A simple explicit bijection between (n,2) Gog and Magog trapezoids

A sub-problem of the open problem of finding an explicit bijection between alternating sign matrices and totally symmetric self-complementary plane partitions consists in finding an explicit bijection between so-called $(n,k)$ Gog trapezoids and $(n,k)$ Magog trapezoids. A quite involved bijection was found by Biane and Cheballah in the case $k=2$. We give here a simpler bijection for this case

Scaling Limits for Random Quadrangulations of Positive Genus

We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given $g$, we consider, for every $n \ge 1$, a random quadrangulation \q_n uniformly distributed over the set of all rooted bipartite quadrangulations of genus $g$ with $n$ faces. We view it as a metric space by endowing its set of vertices with the graph distance. We show that, as $n$ tends to infinity, this metric space, with distances rescaled by the factor $n^{-1/4}$, converges in distribution, at least along some subsequence, toward a limiting random metric space. This convergence holds in the sense of the Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of the choice of the subsequence, the Hausdorff dimension of the limiting space is almost surely equal to 4. Our main tool is a bijection introduced by Chapuy, Marcus, and Schaeffer between the quadrangulations we consider and objects they call well-labeled $g$-trees. An important part of our study consists in determining the scaling limits of the latter

Stochastic calculus for fractional Brownian motion with Hurst exponent H>1/4H>1/4: A rough path method by analytic extension

The $d$-dimensional fractional Brownian motion (FBM for short) $B_t=((B_t^{(1)},...,B_t^{(d)}),t\in\mathbb{R})$ with Hurst exponent $\alpha$, $\alpha\in(0,1)$, is a $d$-dimensional centered, self-similar Gaussian process with covariance ${\mathbb{E}}[B_s^{(i)}B _t^{(j)}]={1/2}\delta_{i,j}(|s|^{2\alpha}+|t|^{2\alpha}-|t-s|^{2 \alpha}).$ The long-standing problem of defining a stochastic integration with respect to FBM (and the related problem of solving stochastic differential equations driven by FBM) has been addressed successfully by several different methods, although in each case with a restriction on the range of either $d$ or $\alpha$. The case $\alpha={1/2}$ corresponds to the usual stochastic integration with respect to Brownian motion, while most computations become singular when $\alpha$ gets under various threshhold values, due to the growing irregularity of the trajectories as $\alpha\to0$. We provide here a new method valid for any $d$ and for $\alpha>{1/4}$ by constructing an approximation $\Gamma(\varepsilon)_t$, $\varepsilon\to0$, of FBM which allows to define iterated integrals, and then applying the geometric rough path theory. The approximation relies on the definition of an analytic process $\Gamma_z$ on the cut plane $z\in\mathbb{C}\setminus\mathbb{R}$ of which FBM appears to be a boundary value, and allows to understand very precisely the well-known (see \citeCQ02) but as yet a little mysterious divergence of L\'evy's area for $\alpha\to{1/4}$.Comment: Published in at http://dx.doi.org/10.1214/08-AOP413 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal Discrete Uniform Generation from Coin Flips, and Applications

This article introduces an algorithm to draw random discrete uniform variables within a given range of size n from a source of random bits. The algorithm aims to be simple to implement and optimal both with regards to the amount of random bits consumed, and from a computational perspective---allowing for faster and more efficient Monte-Carlo simulations in computational physics and biology. I also provide a detailed analysis of the number of bits that are spent per variate, and offer some extensions and applications, in particular to the optimal random generation of permutations.Comment: first draft, 22 pages, 5 figures, C code implementation of algorith

A bijection for nonorientable general maps

We give a different presentation of a recent bijection due to Chapuy and Dol\k{e}ga for nonorientable bipartite quadrangulations and we extend it to the case of nonorientable general maps. This can be seen as a Bouttier--Di Francesco--Guitter-like generalization of the Cori--Vauquelin--Schaeffer bijection in the context of general nonorientable surfaces. In the particular case of triangulations, the encoding objects take a particularly simple form and this allows us to recover a famous asymptotic enumeration formula found by Gao