Phase Transition and Uniqueness of Levelset Percolation

The main purpose of this paper is to introduce and establish basic results of a natural extension of the classical Boolean percolation model (also known as the Gilbert disc model). We replace the balls of that model by a positive non-increasing attenuation function \(l:(0,\infty ) \rightarrow [0,\infty )\) to create the random field \(\Psi (y)=\sum _{x\in \eta }l(|x-y|),\) where \(\eta \) is a homogeneous Poisson process in \({\mathbb {R}}^d.\) The field \(\Psi \) is then a random potential field with infinite range dependencies whenever the support of the function l is unbounded. In particular, we study the level sets \(\Psi _{\ge h}(y)\) containing the points \(y\in {\mathbb {R}}^d\) such that \(\Psi (y)\ge h.\) In the case where l has unbounded support, we give, for any \(d\ge 2,\) a necessary and sufficient condition on l for \(\Psi _{\ge h}(y)\) to have a percolative phase transition as a function of h. We also prove that when l is continuous then so is \(\Psi \) almost surely. Moreover, in this case and for \(d=2,\) we prove uniqueness of the infinite component of \(\Psi _{\ge h}\) when such exists, and we also show that the so-called percolation function is continuous below the critical value \(h_c\).     

Noise sensitivity in continuum percolation

We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first which involves a percolation model with critical probability p _c ≠ 1/2. Our proof uses a version of the Benjamini-Kalai-Schramm Theorem for biased product measures. A quantitative version of this result was recently proved by Keller and Kindler. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete models with p _c bounded away from zero; this method is based on an extremal result on non-uniform hypergraphs.     

Stochastic Ordering of Infinite Geometric Galton–Watson Trees

We consider Galton–Watson trees with Geom\((p)\) offspring distribution. We let \(T_{\infty }(p)\) denote such a tree conditioned on being infinite. We prove that for any \(1/2\le p_1 <p_2 \le 1\), there exists a coupling between \(T_{\infty }(p_1)\) and \(T_{\infty }(p_2)\) such that \({\mathbb {P}}(T_{\infty }(p_1) \subseteq T_{\infty }(p_2))=1\).     

Linear Free‐Energy Correlations for the Vinylheptafulvene Ring Closure: A Probe for Hammett σ Values

Linear free‐energy relationships, like Hammett correlations, are fundamental in physical organic chemistry for the elucidation of reaction mechanisms. In this work, we show that Hammett correlations exist for the ring closure of six different model systems of vinylheptafulvenes (VHFs) to their corresponding dihydroazulenes (DHAs). These first‐order reactions were easily followed by UV/Vis absorption spectroscopy on account of the significantly different absorption characteristics between VHFs and DHAs. Opposing effects displayed by substituent groups at two different positions are conveniently accounted for by simply subtracting the two Hammett σ values of each group. The linear correlations readily allow us to obtain unknown and approximate Hammett σ values for previously uninvestigated substituents. We also show that they can provide alternative values to the standard ones. We present values for a variety of substituent groups ranging from alkynes, sulfones, sulfoxides, and different heteroaromatics. The electronic effects exerted by substituent groups on VHFs are also reflected in their absorption maxima. Thus, we have established an empirical relationship between the absorption maximum of the VHF and the Hammett σ values of its substituents. This fine‐tuning of electronic properties is particularly important for the ongoing efforts of using the DHA/VHF molecular switch in molecular electronics devices.     

Dimension (In)equalities and Hölder Continuous Curves in Fractal Percolation

We relate various concepts of fractal dimension of the limiting set $\mathcal{C}$ in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in  $\mathcal{C}$ (the “dust”). In two dimensions, we also show that the set consisting of connected components larger than one point is almost surely the union of non-trivial Hölder continuous curves, all with the same exponent. Finally, we give a short proof of the fact that in two dimensions, any curve in the limiting set must have Hausdorff dimension strictly larger than 1.     

New synthetic route to substituted dihydroazulene photoswitches

A new procedure for functionalization of the dihydroazulene photoswitch on its seven-membered ring was developed, which has allowed isolation of the first dihydroazulene with a phenyl substituent at position 5 from a mixture of regioisomers. Light-induced ring-opening to the corresponding vinylheptafulvene and the thermal back-reaction was studied in detail.