scholarly journals Some generalized juggling processes (extended abstract)

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Arvind Ayyer ◽  
Jérémie Bouttier ◽  
Svante Linusson ◽  
François Nunzi
Keyword(s):  
Markov Chains ◽  
Closed Form ◽  
Stationary Probability ◽  
Normalization Factor ◽  
Finite Models ◽  
International Audience ◽  
Product Formulas

International audience We consider generalizations of juggling Markov chains introduced by Ayyer, Bouttier, Corteel and Nunzi. We first study multispecies generalizations of all the finite models therein, namely the MJMC, the add-drop and the annihilation models. We then consider the case of several jugglers exchanging balls. In all cases, we give explicit product formulas for the stationary probability and closed-form expressions for the normalization factor if known. On s’intéresse à des généralisations des chaînes de Markov de jonglage introduites par Ayyer, Bouttier, Corteel et Nunzi. On étudie d’abord des généralisations multiespèces de tous les modèles finis, à savoir le MJMC et les modèles d’add-drop et d’annihilation. On considère ensuite le cas de plusieurs jongleurs échangeant des balles entre eux. Dans chacun des cas, on donne une formule explicite sous forme de produit pour l’état stationnaire, ainsi qu’une forme réduite pour le facteur de normalisation dans les cas où l’on en connaît une.

scholarly journals Random sampling of lattice paths with constraints, via transportation

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Lucas Gerin
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Random Sampling ◽  
Optimal Transport ◽  
Lattice Paths ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Contraction Property ◽  
International Audience

International audience We build and analyze in this paper Markov chains for the random sampling of some one-dimensional lattice paths with constraints, for various constraints. These chains are easy to implement, and sample an "almost" uniform path of length $n$ in $n^{3+\epsilon}$ steps. This bound makes use of a certain $\textit{contraction property}$ of the Markov chain, and is proved with an approach inspired by optimal transport.

scholarly journals On explicit form of the stationary distributions for a class of bounded Markov chains

2016 ◽  
Vol 53 (1) ◽  
pp. 231-243 ◽  
Author(s):  
S. McKinlay ◽  
K. Borovkov
Keyword(s):  
Markov Chains ◽  
Closed Form ◽  
State Space ◽  
Explicit Form ◽  
Discrete Time ◽  
Unit Interval ◽  
Time Step ◽  
Stationary Distributions ◽  
End Point ◽  
Current Location

AbstractWe consider a class of discrete-time Markov chains with state space [0, 1] and the following dynamics. At each time step, first the direction of the next transition is chosen at random with probability depending on the current location. Then the length of the jump is chosen independently as a random proportion of the distance to the respective end point of the unit interval, the distributions of the proportions being fixed for each of the two directions. Chains of that kind were the subjects of a number of studies and are of interest for some applications. Under simple broad conditions, we establish the ergodicity of such Markov chains and then derive closed-form expressions for the stationary densities of the chains when the proportions are beta distributed with the first parameter equal to 1. Examples demonstrating the range of stationary distributions for processes described by this model are given, and an application to a robot coverage algorithm is discussed.

scholarly journals Light-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type

2005 ◽  
Vol 37 (04) ◽  
pp. 1075-1093 ◽  
Author(s):  
Quan-Lin Li ◽  
Yiqiang Q. Zhao
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chain ◽  
Markov Chains ◽  
Asymptotic Analysis ◽  
Positive Solution ◽  
Generating Function ◽  
Spectral Properties ◽  
Stationary Probability ◽  
Probability Vector ◽  
Stationary Probability Vector

In this paper, we consider the asymptotic behavior of stationary probability vectors of Markov chains of GI/G/1 type. The generating function of the stationary probability vector is explicitly expressed by theR-measure. This expression of the generating function is more convenient for the asymptotic analysis than those in the literature. TheRG-factorization of both the repeating row and the Wiener-Hopf equations for the boundary row are used to provide necessary spectral properties. The stationary probability vector of a Markov chain of GI/G/1 type is shown to be light tailed if the blocks of the repeating row and the blocks of the boundary row are light tailed. We derive two classes of explicit expression for the asymptotic behavior, the geometric tail, and the semigeometric tail, based on the repeating row, the boundary row, or the minimal positive solution of a crucial equation involved in the generating function, and discuss the singularity classes of the stationary probability vector.

