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

Abrupt Convergence and Escape Behavior for Birth and Death Chains

We link two phenomena concerning the asymptotical behavior of stochastic processes: (i) abrupt convergence or cut-off phenomenon, and (ii) the escape behavior usually associated to exit from metastability. The former is characterized by convergence at asymptotically deterministic times, while the convergence times for the latter are exponentially distributed. We compare and study both phenomena for discrete-time birth-and-death chains on Z with drift towards zero. In particular, this includes energy-driven evolutions with energy functions in the form of a single well. Under suitable drift hypotheses, we show that there is both an abrupt convergence towards zero and escape behavior in the other direction. Furthermore, as the evolutions are reversible, the law of the final escape trajectory coincides with the time reverse of the law of cut-off paths. Thus, for evolutions defined by one-dimensional energy wells with sufficiently steep walls, cut-off and escape behavior are related by time inversion.Comment: 2 figure

First exit times of solutions of stochastic differential equations driven by multiplicative Levy noise with heavy tails

In this paper we study first exit times from a bounded domain of a gradient dynamical system $\dot Y_t=-\nabla U(Y_t)$ perturbed by a small multiplicative L\'evy noise with heavy tails. A special attention is paid to the way the multiplicative noise is introduced. In particular we determine the asymptotics of the first exit time of solutions of It\^o, Stratonovich and Marcus canonical SDEs.Comment: 19 pages, 2 figure

Typical profiles of the Kac-Hopfield model

Mean field models, random or not, are very important for explaining simply the general phenomenon of phase transitions. However, for random systems, in general, their analysis, as many of the contributions in this volume confirm, is not simple at all, a fact which may justify the amount of effort spent on them. In spite of all that, mean fields models, in many respects, are only poor caricatures of realistic systems1 and are unable to feature some of their most important aspects; in particular, in a phase-transition regime, they are unable to properly account for the phenomenon of phase separation, that is, the observed feature that states of the system with two or more phases coexist in separate regions of space. This deficiency manifests itself also in the fact that the canonical free energy is generally not a convex function of the order parameters, which in turn means that the usual formalism of thermodynamics cannot be immediately used (e.g., the isotherms are not monotone, thus cannot directly be used to determine the equations of state, and insisting on doing so would produce a totally unphysical effect, like regions of parameters where pressure is a decreasing function of density). This problem is solved by the Maxwell construction, by which free energy is simply replaced ad hoc by its convex hull

Field evaluation of the CATT/Trypanosoma brucei gambiense on blood-impregnated filter papers for diagnosis of human African trypanosomiasis in southern Sudan.

Most Human African Trypanosomiasis (HAT) control programmes in areas endemic for Trypanosoma brucei gambiense rely on a strategy of active mass screening with the Card Agglutination Test for Trypanosomiasis (CATT)/T. b. gambiense. We evaluated the performance, stability and reproducibility of the CATT/T. b. gambiense on blood-impregnated filter papers (CATT-FP) in Kajo-Keji County, South-Sudan, where some areas are inaccessible to mobile teams. The CATT-FP was performed with a group of 100 people with a positive CATT on whole blood including 17 confirmed HAT patients and the results were compared with the CATT on plasma (CATT-P). The CATT-FP was repeated on impregnated filter papers stored at ambient and refrigerated temperature for 1, 3, 7 and 14 days. Another 82 patients with HAT, including 78 with a positive parasitology, were tested with the CATT-FP and duplicate filter paper samples were sent to a reference laboratory to assess reproducibility. The CATT-FP was positive in 90 of 99 patients with HAT (sensitivity: 91%). It was less sensitive than the CATT-P (mean dilution difference: -2.5). There was no significant loss of sensitivity after storage for up to 14 days both at ambient and cool temperature. Reproducibility of the CATT-FP was found to be excellent (kappa: 0.84). The CATT-FP can therefore be recommended as a screening test for HAT in areas where the use of CATT-P is not possible. Further studies on larger population samples in different endemic foci are still needed before the CATT-FP can be recommended for universal use

Eflornithine is Safer Than Melarsoprol for the Treatment of Second-Stage Trypanosoma Brucei Gambiense Human African Trypanosomiasis.

Patients with second-stage human African trypanosomiasis treated with eflornithine (n = 251) in 2003 in Kiri, southern Sudan, had an adjusted relative risk of death of 0.2 and experienced significantly fewer cutaneous and neurological adverse effects than did patients who were treated with melarsoprol in 2001 and 2002 (n = 708)

Tunneling and Metastability of continuous time Markov chains

We propose a new definition of metastability of Markov processes on countable state spaces. We obtain sufficient conditions for a sequence of processes to be metastable. In the reversible case these conditions are expressed in terms of the capacity and of the stationary measure of the metastable states

Metastates in mean-field models with random external fields generated by Markov chains

We extend the construction by Kuelske and Iacobelli of metastates in finite-state mean-field models in independent disorder to situations where the local disorder terms are are a sample of an external ergodic Markov chain in equilibrium. We show that for non-degenerate Markov chains, the structure of the theorems is analogous to the case of i.i.d. variables when the limiting weights in the metastate are expressed with the aid of a CLT for the occupation time measure of the chain. As a new phenomenon we also show in a Potts example that, for a degenerate non-reversible chain this CLT approximation is not enough and the metastate can have less symmetry than the symmetry of the interaction and a Gaussian approximation of disorder fluctuations would suggest.Comment: 20 pages, 2 figure

Tight Binding Hamiltonians and Quantum Turing Machines

This paper extends work done to date on quantum computation by associating potentials with different types of computation steps. Quantum Turing machine Hamiltonians, generalized to include potentials, correspond to sums over tight binding Hamiltonians each with a different potential distribution. Which distribution applies is determined by the initial state. An example, which enumerates the integers in succession as binary strings, is analyzed. It is seen that for some initial states the potential distributions have quasicrystalline properties and are similar to a substitution sequence.Comment: 4 pages Latex, 2 postscript figures, submitted to Phys Rev Letter