Selberg integrals in 1D random Euclidean optimization problems

We consider a set of Euclidean optimization problems in one dimension, where the cost function associated to the couple of points $x$ and $y$ is the Euclidean distance between them to an arbitrary power $p\ge1$, and the points are chosen at random with flat measure. We derive the exact average cost for the random assignment problem, for any number of points, by using Selberg's integrals. Some variants of these integrals allows to derive also the exact average cost for the bipartite travelling salesman problem.Comment: 9 pages, 2 figure

Formation of waterfalls by intermittent burial of active faults

Waterfalls commonly exist near bounding faults of mountain ranges, where erosional bedrock catchments transition to depositional alluvial fans. We hypothesize that aggradation on alluvial fans can bury active faults, and that the faults accumulate slip in the subsurface to produce a bedrock scarp. Following entrenchment of the alluvial fan, the scarp can be exposed as a waterfall. To explore this hypothesis, we derived a geometric model for waterfall height that depends on alluvial fan length and the relative time scales of (1) tectonic uplift, (2) a forcing mechanism for cycles of fan aggradation and incision, and (3) a response of fan aggradation to changes in sediment flux. We find that the model is consistent with observations at Gower Gulch, Death Valley, California, where a man-made drainage capture event in 1941 caused rapid fan incision and exposed a waterfall at the canyon-fan transition. We also compared the model to 62 waterfalls in 18 catchments of the Death Valley area and found that at least 15 of the waterfalls are best explained by the fault-burial mechanism. Using field measurements of grain size and channel geometries, we show that the fault-burial mechanism can produce the observed waterfall heights, measuring 4−19 m, under a uniform climatic forcing scenario requiring variations of 20% in precipitation during the late Pleistocene. The fault-burial mechanism, through the creation of upstream propagating waterfalls, may allow catchment-fan systems to experience frequent cycles of enhanced erosion in catchments and deposition on fans that likely convolve tectonic and climatic signals

Impact of Climate and Tectonics on the Morphodynamics of Alluvial Piedmonts, Implications for Sediment Transfer and the Stratigraphic Record

Alluvial piedmonts encircle most of the Earth mountain ranges. The erosion product of these mountains must cross the piedmont domain before reaching the basins where they can enter the sedimentary record. The flux of sediment transfers environmental signals, e.g. tectonics and climate, from source to sink, their preservation is critical for the sedimentary record. Alluvial piedmonts are very reactive to external and internal forcing often incise by tens to hundreds of vertical meters over a few thousand years only to subsequently aggrade by roughly the same amount. In my thesis, I set to study the morphodynamics of alluvial piedmonts in two areas, Death Valley, California, and the Eastern Tian Shan. In Death Valley I show that cycles of aggradation and incision repeatedly bury active fault scarps and that leads to the accumulation of tectonic slip until the next incision episode when a tall waterfall is released in the catchment. This process links the release of tectonic offset to climatic periods and it also accounts for many unexplained 10 to 30 m waterfalls in the Death Valley area. In the Eastern Tian Shan, I establish that the northern alluvial piedmont, that is incised by 100 to 300 m in the Holocene, undergoes repeated cycles of aggradation and incision driven by fluctuations in Westerlies moisture and insolation at a 21 kyr period without the need for monsoonal moisture to drive the system. The varying extent of glacial overprint in the Eastern Tian Shan is responsible for the morphological contrast between the incised north and aggraded south piedmonts. I produce more terrace abandonment ages that show that all rivers of the northern piedmont incised since the last deglaciation but did so out of sync because of the geometry of the upstream glaciated valleys, and internal feedbacks with alluvial valley walls during river entrenchment. With new luminescence ages sampled in the stratigraphy I establish that sediments from the last 0.5 Myr are remobilized in the modern sediment flux by the alluvial incision. I develop a numerical tool that predicts locally the probability of sediment mixing based on the essential length- and timescales of the individual rivers of an alluvial piedmont.</p

Evidence of Facilitation Cascade Processes as Drivers of Successional Patterns of Ecosystem Engineers at the Upper Altitudinal Limit of the Dry Puna

Facilitation processes constitute basic elements of vegetation dynamics in harsh systems. Recent studies in tropical alpine environments demonstrated how pioneer plant species defined as "ecosystem engineers" are capable of enhancing landscape-level richness by adding new species to the community through the modification of microhabitats, and also provided hints about the alternation of different ecosystem engineers over time. Nevertheless, most of the existing works analysed different ecosystem engineers separately, without considering the interaction of different ecosystem engineers. Focusing on the altitudinal limit of Peruvian Dry Puna vegetation, we hypothesized that positive interactions structure plant communities by facilitation cascades involving different ecosystem engineers, determining the evolution of the microhabitat patches in terms of abiotic resources and beneficiary species hosted. To analyze successional mechanisms, we used a "space-for-time" substitution to account for changes over time, and analyzed data on soil texture, composition, and temperature, facilitated species and their interaction with nurse species, and surface area of engineered patches by means of chemical analyses, indicator species analysis, and rarefaction curves. A successional process, resulting from the dynamic interaction of different ecosystem engineers, which determined a progressive amelioration of soil conditions (e.g. nitrogen and organic matter content, and temperature), was the main driver of species assemblage at the community scale, enhancing species richness. Cushion plants act as pioneers, by starting the successional processes that continue with shrubs and tussocks. Tussock grasses have sometimes been found to be capable of creating microhabitat patches independently. The dynamics of species assemblage seem to follow the nested assemblage mechanism, in which the first foundation species to colonize a habitat provides a novel substrate for colonization by other foundation species through a facilitation cascade process

