_N$ which is essentially equal to the inverse mean exit time from this state. The corresponding eigenfunctions are close to the indicator function of the support of the metastable state. Moreover, these results imply very sharp uniform control of the deviation of the probability distribution of metastable exit times from the exponential distribution.Comment: 44pp, AMSTe

Metastability in stochastic dynamics of disordered mean-field models

We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding Gibbs measures. We derive the analog of the Wentzell-Freidlin theory in this case, showing that any transition can be decomposed, with probability exponentially close to one, into a deterministic sequence of ``admissible transitions''. For these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant. The distribution rescaled transition times are shown to converge to the exponential distribution. We exemplify our results in the context of the random field Curie-Weiss model.Comment: 73pp, AMSTE

Dynamic phase diagram of the REM

By studying the two-time overlap correlation function, we give a comprehensive analysis of the phase diagram of the Random Hopping Dynamics of the Random Energy Model (REM) on time-scales that are exponential in the volume. These results are derived from the convergence properties of the clock process associated to the dynamics and fine properties of the simple random walk in the $n$-dimensional discrete cube.Comment: This paper is in large part based on the unpublished work arXiv:1008.3849. In particular, the analysis of the overlap correlation function is new as well as the study of the high temperature and short time-scale transition line between aging and stationarit

Infinitely many states and stochastic symmetry in a Gaussian Potts-Hopfield model

We study a Gaussian Potts-Hopfield model. Whereas for Ising spins and two disorder variables per site the chaotic pair scenario is realized, we find that for q-state Potts spins [{q(q-1} \over 2]-tuples occur. Beyond the breaking of a continous stochastic symmetry, we study the fluctuations and obtain the Newman-Stein metastate description for our model.Comment: latex, 17 page

Assessment of metabolomic and proteomic biomarkers in detection and prognosis of progression of renal function in chronic kidney disease

Chronic kidney disease (CKD) is part of a number of systemic and renal diseases and may reach epidemic proportions over the next decade. Efforts have been made to improve diagnosis and management of CKD. We hypothesised that combining metabolomic and proteomic approaches could generate a more systemic and complete view of the disease mechanisms. To test this approach, we examined samples from a cohort of 49 patients representing different stages of CKD. Urine samples were analysed for proteomic changes using capillary electrophoresis-mass spectrometry and urine and plasma samples for metabolomic changes using different mass spectrometry-based techniques. The training set included 20 CKD patients selected according to their estimated glomerular filtration rate (eGFR) at mild (59.9±16.5 mL/min/1.73 m2; n = 10) or advanced (8.9±4.5 mL/min/1.73 m2; n = 10) CKD and the remaining 29 patients left for the test set. We identified a panel of 76 statistically significant metabolites and peptides that correlated with CKD in the training set. We combined these biomarkers in different classifiers and then performed correlation analyses with eGFR at baseline and follow-up after 2.8±0.8 years in the test set. A solely plasma metabolite biomarker-based classifier significantly correlated with the loss of kidney function in the test set at baseline and follow-up (ρ = −0.8031; p<0.0001 and ρ = −0.6009; p = 0.0019, respectively). Similarly, a urinary metabolite biomarker-based classifier did reveal significant association to kidney function (ρ = −0.6557; p = 0.0001 and ρ = −0.6574; p = 0.0005). A classifier utilising 46 identified urinary peptide biomarkers performed statistically equivalent to the urinary and plasma metabolite classifier (ρ = −0.7752; p<0.0001 and ρ = −0.8400; p<0.0001). The combination of both urinary proteomic and urinary and plasma metabolic biomarkers did not improve the correlation with eGFR. In conclusion, we found excellent association of plasma and urinary metabolites and urinary peptides with kidney function, and disease progression, but no added value in combining the different biomarkers data

Metastability and small eigenvalues in Markov chains

In this letter we announce rigorous results that elucidate the relation between metastable states and low-lying eigenvalues in Markov chains in a much more general setting and with considerable greater precision as was so far available. This includes a sharp uncertainty principle relating all low-lying eigenvalues to mean times of metastable transitions, a relation between the support of eigenfunctions and the attractor of a metastable state, and sharp estimates on the convergence of probability distribution of the metastable transition times to the exponential distribution.Comment: 5pp, AMSTe

Toxicokinetics of bisphenol-S and its glucuronide in plasma and urine following oral and dermal exposure in volunteers for the interpretation of biomonitoring data

The measurement of bisphenol-S (BPS) and its glucurono-conjugate (BPSG) in urine may be used for the biomonitoring of exposure in populations. However, this requires a thorough knowledge of their toxicokinetics. The time courses of BPS and BPSG were assessed in accessible biological matrices of orally and dermally exposed volunteers. Under the approval of the Research Ethics Committee of the University of Montreal, six volunteers were orally exposed to a BPS-d8 deuterated dose of 0.1 mg/kg body weight (bw). One month later, 1 mg/kg bw of BPS-d8 were applied on 40 cm2 of the forearm and then washed 6 h after application. Blood samples were taken prior to dosing and at fixed time periods over 48 h after treatment; complete urine voids were collected pre-exposure and at pre-established intervals over 72 h postdosing. Following oral exposure, the plasma concentration–time courses of BPS-d8 and BPSG-d8 over 48 h evolved in parallel, and showed a rapid appearance and elimination. Average peak values (±SD) were reached at 0.7 ± 0.1 and 1.1 ± 0.4 h postdosing and mean (±SD) apparent elimination half-lives (t½) of 7.9 ± 1.1 and 9.3 ± 7.0 h were calculated from the terminal phase of BPS-d8 and BPSG-d8 in plasma, respectively. The fraction of BPS-d8 reaching the systemic circulation unchanged (i.e. bioavailability) was further estimated at 62 ± 5% on average (±SD) and the systemic plasma clearance at 0.57 ± 0.07 L/kg bw/h. Plasma concentration–time courses and urinary excretion rate profiles roughly evolved in parallel for both substances, as expected. The average percent (±SD) of the administered dose recovered in urine as BPS-d8 and BPSG-d8 over the 0–72 h period postdosing was 1.72 ± 1.3 and 54 ± 10%. Following dermal application, plasma levels were under the lower limit of quantification (LLOQ) at most time points. However, peak values were reached between 5 and 8 h depending on individuals, suggesting a slower absorption rate compared to oral exposure. Similarly, limited amounts of BPS-d8 and its conjugate were recovered in urine and peak excretion rates were reached between 5 and 11 h postdosing. The average percent (±SD) of the administered dose recovered in urine as BPS-d8 and BPSG-d8 was about 0.004 ± 0.003 and 0.09 ± 0.07%, respectively. This study provided greater precision on the kinetics of this contaminant in humans and, in particular, evidenced major differences between BPA and BPS kinetics with much higher systemic levels of active BPS than BPA, an observation explained by a higher oral bioavailability of BPS than BPA. These data should also be useful in developing a toxicokinetic model for a better interpretation of biomonitoring data

Aging in the random energy model

In this letter we announce rigorous results on the phenomenon of aging in the Glauber dynamics of the random energy model and their relation to Bouchaud's 'REM-like' trap model. We show that, below the critical temperature, if we consider a time-scale that diverges with the system size in such a way that equilibrium is almost, but not quite reached on that scale, a suitably defined autocorrelation function has the same asymptotic behaviour than its analog in the trap model.Comment: 4pp, P