The speed of biased random walk on percolation clusters

 We consider biased random walk on supercritical percolation clusters in ℤ². We show that the random walk is transient and that there are two speed regimes: If the bias is large enough, the random walk has speed zero, while if the bias is small enough, the speed of the random walk is positive. Received: 20 November 2002 / Revised version: 17 January 2003 Published online: 15 April 2003 Research supported by Microsoft Research graduate fellowship. Research partially supported by the DFG under grant SPP 1033. Research partially supported by NSF grant #DMS-0104073 and by a Miller Professorship at UC Berkeley. Mathematics Subject Classification (2000): 60K37; 60K35; 60G50 Key words or phrases: Percolation – Random walk     

On the cover time of random walks on graphs

This article deals with random walks on arbitrary graphs. We consider the cover time of finite graphs. That is, we study the expected time needed for a random walk on a finite graph to visit every vertex at least once. We establish an upper bound ofO(n ²) for the expectation of the cover time for regular (or nearly regular) graphs. We prove a lower bound of (n logn) for the expected cover time for trees. We present examples showing all our bounds to be tight.Mike Saks was supported by NSF-DMS87-03541 and by AFOSR-0271. Jeff Kahn was supported by MCS-83-01867 and by AFOSR-0271.     

Electronic determinants of photoacidity in cyanonaphthols

We present semiempirical AM1 calculations for the ground and excited state of 2-naphthol and some of its cyano derivatives in the gas phase. Following photoexcitation, the Mulliken electron density on the oxygen diminishes slightly for the acid and more conspicuously for the anionic conjugated base. This agrees with the measured solvatochromic parameters for 2-naphthol. In both electronic states, we find a nice correlation with the measured pK values in water. The electronic charge distribution on the distal ring of the anion agrees with the experimental acidity order in both S(0) and S(1). Upon excitation, it increases predominantly in positions 3, 5, and 8. The ring system of the anion assumes an alternate quinoidal structure in the ground state of the anion, which becomes more symmetric in the relaxed excited state. This suggests that the enhanced aromatic character of a 4n electron system in the excited state allows for better delocalization of the oxygen charge within the ring.     

Organo tin nucleophiles iii. Palladium catalyzed reductive cleavage of allylic heterosubstituents with tin hydride

Tributyl tin hydride, serving as an efficient hydride transfer agent, allows highly chemoselective palladium catalyzed reductions of allylic heterosubstituents even in presence of aldehydes, benzyl acetate and benzyl chloride.