Self-similar scaling limits of Markov chains on the positive integers

We are interested in the asymptotic behavior of Markov chains on the set of positive integers for which, loosely speaking, large jumps are rare and occur at a rate that behaves like a negative power of the current state, and such that small positive and negative steps of the chain roughly compensate each other. If $X_{n}$ is such a Markov chain started at $n$, we establish a limit theorem for $\frac{1}{n}X_{n}$ appropriately scaled in time, where the scaling limit is given by a nonnegative self-similar Markov process. We also study the asymptotic behavior of the time needed by $X_{n}$ to reach some fixed finite set. We identify three different regimes (roughly speaking the transient, the recurrent and the positive-recurrent regimes) in which $X_{n}$ exhibits different behavior. The present results extend those of Haas & Miermont who focused on the case of non-increasing Markov chains. We further present a number of applications to the study of Markov chains with asymptotically zero drifts such as Bessel-type random walks, nonnegative self-similar Markov processes, invariance principles for random walks conditioned to stay positive, and exchangeable coalescence-fragmentation processes.Comment: 39 pages, 1 figure. Final version: to appear in Ann. Appl. Proba

Random non-crossing plane configurations: A conditioned Galton-Watson tree approach

We study various models of random non-crossing configurations consisting of diagonals of convex polygons, and focus in particular on uniform dissections and non-crossing trees. For both these models, we prove convergence in distribution towards Aldous' Brownian triangulation of the disk. In the case of dissections, we also refine the study of the maximal vertex degree and validate a conjecture of Bernasconi, Panagiotou and Steger. Our main tool is the use of an underlying Galton-Watson tree structure.Comment: 24 pages, 9 figure

Triangulating stable laminations

We study the asymptotic behavior of random simply generated noncrossing planar trees in the space of compact subsets of the unit disk, equipped with the Hausdorff distance. Their distributional limits are obtained by triangulating at random the faces of stable laminations, which are random compact subsets of the unit disk made of non-intersecting chords coded by stable L\'evy processes. We also study other ways to "fill-in" the faces of stable laminations, which leads us to introduce the iteration of laminations and of trees.Comment: 34 pages, 5 figure

Random stable laminations of the disk

We study large random dissections of polygons. We consider random dissections of a regular polygon with $n$ sides, which are chosen according to Boltzmann weights in the domain of attraction of a stable law of index $\theta\in(1,2]$. As $n$ goes to infinity, we prove that these random dissections converge in distribution toward a random compact set, called the random stable lamination. If $\theta=2$, we recover Aldous' Brownian triangulation. However, if $\theta\in(1,2)$, large faces remain in the limit and a different random compact set appears. We show that the random stable lamination can be coded by the continuous-time height function associated to the normalized excursion of a strictly stable spectrally positive L\'{e}vy process of index $\theta$. Using this coding, we establish that the Hausdorff dimension of the stable random lamination is almost surely $2-1/\theta$.Comment: Published in at http://dx.doi.org/10.1214/12-AOP799 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org