Limit Shape Research Articles

Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes

A Gelfand–Tsetlin scheme of depth $$N$$ is a triangular array with $$m$$ integers at level $$m$$ , $$m=1,\ldots ,N$$ , subject to certain interlacing constraints. We study the ensemble of uniformly random Gelfand–Tsetlin schemes with arbitrary fixed $$N$$ th row. We obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble (and also of its $$q$$ -deformation). This provides new tools for asymptotic analysis of uniformly random lozenge tilings of polygons on the triangular lattice; or, equivalently, of random stepped surfaces. We work with a class of polygons which allows arbitrarily large number of sides. We show that the local limit behavior of random tilings (as all dimensions of the polygon grow) is directed by ergodic translation invariant Gibbs measures. The slopes of these measures coincide with the ones of tangent planes to the corresponding limit shapes described by Kenyon and Okounkov (Acta Math 199(2):263–302, 2007). We also prove that at the edge of the limit shape, the asymptotic behavior of random tilings is given by the Airy process. In particular, our results cover the most investigated case of random boxed plane partitions (when the polygon is a hexagon).

Annealed Brownian motion in a heavy tailed Poissonian potential

Consider a d-dimensional Brownian motion in a random potential defined by attaching a nonnegative and polynomially decaying potential around Poisson points. We introduce a repulsive interaction between the Brownian path and the Poisson points by weighting the measure by the Feynman-Kac functional. We show that under the weighted measure, the Brownian motion tends to localize around the origin. We also determine the scaling limit of the path and also the limit shape of the random potential.

Birth of the shape of a reachable set

We address a linear control system under geometric constraints on control and study its reachable sets starting at zero time from the origin. The main result is the existence of a limit shape of the reachable sets as the terminal time tends to zero. Here, a shape of a set stands for the set regarded up to an invertible linear transformation. Both autonomous and non-autonomous cases are considered.

Examples of nonpolygonal limit shapes in i.i.d. first-passage percolation and infinite coexistence in spatial growth models

We construct an edge-weight distribution for i.i.d. first-passage percolation on $\mathbb{Z}^2$ whose limit shape is not a polygon and whose extreme points are arbitrarily dense in the boundary. Consequently, the associated Richardson-type growth model can support coexistence of a countably infinite number of distinct species, and the graph of infection has infinitely many ends.

Roles of Seed and Establishment Limitation in Determining Patterns of Afrotropical Tree Recruitment

Quantifying the relative importance of the multiple processes that limit recruitment may hold the key to understanding tropical tree diversity. Here we couple theoretical models with a large-scale, multi-species seed-sowing experiment to assess the degree to which seed and establishment limitation shape patterns of tropical tree seedling recruitment in a central African forest. Of five randomly selected species (Pancovia laurentii, Staudtia kamerunensis, Manilkara mabokeensis, Myrianthus arboreas, and Entandophragma utile), seedling establishment and survival were low (means of 16% and 6% at 3 and 24 months, respectively), and seedling density increased with seed augmentation. Seedling recruitment was best explained by species identity and the interaction of site-by-species, suggesting recruitment probabilities vary among species and sites, and supporting the role of niche-based mechanisms. Although seed augmentation enhanced initial seedling density, environmental filtering and post-establishment mortality strongly limited seedling recruitment. The relative importance of seed and establishment limitation changed with seed and seedling density and through time. The arrival of seeds most strongly affected local recruitment when seeds were nearly absent from a site (∼ 1 seed m2), but was also important when seeds arrived in extremely high densities, overwhelming niche-based mortality factors. The strength of seed limitation and density-independent mortality decreased significantly over time, while density-dependent mortality showed the opposite trend. The varying strengths of seed and establishment limitation as a function of juvenile density and time emphasize the need to evaluate their roles through later stages of a tree’s life cycle.

