How to Couple from the Past Using a Read-Once Source of Randomness

We give a new method for generating perfectly random samples from the stationary distribution of a Markov chain. The method is related to coupling from the past (CFTP), but only runs the Markov chain forwards in time, and never restarts it at previous times in the past. The method is also related to an idea known as PASTA (Poisson arrivals see time averages) in the operations research literature. Because the new algorithm can be run using a read-once stream of randomness, we call it read-once CFTP. The memory and time requirements of read-once CFTP are on par with the requirements of the usual form of CFTP, and for a variety of applications the requirements may be noticeably less. Some perfect sampling algorithms for point processes are based on an extension of CFTP known as coupling into and from the past; for completeness, we give a read-once version of coupling into and from the past, but it remains unpractical. For these point process applications, we give an alternative coupling method with which read-once CFTP may be efficiently used.Comment: 28 pages, 2 figure

Random Forests and Networks Analysis

D. Wilson~\cite{[Wi]} in the 1990's described a simple and efficient algorithm based on loop-erased random walks to sample uniform spanning trees and more generally weighted trees or forests spanning a given graph. This algorithm provides a powerful tool in analyzing structures on networks and along this line of thinking, in recent works~\cite{AG1,AG2,ACGM1,ACGM2} we focused on applications of spanning rooted forests on finite graphs. The resulting main conclusions are reviewed in this paper by collecting related theorems, algorithms, heuristics and numerical experiments. A first foundational part on determinantal structures and efficient sampling procedures is followed by four main applications: 1) a random-walk-based notion of well-distributed points in a graph 2) how to describe metastable dynamics in finite settings by means of Markov intertwining dualities 3) coarse graining schemes for networks and associated processes 4) wavelets-like pyramidal algorithms for graph signals.Comment: Survey pape

Markov chains in a Dirichlet Environment and hypergeometric integrals

6 pages, preliminary note.International audienceThe aim of this text is to establish some relations between Markov chains in Dirichlet Environments on directed graphs and certain hypergeometric integrals associated with a particular arrangement of hyperplanes. We deduce from these relations and the computation of the connexion obtained by moving one hyperplane of the arrangement some new relations on important functionals of the Markov chain