Importance of the topological defects for two dimensional phase transitions and their relevance for the renormalization group

For various two dimensional non linear $\sigma$ models, we present a direct comparison between the $\beta$ functions computed with the $2+\epsilon$ renormalization group and the $\beta$ functions measured by Monte Carlo simulations. The theoretical and measured $\beta$ functions match each other nicely for models with a trivial topology, yet they disagree clearly for models containing topological defects. In these later cases, they are compatible with a phase transition at a finite temperature. This indicates that the global properties of the manifold do matter, in contradiction with the assumption used in the $2+\epsilon$ RG computation.Comment: RevTex file, 8 pages, to appears in Phys.Lett.

Heterogeneous volatility cascade in financial markets

Using high frequency data, we have studied empirically the change of volatility, also called volatility derivative, for various time horizons. In particular, the correlation between the volatility derivative and the volatility realized in the next time period is a measure of the response function of the market participants. This correlation shows explicitly the heterogeneous structure of the market according to the characteristic time horizons of the differents agents. It reveals a volatility cascade from long to short time horizons, with a structure different from the one observed in turbulence. Moreover, we have developed a new ARCH-type model which incorporates the different groups of agents, with their characteristic memory. This model reproduces well the empirical response function, and allows us to quantify the importance of each group.Comment: 10 pages, 2 figures, To be published in Physica