The untapped potential of plant thin cell layers

Thin cell layers (TCLs), which contain a small number of cells or tissues, are explants excised from different organs (stems, leaves, roots, inflorescences, flowers, cotyledons, hypocotyls/epicotyls, and embryos). After almost 45 years of research, this culture system has been used for several monocotyledonous and dicotyledonous plants of commercial importance, and for model plants. The limited amount of cells in a TCL is of paramount importance because marker molecules/genes of differentiation can be easily localized in situ in the target/responsive cells. Thus, the use of TCLs has allowed, and continues to allow, for the expansion of knowledge in plant research in a practical and applied manner into the fields of tissue culture and micropropagation, cell and organ genetics, molecular biology, biochemistry, and development. Starting from a brief historical background, the actual and potential uses of the TCL system are briefly reviewed

Percolation and local isoperimetric inequalities

In this paper we establish some relations between percolation on a given graph G and its geometry. Our main result shows that, if G has polynomial growth and satisfies what we call the local isoperimetric inequality of dimension d > 1, then p_c(G) < 1. This gives a partial answer to a question of Benjamini and Schramm. As a consequence of this result we derive, under the additional condition of bounded degree, that these graphs also undergo a non-trivial phase transition for the Ising-Model, the Widom-Rowlinson model and the beach model. Our techniques are also applied to dependent percolation processes with long range correlations. We provide results on the uniqueness of the infinite percolation cluster and quantitative estimates on the size of finite components. Finally we leave some remarks and questions that arise naturally from this work.Comment: 21 pages, 2 figure

On the size of a finite vacant cluster of random interlacements with small intensity

In this paper we establish some properties of percolation for the vacant set of random interlacements, for d at least 5 and small intensity u. The model of random interlacements was first introduced by A.S. Sznitman in arXiv:0704.2560. It is known that, for small u, almost surely there is a unique infinite connected component in the vacant set left by the random interlacements at level u, see arXiv:0808.3344 and arXiv:0805.4106. We estimate here the distribution of the diameter and the volume of the vacant component at level u containing the origin, given that it is finite. This comes as a by-product of our main theorem, which proves a stretched exponential bound on the probability that the interlacement set separates two macroscopic connected sets in a large cube. As another application, we show that with high probability, the unique infinite connected component of the vacant set is `ubiquitous' in large neighborhoods of the origin.Comment: Accepted for publication in Probability Theory and Related Field

Interlacement percolation on transient weighted graphs

In this article, we first extend the construction of random interlacements, introduced by A.S. Sznitman in [arXiv:0704.2560], to the more general setting of transient weighted graphs. We prove the Harris-FKG inequality for this model and analyze some of its properties on specific classes of graphs. For the case of non-amenable graphs, we prove that the critical value u_* for the percolation of the vacant set is finite. We also prove that, once G satisfies the isoperimetric inequality IS_6 (see (1.5)), u_* is positive for the product GxZ (where we endow Z with unit weights). When the graph under consideration is a tree, we are able to characterize the vacant cluster containing some fixed point in terms of a Bernoulli independent percolation process. For the specific case of regular trees, we obtain an explicit formula for the critical value u_*.Comment: 25 pages, 2 figures, accepted for publication in the Elect. Journal of Pro

On the uniqueness of the infinite cluster of the vacant set of random interlacements

We consider the model of random interlacements on $\mathbb{Z}^d$ introduced in Sznitman [Vacant set of random interlacements and percolation (2007) preprint]. For this model, we prove the uniqueness of the infinite component of the vacant set. As a consequence, we derive the continuity in $u$ of the probability that the origin belongs to the infinite component of the vacant set at level $u$ in the supercritical phase $u<u_*$.Comment: Published in at http://dx.doi.org/10.1214/08-AAP547 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org