Geometric bounds on iterative approximations for nearly completely decomposable Markov chains

1990 ◽  
Vol 27 (03) ◽  
pp. 521-529 ◽  
Author(s):  
Guy Louchard ◽  
Guy Latouche
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Perturbation Analysis ◽  
Stochastic Matrix ◽  
Stationary Probability ◽  
Probability Vector ◽  
Finite Markov Chain ◽  
Stationary Probability Vector

We consider a finite Markov chain with nearly-completely decomposable stochastic matrix. We determine bounds for the error, when the stationary probability vector is approximated via a perturbation analysis.

scholarly journals Light-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type

2005 ◽  
Vol 37 (4) ◽  
pp. 1075-1093 ◽  
Author(s):  
Quan-Lin Li ◽  
Yiqiang Q. Zhao
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chain ◽  
Markov Chains ◽  
Asymptotic Analysis ◽  
Positive Solution ◽  
Generating Function ◽  
Spectral Properties ◽  
Stationary Probability ◽  
Probability Vector ◽  
Stationary Probability Vector

In this paper, we consider the asymptotic behavior of stationary probability vectors of Markov chains of GI/G/1 type. The generating function of the stationary probability vector is explicitly expressed by the R-measure. This expression of the generating function is more convenient for the asymptotic analysis than those in the literature. The RG-factorization of both the repeating row and the Wiener-Hopf equations for the boundary row are used to provide necessary spectral properties. The stationary probability vector of a Markov chain of GI/G/1 type is shown to be light tailed if the blocks of the repeating row and the blocks of the boundary row are light tailed. We derive two classes of explicit expression for the asymptotic behavior, the geometric tail, and the semigeometric tail, based on the repeating row, the boundary row, or the minimal positive solution of a crucial equation involved in the generating function, and discuss the singularity classes of the stationary probability vector.

Geometric bounds on iterative approximations for nearly completely decomposable Markov chains

1990 ◽  
Vol 27 (3) ◽  
pp. 521-529 ◽  
Author(s):  
Guy Louchard ◽  
Guy Latouche
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Perturbation Analysis ◽  
Stochastic Matrix ◽  
Stationary Probability ◽  
Probability Vector ◽  
Finite Markov Chain ◽  
Stationary Probability Vector

We consider a finite Markov chain with nearly-completely decomposable stochastic matrix. We determine bounds for the error, when the stationary probability vector is approximated via a perturbation analysis.

scholarly journals On Sampling Colorings of Bipartite Graphs

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
R. Balasubramanian ◽  
C.R. Subramanian
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Mixing Time ◽  
Bipartite Graphs ◽  
Exponential Time ◽  
Negative Results ◽  
Bipartite Subgraph ◽  
Single Transition ◽  
International Audience ◽  
Complete Bipartite Subgraph

International audience We study the problem of efficiently sampling k-colorings of bipartite graphs. We show that a class of markov chains cannot be used as efficient samplers. Precisely, we show that, for any k, 6 ≤ k ≤ n^\1/3-ε \, ε > 0 fixed, \emphalmost every bipartite graph on n+n vertices is such that the mixing time of any markov chain asymptotically uniform on its k-colorings is exponential in n/k^2 (if it is allowed to only change the colors of O(n/k) vertices in a single transition step). This kind of exponential time mixing is called \emphtorpid mixing. As a corollary, we show that there are (for every n) bipartite graphs on 2n vertices with Δ (G) = Ω (\ln n) such that for every k, 6 ≤ k ≤ Δ /(6 \ln Δ ), each member of a large class of chains mixes torpidly. While, for fixed k, such negative results are implied by the work of CDF, our results are more general in that they allow k to grow with n. We also show that these negative results hold true for H-colorings of bipartite graphs provided H contains a spanning complete bipartite subgraph. We also present explicit examples of colorings (k-colorings or H-colorings) which admit 1-cautious chains that are ergodic and are shown to have exponential mixing time. While, for fixed k or fixed H, such negative results are implied by the work of CDF, our results are more general in that they allow k or H to vary with n.

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