Two-dimensional Poisson Trees converge to the Brownian web

The Brownian web can be roughly described as a family of coalescing one-dimensional Brownian motions starting at all times in $\R$ and at all points of $\R$. It was introduced by Arratia; a variant was then studied by Toth and Werner; another variant was analyzed recently by Fontes, Isopi, Newman and Ravishankar. The two-dimensional \emph{Poisson tree} is a family of continuous time one-dimensional random walks with uniform jumps in a bounded interval. The walks start at the space-time points of a homogeneous Poisson process in $\R^2$ and are in fact constructed as a function of the point process. This tree was introduced by Ferrari, Landim and Thorisson. By verifying criteria derived by Fontes, Isopi, Newman and Ravishankar, we show that, when properly rescaled, and under the topology introduced by those authors, Poisson trees converge weakly to the Brownian web.Comment: 22 pages, 1 figure. This version corrects an error in the previous proof. The results are the sam

Factors and Connected Factors in Tough Graphs with High Isolated Toughness

In this paper, we show that every $1$-tough graph with order and isolated toughness at least $r+1$ has a factor whose degrees are $r$, except for at most one vertex with degree $r+1$. Using this result, we conclude that every $3$-tough graph with order and isolated toughness at least $r+1$ has a connected factor whose degrees lie in the set $\{r,r+1\}$, where $r\ge 3$. Also, we show that this factor can be found $m$-tree-connected, when $G$ is a $(2m+\epsilon)$-tough graph with order and isolated toughness at least $r+1$, where $r\ge (2m-1)(2m/\epsilon+1)$ and $\epsilon > 0$. Next, we prove that every $(m+\epsilon)$-tough graph of order at least $2m$ with high enough isolated toughness admits an $m$-tree-connected factor with maximum degree at most $2m+1$. From this result, we derive that every $(2+\epsilon)$-tough graph of order at least three with high enough isolated toughness has a spanning Eulerian subgraph whose degrees lie in the set $\{2,4\}$. In addition, we provide a family of $5/3$-tough graphs with high enough isolated toughness having no connected even factors with bounded maximum degree

Broken R Parity Contributions to Flavor Changing Rates and CP Asymmetries in Fermion Pair Production at Leptonic Colliders

We examine the effects of the R parity odd renormalizable interactions on flavor changing rates and CP violation asymmetries in the production of fermion-antifermion pairs at $e^-- e^+$ leptonic colliders. The produced fermions may be leptons, down-quarks or up-quarks, and the center of mass energies may range from the Z-boson pole up to $1000$ GeV. Off the Z-boson pole, the flavor changing rates are controlled by tree level amplitudes and the CP asymmetries by interference terms between tree and loop level amplitudes. At the Z-boson pole, both observables involve loop amplitudes. The lepton number violating interactions, associated with the coupling constants, \l_{ijk}, \l'_{ijk}, are only taken into account. The consideration of loop amplitudes is restricted to the photon and Z-boson vertex corrections. We briefly review flavor violation physics at colliders. We present numerical results using a single, species and family independent, mass parameter, $\tilde m$, for all the scalar superpartners and considering simple assumptions for the family dependence of the R parity odd coupling constants.Comment: Latex File. 23 pages. 4 postscript figures. 1 table. Revised version with new results and several corrections in numerical result