Modes of extensional faulting controlled by surface processes

We investigate the feedbacks between surface processes and tectonics in an extensional setting by coupling a 2-D geodynamical model with a landscape evolution law. Focusing on the evolution of a single normal fault, we show that surface processes significantly enhance the amount of horizontal extension a fault can accommodate before being abandoned in favor of a new fault. In simulations with very slow erosion rates, a 15 km thick brittle layer extends via a succession of crosscutting short-lived faults (heave 10 km). Using simple scaling arguments, we quantify the effect of surface mass removal on the force balance acting on a growing normal fault. This leads us to propose that the major range-bounding normal faults observed in many continental rifts owe their large offsets to erosional and depositional processes

Autogenic entrenchment patterns and terraces due to coupling with lateral erosion in incising alluvial channels

The abandonment of terraces in incising alluvial rivers can be used to infer tectonic and climatic histories. A river incising into alluvium erodes both vertically and laterally as it abandons fill-cut terraces. We argue that the input of sediment from the valley walls during entrenchment can alter the incision dynamics of a stream by promoting vertical incision over lateral erosion. Using a numerical model, we investigate how valley wall feedbacks may affect incision rates and terrace abandonment as the channel becomes progressively more entrenched in its valley. We postulate that erosion of taller valley walls delivers large pulses of sediment to the incising channel, potentially overwhelming the local sediment transport capacity. Based on field observations, we propose that these pulses of sediment can form talus piles that shield the valley wall from subsequent erosion and potentially force progressive channel narrowing. Our model shows that this positive feedback mechanism can enhance vertical incision relative to 1-D predictions that ignore lateral erosion. We find that incision is most significantly enhanced when sediment transport rates are low relative to the typical volume of material collapsed from the valley walls. The model also shows a systematic erosion of the youngest terraces when river incision slows down. The autogenic entrenchment due to lateral feedbacks with valley walls should be taken into account in the interpretation of complex-response terraces

The twin peaks of learning neural networks

Recent works demonstrated the existence of a double-descent phenomenon for the generalization error of neural networks, where highly overparameterized models escape overfitting and achieve good test performance, at odds with the standard bias-variance trade-off described by statistical learning theory. In the present work, we explore a link between this phenomenon and the increase of complexity and sensitivity of the function represented by neural networks. In particular, we study the Boolean mean dimension (BMD), a metric developed in the context of Boolean function analysis. Focusing on a simple teacher-student setting for the random feature model, we derive a theoretical analysis based on the replica method that yields an interpretable expression for the BMD, in the high dimensional regime where the number of data points, the number of features, and the input size grow to infinity. We find that, as the degree of overparameterization of the network is increased, the BMD reaches an evident peak at the interpolation threshold, in correspondence with the generalization error peak, and then slowly approaches a low asymptotic value. The same phenomenology is then traced in numerical experiments with different model classes and training setups. Moreover, we find empirically that adversarially initialized models tend to show higher BMD values, and that models that are more robust to adversarial attacks exhibit a lower BMD.Comment: 37 pages, 31 figure

Formation of waterfalls by intermittent burial of active faults

Waterfalls commonly exist near bounding faults of mountain ranges, where erosional bedrock catchments transition to depositional alluvial fans. We hypothesize that aggradation on alluvial fans can bury active faults, and that the faults accumulate slip in the subsurface to produce a bedrock scarp. Following entrenchment of the alluvial fan, the scarp can be exposed as a waterfall. To explore this hypothesis, we derived a geometric model for waterfall height that depends on alluvial fan length and the relative time scales of (1) tectonic uplift, (2) a forcing mechanism for cycles of fan aggradation and incision, and (3) a response of fan aggradation to changes in sediment flux. We find that the model is consistent with observations at Gower Gulch, Death Valley, California, where a man-made drainage capture event in 1941 caused rapid fan incision and exposed a waterfall at the canyon-fan transition. We also compared the model to 62 waterfalls in 18 catchments of the Death Valley area and found that at least 15 of the waterfalls are best explained by the fault-burial mechanism. Using field measurements of grain size and channel geometries, we show that the fault-burial mechanism can produce the observed waterfall heights, measuring 4−19 m, under a uniform climatic forcing scenario requiring variations of 20% in precipitation during the late Pleistocene. The fault-burial mechanism, through the creation of upstream propagating waterfalls, may allow catchment-fan systems to experience frequent cycles of enhanced erosion in catchments and deposition on fans that likely convolve tectonic and climatic signals