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

The Influence of External Factors Upon the Functions of the Family Since 1929

The purpose of this thesis is to show the influence of external factors upon the functions of the family since 1929. The writer will consider the affectional, religious, economic, protective, recreational and educational functions, and will endeavor to determine the extent to which external factors have affected the functions of the family and the family itself. Studies on the social aspects of the depression, published by the Social Science Research Council, have been particularly helpful in the preparation of this thesis. The writer refers especially to the Research Memorandum on the Family in the Depression, Research Memorandum on Religion in the Depression, Research Memorandum on Social Aspects of Consumption in the Depression, Research Memorandum on Recreation in the Depression, and the Research .Memorandum on Education in the Depression. All tables referred to in the chapters of this thesis are in the appendix

Uniqueness of the mixing measure for a random walk in a random environment on the positive integers

Consider a random walk in an irreducible random environment on the positive integers. We prove that the annealed law of the random walk determines uniquely the law of the random environment. An application to linearly edge-reinforced random walk is given

Optimal population-level infection detection strategies for malaria control and elimination in a spatial model of malaria transmission

Mass campaigns with antimalarial drugs are potentially a powerful tool for local elimination of malaria, yet current diagnostic technologies are insufficiently sensitive to identify all individuals who harbor infections. At the same time, overtreatment of uninfected individuals increases the risk of accelerating emergence of drug resistance and losing community acceptance. Local heterogeneity in transmission intensity may allow campaign strategies that respond to index cases to successfully target subpatent infections while simultaneously limiting overtreatment. While selective targeting of hotspots of transmission has been proposed as a strategy for malaria control, such targeting has not been tested in the context of malaria elimination. Using household locations, demographics, and prevalence data from a survey of four health facility catchment areas in southern Zambia and an agent-based model of malaria transmission and immunity acquisition, a transmission intensity was fit to each household based on neighborhood age-dependent malaria prevalence. A set of individual infection trajectories was constructed for every household in each catchment area, accounting for heterogeneous exposure and immunity. Various campaign strategies (mass drug administration, mass screen and treat, focal mass drug administration, snowball reactive case detection, pooled sampling, and a hypothetical serological diagnostic) were simulated and evaluated for performance at finding infections, minimizing overtreatment, reducing clinical case counts, and interrupting transmission. For malaria control, presumptive treatment leads to substantial overtreatment without additional morbidity reduction under all but the highest transmission conditions. Selective targeting of hotspots with drug campaigns is an ineffective tool for elimination due to limited sensitivity of available field diagnostics

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

Malaria elimination campaigns in the Lake Kariba region of Zambia: a spatial dynamical model

Background As more regions approach malaria elimination, understanding how different interventions interact to reduce transmission becomes critical. The Lake Kariba area of Southern Province, Zambia, is part of a multi-country elimination effort and presents a particular challenge as it is an interconnected region of variable transmission intensities. Methods In 2012-13, six rounds of mass-screen-and-treat drug campaigns were carried out in the Lake Kariba region. A spatial dynamical model of malaria transmission in the Lake Kariba area, with transmission and climate modeled at the village scale, was calibrated to the 2012-13 prevalence survey data, with case management rates, insecticide-treated net usage, and drug campaign coverage informed by surveillance. The model was used to simulate the effect of various interventions implemented in 2014-22 on reducing regional transmission, achieving elimination by 2022, and maintaining elimination through 2028. Findings The model captured the spatio-temporal trends of decline and rebound in malaria prevalence in 2012-13 at the village scale. Simulations predicted that elimination required repeated mass drug administrations coupled with simultaneous increase in net usage. Drug campaigns targeted only at high-burden areas were as successful as campaigns covering the entire region. Interpretation Elimination in the Lake Kariba region is possible through coordinating mass drug campaigns with high-coverage vector control. Targeting regional hotspots is a viable alternative to global campaigns when human migration within an interconnected area is responsible for maintaining transmission in low-burden areas

Long paths in first passage percolation on the complete graph I. Local PWIT dynamics

We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed edge weights. We find classes with different behaviour depending on a sequence of parameters (sn)n≥1 that quantifies the extreme-value behavior of small weights. We consider both n-independent as well as n-dependent edge weights and illustrate our results in many examples. In particular, we investigate the case where sn → ∞, and focus on the exploration process that grows the smallest-weight tree from a vertex. We establish that the smallest-weight tree process locally converges to the invasion percolation cluster on the Poisson-weighted infinite tree, and we identify the scaling limit of the weight of the smallest-weight path between two uniform vertices. In addition, we show that over a long time interval, the growth of the smallest-weight tree maintains the same volume-height scaling exponent – volume proportional to the square of the height – found in critical Galton–Watson branching trees and critical Erdős-Rényi random graphs