Effects of environmental filtering and dispersal limitation on species and phylogenetic beta diversity in Gutianshan National Nature Reserve

Phylobetadiversity can provide new insights into assembly mechanisms of community ecology because it incorporates phylogenetic information. The combination of phylobetadiversity and traditional species beta diversity is of great benefit to research that attempts to explain the mechanisms of community assemblages. To reveal the community assembly mechanisms of a 24-hm2 forest plot established in Gutianshan National Nature Reserve, Kaihua County, Zhejiang, China, our study used four spatial statistical models to simulate model communities created in four ways: purely random processes, environmental filtering, dispersal limitations and the combination of environmental filtering and dispersal limitation. Then we analyzed the distance-decay of similarity of these four model communities and observed communities based on the plot in the Gutianshan National Nature Reserve. Our results showed (1) an obvious distance-decay relationship exists for species and phylogenetic similarity in the Gutianshan plot; (2) the combined effects of environmental filtering and dispersal limitation can best explain the mechanisms involved in structuring species beta diversity in the plot; and (3) different results were produced using pairwise ( D pw) and nearest neighbor ( D nn) metrics. The results from the nearest neighbor metric ( D nn) suggested that the combined effects of environmental filtering and dispersal limitation shape the community phylogenetic structure of the plot. Additionally, the results from the pairwise metric ( D pw) showed that the combined effect of environmental filtering and dispersal limitation can best explain the community phylogenetic structure at the scales of 20 m2×20 m2, while environmental filtering is the main force shaping community phylogenetic structure at scales of 40 m2×40 m2 and 50 m2×50 m2. The combined patterns observed when considering both phylobetadiversity and traditional species beta diversity reflected the important roles of habitat filtering and dispersal limitation in diversity maintenance of this subtropical forest habitat.

Performance of miniaturised thick-film solid state pH sensors

The performance of novel solid state thick-film pH sensors for water quality sampling that are designed for deployment in remote catchment areas is described. These miniaturised screen printed planar pH sensors are an alternative to conventional glass pH electrodes that have many disadvantages such as high-cost, large size, mechanical fragility and limited shape. The approach adopted here for improvement of the ruggedness of the pH electrode is through its implementation as a screen printed metal oxide (RuO2) ion selective electrode used in combination with a screen printed silver/silver chloride (Ag/AgCl) reference electrode. Various choices of materials have been evaluated for the fabrication of the thick-film pH sensors resulting in a device with a sensitivity of approximately 30mV/pH at ambient temperature.

Study on physical and chemical properties of external lime–sand plasters of some historical buildings

The article presents the results of laboratory tests of facade plasters from two selected sacred objects from the fourteenth and eighteenth centuries. In both buildings lime plaster was used, both internal and external facades. Plasters have different strength depending on the destination. In spite of the use of lime binders only, some of plasters, i.e. on plinths, are of considerable hardness and high resistance against weathering agents in comparison to other ones. In order to identify differences in the internal structure of different strengths plaster, samples were collected from different places and tests on physico-chemical characteristics were carried out. Due to the limited size and shape of the samples, physical properties such as absorbability, density and total porosity and granulometric analysis of the aggregate were marked. In addition, studies were made of mineral composition by differential thermal analysis and X-ray diffraction and chemical tests. The results obtained have allowed to clearly identify the nature of the binder and the mineral composition of the original plaster existing hikes, as well as reasons for the high hardness of some plaster. These results are of cognitive importance, as well as in the practical work of the revaluation of historical objects.

Multiplicity of summands in the random partitions of an integer

In this paper, we prove a conjecture of Yakubovich regarding limit shapes of ‘slices’ of two-dimensional (2D) integer partitions and compositions of n when the number of summands m ~Anα for some A > 0 and \(\alpha < \frac{1}{2}\). We prove that the probability that there is a summand of multiplicity j in any randomly chosen partition or composition of an integer n goes to zero asymptotically with n provided j is larger than a critical value. As a corollary, we strengthen a result due to Erdos and Lehner (Duke Math. J.8 (1941) 335–345) that concerns the relation between the number of integer partitions and compositions when \(\alpha = \frac{1}{3}\).

