Is the critical percolation probability local?

We show that the critical probability for percolation on a d-regular non-amenable graph of large girth is close to the critical probability for percolation on an infinite d-regular tree. We also prove a finite analogue of this statement, valid for expander graphs, without any girth assumption.     

A generalized Pólya's urn with graph based interactions

Given a finite connected graph G, place a bin at each vertex. Two bins are called a pair if they share an edge of G. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls raised by some fixed power . We characterize the limiting behavior of the proportion of balls in the bins. The proof uses a dynamical approach to relate the proportion of balls to a vector field. Our main result is that the limit set of the proportion of balls is contained in the equilibria set of the vector field. We also prove that if then there is a single point with non‐zero entries such that the proportion converges to v almost surely. A special case is when G is regular and . We show e.g. that if G is non‐bipartite then the proportion of balls in the bins converges to the uniform measure almost surely.Copyright © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46,614–634, 2015     

The Best Constant in the Embedding of WN,1(ℝN) into L ∞ ( ℝ N ) $L^{\infty }(\mathbb {R} ^{N})$

Potential Analysis - We compute the best constant in the embedding of $W^{N,1}(\mathbb {R} ^{N})$ into $L^{\infty }(\mathbb {R} ^{N})$ , extending a result of Humbert and Nazaret in dimensions one...     

Submean Variance Bound for Effective Resistance of Random Electric Networks

We study a model of random electric networks with Bernoulli resistances. In the case of the lattice \({\mathbb{Z}^2}\) , we show that the point-to-point effective resistance between 0 and a vertex v has a variance of order at most \({({\rm log} |v|)^{\frac{2}{3}}}\) , whereas its expected value is of order log | v|, when v goes to infinity. When d ≠ 2, expectation and variance are of the same order. Similar results are obtained in the context of p- resistance. The proofs rely on a modified Poincaré inequality due to Falik and Samorodnitsky [7].     

Selective Terminal Functionalization of Linear Alkanes**

A two-step sequential strategy involving a biocatalytic dehydrogenation/remote hydrofunctionalization, as a unified and versatile approach to selectively convert linear alkanes into a large array of valuable functionalized aliphatic derivatives is reported. The dehydrogenation is carried out by a mutant strain of a bacteria Rhodococcus and the produced alkenes are subsequently engaged in a remote functionalization through a metal-catalyzed hydrometalation/migration sequence that subsequently react with a large variety of electrophiles. The judicious implementation of this combined biocatalytic and organometallic approach enabled us to develop a high-yielding protocol to site-selectively functionalize unreactive primary C−H bonds.