Inequalities for rare events in time-reversible Markov chains. I

Aldous, David; Brown, Mark

doi:10.1214/lnms/1215461937

Cited by 51 publications

(79 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will give simple and practical hypotheses to ensure that they are close in some sense, then we will be able to prove some kind of recurrence property for µ * R . In doing so we will also answer some problems left open in [9] (see our comment after formula (2.31)). All this will be done in the simplest possible setup: considering a Markov process on a finite configuration space in some asymptotic regime (including the possibility of sending to infinity the cardinality of the configuration space).…”

Section: Starting Ideasmentioning

confidence: 84%

“…Since 1 − φ * R is the largest eigenvalue of p * R , we have 9) then the equality in Lemma 2.2 is a consequence of Lemma 3.1. Taking f = 1 R as test function in (3.9), we get 10) and it remains to prove that E µ R τ X\R lies between 1/φ R and 1/φ * R .…”

Section: Proof Of Lemma 22 Let Us Denote By L *mentioning

confidence: 89%

“…The main advantage of µ * R is that the exit law of R for the system started in µ * R is an exponential law. Our first results will then start, as in [9], with a comparison between µ R and µ * R . We will give simple and practical hypotheses to ensure that they are close in some sense, then we will be able to prove some kind of recurrence property for µ * R .…”

Section: Starting Ideasmentioning

confidence: 99%

“…This can be seen in a different way by following [9]. Consider, for any t > 0 the natural coupling up to time t ∧ τ X\R between X started from a measure ν and a process that starts from X(0), follows the law of the reflected process up to time t, and then the same law as the original process.…”

Section: Lemma 27 It Holdsmentioning

confidence: 99%

“…One of the questions raised in [9] is that of upper bounds on mean exit times, i.e., that of lower bounds for φ * R . This is the question we will now adress.…”

Section: Lemma 27 It Holdsmentioning

confidence: 99%

See 4 more Smart Citations

Metastable states, quasi-stationary distributions and soft measures

Bianchi

Gaudillière

2016

Stochastic Processes and their Applications

View full text Add to dashboard Cite

ABSTRACT. We establish metastability in the sense of Lebowitz and Penrose under practical and simple hypotheses for Markov chains on a finite configuration space in some asymptotic regime. By comparing restricted ensemble and quasi-stationary measures, and introducing soft measures as interpolation between the two, we prove asymptotic exponential exit law and, on a generally different time scale, asymptotic exponential transition law. By using potential-theoretic tools, and introducing "(κ, λ)-capacities", we give sharp estimates on relaxation time, as well as mean exit time and transition time. We also establish local thermalization on shorter time scales. (a) only one thermodynamic phase is present, (b) a system that starts in this state is likely to take a long time to get out, (c) once the system has gotten out, it is unlikely to return. We can think, for example, about freezing fog made of small droplets in which only one phase is present (liquid phase) that remains for a long time in such a state (until collision with ground or trees, forming then hard rime) and that once frozen will typically not return to liquid state before pressure or temperature have changed.To model such a state they considered in [4] a deterministic dynamics with equilibrium measure µ. First, they associated with the metastable phase a subset R of the configuration space, and described this metastable state by the restricted ensemble µ R = µ(·|R). Second, they proved that the escape rate from R of the system started in µ R is maximal at time t = 0, and that this initial escape rate is very small. Last, they used standard methods of equilibrium statistical mechanics to deal with (c). As an estimate of the returning probability to the metastable state they used the fraction of members of the equilibrium ensemble that have configurations in R and they noted ([4], Section 8):This amounts to assuming that a system whose dynamical state has just left R is no more likely to return to it than one whose dynamical state was never anywhere near R. The validity of this assumption, at least in the short run, is dubious, but at least it provides us with some indication of what to expect. In this paper we want to give a different model for the same phenomenology that overcomes the last difficulty. We will work with stochastic processes rather than deterministic dynamics, but the Lebowitz-Penrose modelization will be our guideline. We will try to recover this phenomenology under simple and practical hypotheses only. Since the study of metastability has been considerably enriched after the Lebowitz and Penrose work, we want also to incorporate in our modeling as much as possible of what was previously achieved. We will then make a brief and partial review of these achievements. Our goals and starting ideas will depend on this review but not our proofs, since we want to make this paper as self-contained as possible. Our model and results are presented in Section 2. Examples of applications are given in Section 7.

show abstract

Section: Starting Ideasmentioning

confidence: 84%

Section: Proof Of Lemma 22 Let Us Denote By L *mentioning

confidence: 89%

Section: Starting Ideasmentioning

confidence: 99%

Section: Lemma 27 It Holdsmentioning

confidence: 99%

“…One of the questions raised in [9] is that of upper bounds on mean exit times, i.e., that of lower bounds for φ * R . This is the question we will now adress.…”

Section: Lemma 27 It Holdsmentioning

confidence: 99%

See 3 more Smart Citations

Metastable states, quasi-stationary distributions and soft measures

Bianchi

Gaudillière

2016

Stochastic Processes and their Applications

View full text Add to dashboard Cite

show abstract

On the fragmentation of a torus by random walk

Teixeira

Windisch

2011

Comm Pure Appl Math

View full text Add to dashboard Cite

We consider a simple random walk on a discrete torus (Z/NZ)^d with dimension d at least 3 and large side length N. For a fixed constant u > 0, we study the percolative properties of the vacant set, consisting of the set of vertices not visited by the random walk in its first [uN^d] steps. We prove the existence of two distinct phases of the vacant set in the following sense: if u > 0 is chosen large enough, all components of the vacant set contain no more than a power of log(N) vertices with high probability as N tends to infinity. On the other hand, for small u > 0, there exists a macroscopic component of the vacant set occupying a non degenerate fraction of the total volume N^d. In dimensions d at least 5, we additionally prove that this macroscopic component is unique, by showing that all other components have volumes of order at most a power of log(N). Our results thus solve open problems posed by Benjamini and Sznitman in arXiv:math/0610802, who studied the small u regime in high dimension. The proofs are based on a coupling of the random walk with random interlacements on Z^d. Among other techniques, the construction of this coupling employs a refined use of discrete potential theory. By itself, this coupling strengthens a result in arXiv:0802.3654.Comment: 37 pages, 3 figure

show abstract

Distinctness of compositions of an integer: A probabilistic analysis

Hitczenko

Louchard

2001

Random Struct Algorithms

View full text Add to dashboard Cite

ABSTRACT:Compositions of integers are used as theoretical models for many applications.The degree of distinctness of a composition is a natural and important parameter. In this article, we use as measure of distinctness the number of distinct parts (or components). We investigate, from a probabilistic point of view, the first empty part, the maximum part size and the distribution of the number of distinct part sizes. We obtain asymptotically, for the classical composition of an integer, the moments and an expression for a continuous distribution F, the (discrete) distribution of the number of distinct part sizes being computable from F. We next analyze another composition: the Carlitz one, where two successive parts are different. We use tools such as analytical depoissonization, Mellin transforms, Markov chain potential theory, limiting hitting times, singularity analysis and perturbation analysis.

show abstract

Inequalities for rare events in time-reversible Markov chains. I

Cited by 51 publications

References 0 publications

Metastable states, quasi-stationary distributions and soft measures

Metastable states, quasi-stationary distributions and soft measures

On the fragmentation of a torus by random walk

Distinctness of compositions of an integer: A probabilistic analysis

Contact Info

Product

Resources

About