Templates for geodesic flows

The fact that the modular template coincides with the Lorenz template, discovered by Ghys, implies modular knots have very peculiar properties. We obtain a generalization of these results to other Hecke triangle groups. In this context, the geodesic flow can never be seen as a flow on a subset of $S^3$, and one is led to consider embeddings into lens spaces. We will geometrically construct homeomorphisms from the unit tangent bundles of the orbifolds into the lens spaces, elliminating the need for elliptic functions. Finally we will use these homeomorphisms to compute templates for the geodesic flows. This offers a tool for topologically investigating their otherwise well studied periodic orbits.Comment: 29 pages, 21 figure

When the law of large numbers fails for increasing subsequences of random permutations

Let the random variable $Z_{n,k}$ denote the number of increasing subsequences of length $k$ in a random permutation from $S_n$, the symmetric group of permutations of $\{1,...,n\}$. In a recent paper [Random Structures Algorithms 29 (2006) 277--295] we showed that the weak law of large numbers holds for $Z_{n,k_n}$ if $k_n=o(n^{2/5})$; that is, $$\lim_{n\to\infty}\frac{Z_{n,k_n}}{EZ_{n,k_n}}=1\qquad in probability.$$ The method of proof employed there used the second moment method and demonstrated that this method cannot work if the condition $k_n=o(n^{2/5})$ does not hold. It follows from results concerning the longest increasing subsequence of a random permutation that the law of large numbers cannot hold for $Z_{n,k_n}$ if $k_n\ge cn^{1/2}$, with $c>2$. Presumably there is a critical exponent $l_0$ such that the law of large numbers holds if $k_n=O(n^l)$, with $l<l_0$, and does not hold if $\limsup_{n\to\infty}\frac{k_n}{n^l}>0$, for some $l>l_0$. Several phase transitions concerning increasing subsequences occur at $l=1/2$, and these would suggest that $l_0={1/2}$. However, in this paper, we show that the law of large numbers fails for $Z_{n,k_n}$ if $\limsup_{n\to\infty}\frac{k_n}{n^{4/9}}=\infty$. Thus, the critical exponent, if it exists, must satisfy $l_0\in[{2/5},{4/9}]$.Comment: Published at http://dx.doi.org/10.1214/009117906000000728 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org