On the width of handles in two-dimensional quantum gravity

We discuss the average length l of the shortest non-contractible loop on surfaces in the two-dimensional pure quantum gravity ensemble. The value of $\gamma_{str}$ and the explicit form of the loop functions indicate that l diverges at the critical point. Scaling arguments suggest that the critical exponent of l is 1/2. We show that this value of the critical exponent is also obtained for branched polymers where the calculation is straightforward.Comment: 7 pages, 1 ps figure, late

Nanophotonics with the scanning electron microscope

Physical size and power consumption are both increasingly important issues in increasing the data throughput of future optical interconnects, switches and ultimately even optical memory elements. In this respect, phase-change memories have proven to be strong candidates, with data recording done by switching the material between amorphous and crystalline phases, much in line with today's DVD/DVR technology. However, polymorphic systems exist in which crystalline-to-crystalline transitions can provide for higher-base logics as well. In particular, by coding each distinct optical characteristic by a unique label, the different optical cross-sections of absorption and scattering of the crystalline phases of a single nanoparticle can be used as a logical element

Stabilization of monomial maps

A monomial (or equivariant) selfmap of a toric variety is called stable if its action on the Picard group commutes with iteration. Generalizing work of Favre to higher dimensions, we show that under suitable conditions, a monomial map can be made stable by refining the underlying fan. In general, the resulting toric variety has quotient singularities; in dimension two we give criteria for when it can be chosen smooth, as well as examples when it cannot.Comment: To appear in Michigan Math.