Rectangular Young tableaux and the Jacobi ensemble

It has been shown by Pittel and Romik that the random surface associated with a large rectangular Young tableau converges to a deterministic limit. We study the fluctuations from this limit along the edges of the rectangle. We show that in the corner, these fluctuations are gaussian wheras, away from the corner and when the rectangle is a square, the fluctuations are given by the Tracy-Widom distribution. Our method is based on a connection with the Jacobi ensemble

On the concentration of measure phenomenon for stable and related random vectors

Concentration of measure is studied, and obtained, for stable and related random vectors.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000028

Random-bit optimal uniform sampling for rooted planar trees with given sequence of degrees and Applications

In this paper, we redesign and simplify an algorithm due to Remy et al. for the generation of rooted planar trees that satisfies a given partition of degrees. This new version is now optimal in terms of random bit complexity, up to a multiplicative constant. We then apply a natural process "simulate-guess-and-proof" to analyze the height of a random Motzkin in function of its frequency of unary nodes. When the number of unary nodes dominates, we prove some unconventional height phenomenon (i.e. outside the universal square root behaviour.)Comment: 19 page

Permutations with a prescribed descent set

We give a formula to compute the number of permutations with a prescribed descent set in quadratic time. We give the generating function of the number of permutations with a periodic descent set. We introduce an algorithm generating uniformly distributed random permutations with a prescribed descent set

Generating random alternating permutations in time nlog⁡nn\log n

We introduce an algorithm generating uniformly distributed random alternating permutations of length $n$ in time $n\log n$

Concentration for norms of infinitely divisible vectors with independent components

We obtain dimension-free concentration inequalities for $\ell^p$-norms, $p\geq2$, of infinitely divisible random vectors with independent coordinates and finite exponential moments. Besides such norms, the methods and results extend to some other classes of Lipschitz functions.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ131 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A class of special subordinators with nested ranges

We construct, on a single probability space, a class of special subordinators $S^{(\alpha)}$, indexed by all measurable functions $\alpha: [0,1]\to [0,1]$. Constant functions correspond to stable subordinators. If $\alpha\leq \beta$, then the range of $S^{(\alpha)}$ is contained in the range of $S^{(\beta)}$. Other examples of special subordinators are given in the lattice case

A MDL-based Model of Gender Knowledge Acquisition

This paper presents an iterative model of\ud knowledge acquisition of gender information\ud associated with word endings in\ud French. Gender knowledge is represented\ud as a set of rules containing exceptions.\ud Our model takes noun-gender pairs as input\ud and constantly maintains a list of\ud rules and exceptions which is both coherent\ud with the input data and minimal with\ud respect to a minimum description length\ud criterion. This model was compared to\ud human data at various ages and showed a\ud good fit. We also compared the kind of\ud rules discovered by the model with rules\ud usually extracted by linguists and found\ud interesting discrepancies

Fluctuations of lattice zonotopes and polygons

Following Barany et al., who proved that large random lattice zonotopes converge to a deterministic shape in any dimension after rescaling, we establish a central limit theorem for finite-dimensional marginals of the boundary of the zonotope. In dimension 2, for large random convex lattice polygons contained in a square, we prove a Donsker-type theorem for the boundary fluctuations, which involves a two-dimensional Brownian bridge and a drift term that we identify as a random cubic curve