Cone types and asymptotic invariants for the random walk on the modular group

We compute the cone types of the Cayley graph of the modular group $\mathrm{PSL}(2,\mathbf{Z})$ associated with the standard system of generators ${\small\left\{\left(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right),\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right)\right\}}$. We do this by showing that, in general, there is a set of suffixes of each element that completely determines the cone type of the element, and such suffixes are subwords of primitive relators. Then, using J. W. Cannon's seminal ideas (1984), we compute its growth function. We estimate from above and below the spectral radius of the random walk using ideas from T. Nagnibeda (1999) and S. Gou\"ezel (2015). Finally, using results of Y. Guivarc'h (1980) and S. Gou\"ezel, F. Math\'{e}us and F. Maucourant (2015), we estimate other asymptotic invariants of the random walk, namely, the entropy and the drift.Comment: 31 pages, 6 figures, 11 table

Number of cyclic square-tiled tori

We study cyclic square-tiled tori in $\mathcal{H}(0)$, answering a question by M. Bolognesi (by personal communication to A. Zorich). We give the exact number of cyclic tori tiled by $n\in\mathbb{N}$ squares. We also give the asymptotic proportion of cyclic square-tiled tori over all square-tiled tori.Comment: 6 pages, 1 figur

Counting problem on wind-tree models

We study periodic wind-tree models, billiards in the plane endowed with $\mathbb{Z}^2$-periodically located identical connected symmetric right-angled obstacles. We show asymptotic formulas for the number of (isotopy classes of) closed billiard trajectories (up to $\mathbb{Z}^2$-translations) on the wind-tree billiard. We also compute explicitly the associated Siegel-Veech constant for generic wind-tree billiards depending on the number of corners on the obstacle.Comment: 41 pages, 15 figures. arXiv admin note: substantial text overlap with arXiv:1502.06405 by other author