Trapping in the random conductance model

We consider random walks on $\Z^d$ among nearest-neighbor random conductances which are i.i.d., positive, bounded uniformly from above but whose support extends all the way to zero. Our focus is on the detailed properties of the paths of the random walk conditioned to return back to the starting point at time $2n$. We show that in the situations when the heat kernel exhibits subdiffusive decay --- which is known to occur in dimensions $d\ge4$ --- the walk gets trapped for a time of order $n$ in a small spatial region. This shows that the strategy used earlier to infer subdiffusive lower bounds on the heat kernel in specific examples is in fact dominant. In addition, we settle a conjecture concerning the worst possible subdiffusive decay in four dimensions.Comment: 21 pages, version to appear in J. Statist. Phy

Adult and larval photoreceptors use different mechanisms to specify the same rhodopsin fates

Although development of the adult Drosophila compound eye is very well understood, little is known about development of photoreceptors (PRs) in the simple larval eye. We show here that the larval eye is composed of 12 PRs, four of which express blue-sensitive rhodopsin5 (rh5) while the other eight contain green-sensitive rh6. This is similar to the 30:70 ratio of adult blue and green R8 cells. However, the stochastic choice of adult color PRs and the bistable loop of the warts and melted tumor suppressor genes that unambiguously specify rh5 and rh6 in R8 PRs are not involved in specification of larval PRs. Instead, primary PR precursors signal via EGFR to surrounding tissue to develop as secondary precursors, which will become Rh6-expressing PRs. EGFR signaling is required for the survival of the Rh6 subtype. Primary precursors give rise to the Rh5 subtype. Furthermore, the combinatorial action of the transcription factors Spalt, Seven-up, and Orthodenticle specifies the two PR subtypes. Therefore, even though the larval PRs and adult R8 PRs express the same rhodopsins (rh5 and rh6), they use very distinct mechanisms for their specification

Compounding effects of sea level rise and fluvial flooding

Sea level rise (SLR), a well-documented and urgent aspect of anthropogenic global warming, threatens population and assets located in low-lying coastal regions all around the world. Common flood hazard assessment practices typically account for one driver at a time (e.g., either fluvial flooding only or ocean flooding only), whereas coastal cities vulnerable to SLR are at risk for flooding from multiple drivers (e.g., extreme coastal high tide, storm surge, and river flow). Here, we propose a bivariate flood hazard assessment approach that accounts for compound flooding from river flow and coastal water level, and we show that a univariate approach may not appropriately characterize the flood hazard if there are compounding effects. Using copulas and bivariate dependence analysis, we also quantify the increases in failure probabilities for 2030 and 2050 caused by SLR under representative concentration pathways 4.5 and 8.5. Additionally, the increase in failure probability is shown to be strongly affected by compounding effects. The proposed failure probability method offers an innovative tool for assessing compounding flood hazards in a warming climate