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

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

Direct Detection of WIMP Dark Matter

The status of the recent efforts in the direct search for Weak Interacting Massive Particle (WIMP) Dark Matter is briefly reviewed and the main achievements illustrated by the contributions presented to TAUP 99. The strategies followed in the quest for WIMPs will be first revisited and then the new results and the future prospects reported.Comment: 12 pages, 11 figures. To be published in the TAUP 99 Proceedings, Nucl. Phys. B (Proc. Suppl.), ed. by M. Froissart, J. Dumarchez and D. Vignau