Abrupt Convergence and Escape Behavior for Birth and Death Chains

Barrera, Javiera; Bertoncini, Olivier; Fernández, Roberto

doi:10.1007/s10955-009-9861-7

Cited by 30 publications

(53 citation statements)

References 51 publications

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“…In this section we recall general results on this birth and death model and in the next section we discuss how to apply such results to our 2D model. We mainly follow [6] for the general discussion. We derive explicit formulas in two specific simple cases, that will turn out to be very important from the physical point of view.…”

Section: A Biased Birth and Death Modelmentioning

confidence: 99%

Residence time estimates for asymmetric simple exclusion dynamics on strips

Cirillo

Krehel

Muntean

et al. 2016

Physica A: Statistical Mechanics and its Applications

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The target of our study is to approximate numerically and, in some particular physically relevant cases, also analytically, the residence time of particles undergoing an asymmetric simple exclusion dynamics on a vertical strip. The source of asymmetry is twofold: (i) the choice of boundary conditions (different reservoir levels) and (ii) the strong anisotropy from a nonlinear drift with prescribed directionality. We focus on the effect of the choice of anisotropy in the flux on the asymptotic behavior of the residence time with respect to the length of the strip. The topic is relevant for situations occurring in pedestrian flows or biological transport in crowded environments, where lateral displacements of the particles occur predominantly affecting therefore in an essentially way the efficiency of the ENMC thanks Kerry Landman (Melbourne), Roberto Fernández (Utrecht), and Lorenzo Bertini (Roma) for very stimulating discussions and useful references. cms-serw001.tex -1 agosto 2018 22:05 arXiv:1411.5490v2 [nlin.CG] 9 Feb 2015 overall transport mechanism.

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Section: A Biased Birth and Death Modelmentioning

confidence: 99%

Residence time estimates for asymmetric simple exclusion dynamics on strips

Cirillo

Krehel

Muntean

et al. 2016

Physica A: Statistical Mechanics and its Applications

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“…Ding, Lubetzky and Peres [29] proved that the observation held without these caveats as well (in discrete time, from any start, and in total variation) so long as the chain is lazy, namely the probability of holding at any given point is at least δ > 0, where δ is a constant. Further developments on birth/death cutoffs are seen in Barrera, Bertoncini, and Fernández [3], Diehl [28], and Chen and Saloff‐Coste [11,12,13] Other related results may be found in Lubetzky and Sly [39] and in Pak and Vu [42]. Another step forward: Chen and Saloff‐Coste [11,12,13] have proved that the Peres observation is true in l p distances, p > 1, for any sequence of reversible Markov chains.…”

Section: Introductionmentioning

confidence: 99%

Random doubly stochastic tridiagonal matrices

Diaconis

Wood

2012

Random Struct Algorithms

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Let \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathcal T}\end{align*} \end{document}n be the compact convex set of tridiagonal doubly stochastic matrices. These arise naturally in probability problems as birth and death chains with a uniform stationary distribution. We study ‘typical’ matrices T∈ \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathcal T}\end{align*} \end{document}n chosen uniformly at random in the set \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathcal T}\end{align*} \end{document}n. A simple algorithm is presented to allow direct sampling from the uniform distribution on \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathcal T}\end{align*} \end{document}n. Using this algorithm, the elements above the diagonal in T are shown to form a Markov chain. For large n, the limiting Markov chain is reversible and explicitly diagonalizable with transformed Jacobi polynomials as eigenfunctions. These results are used to study the limiting behavior of such typical birth and death chains, including their eigenvalues and mixing times. The results on a uniform random tridiagonal doubly stochastic matrices are related to the distribution of alternating permutations chosen uniformly at random.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 403–437, 2013

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“…The results presented in this section deserve interest in some physical contexts. We recall that a family of random variables U (λ) exhibits cut-off behavior at mean times if (see, for instance, Definition 1 of [Barrera (2009)…”

Section: Convergence Resultsmentioning

confidence: 99%

“…We finally remark that in this context the sufficient condition given in Proposition 1 of [Barrera (2009)] is not useful to prove Proposition 4.1, since such condition holds only when ν = 1.…”

Section: Convergence Resultsmentioning

confidence: 99%

A fractional counting process and its connection with the Poisson process

Crescenzo

Martinucci²,

Meoli³

2016

ALEA

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We consider a fractional counting process with jumps of amplitude 1, 2, . . . , k, with k ∈ N, whose probabilities satisfy a suitable system of fractional difference-differential equations. We obtain the moment generating function and the probability law of the resulting process in terms of generalized Mittag-Leffler functions. We also discuss two equivalent representations both in terms of a compound fractional Poisson process and of a subordinator governed by a suitable fractional Cauchy problem. The first occurrence time of a jump of fixed amplitude is proved to have the same distribution as the waiting time of the first event of a classical fractional Poisson process, this extending a well-known property of the Poisson process. When k = 2 we also express the distribution of the first passage time of the fractional counting process in an integral form. Finally, we show that the ratios given by the powers of the fractional Poisson process and of the counting process over their means tend to 1 in probability.

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Abrupt Convergence and Escape Behavior for Birth and Death Chains

Cited by 30 publications

References 51 publications

Residence time estimates for asymmetric simple exclusion dynamics on strips

Residence time estimates for asymmetric simple exclusion dynamics on strips

Random doubly stochastic tridiagonal matrices

A fractional counting process and its connection with the Poisson process

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