Spectroscopic measurements of Be erosion at JET ILW and interpretation with ERO modelling

Beryllium (Be) erosion has been studied in dedicated limiter discharges in JET with the recently installed ITER-like wall [1]. Passive spectroscopy [2] in the vicinity of the solid Be limiter is used for measurement of the Be physical sputtering as the main erosion mechanism. To consider the 3D configuration of plasma parameters and the electromagnetic field, the actual limiter shape and the local transport affecting the fraction of Be coming into the observation volume, a detailed modelling with the ERO code has been applied to interpret the experimental data. The observed dependence of BeI and BeII line intensities on plasma parameters during the density scan and various line ratios are used to validate the model and the underlying data including the recently introduced assumptions for Be physical sputtering, the very same which were used before for ITER predictive modelling [3].

Certification, self-calibration and uncertainty in optical surface testing

Conventional approaches to achieving traceability in measurements of optical surfaces are based on a series of intercomparisons ultimately linking the measurement result to a realisation of the unit (metre) at a national measurement institution. Measurement services offered by national measurement institutes (NMIs) cover a limited size and shape range and the uncertainties associated with the transfer of a calibration must be considered when data from any testing service is used in quality assurance. Local realisation of the unit, typically using CIPM recognised wavelengths and their associated uncertainties, combined with in–situ self–calibration of geometry and the development of a rigorous uncertainty analysis meet the conditions to achieve traceability according to the requirements of ISO 17025. This paper reviews the underlying basis for achieving traceability without a direct chain of comparisons to NMIs and gives some examples using a range of optical measurements.

Reconfigurable Closed-Loop Digital ΔΣ CapacitiveMEMS Accelerometer for Wide Dynamic Range, HighLinearity Applications

Closed-loop capacitive MEMS accelerometer is similar to a ΔΣ ADC where the MEMS sensor introduces second-order dynamics in the loop. The sensor has limited signal gain, hence provides very limited noise shaping capability to other noise contributions within the loop. Furthermore, fabrication non-idealities can cause unwanted stability issues in higher order closed-loop operation. We present a 4 th order ΔΣ closed-loop system with a digital 2 nd order ΔΣ modulator. The digital ΔΣ modulator allows fine accuracy in choosing the coefficients for the ΔΣ modulators. It also provides flexibility in the placement of poles and zeros for compensating the unforeseen phase delays caused by fabrication non-idealities. Our system is able to achieve an SNR > 120dB in simulation and > 90dB from experiment with an evaluation board. The digital implementation of ΔΣ modulator allows us to build reconfigurable closed-loop system with the potential for even higher order system which can be optimized to suit different MEMS sensor characteristics and performance.

The Gaussian free field and strict plane partitions

We study height fluctuations around the limit shape of a measure on strict plane partitions. It was shown in our earlier work that this measure is a Pfaffian process. We show that the height fluctuations converge to a pullback of the Green's function for the Laplace operator with Dirichlet boundary conditions on the first quadrant. We use a Pfaffian formula for higher moments to show that the height fluctuations are governed by the Gaussian free field. The results follow from the correlation kernel asymptotics which is obtained by the steepest descent method.

A study on the optimal fabrication method for micro-scale gyroscopes using a hybrid process consisting of electric discharge machining, chemical etching or micro-mechanical milling

For productive manufacturing of an accurate small-scale wine-glass gyroscope, a hybrid fabrication process consisting of either electric discharge machining, chemical etching, or micro-mechanical milling have been proposed. A comparison of silicon cavity fabrication processes has been conducted in terms of productivity, quality and geometrical accuracy, aiming at the use of the cavity as a mold for creating a thin wall diamond hemisphere, which is the main component of a wine-glass gyroscope. The results have shown that the EDM process, combined with chemical etching, can yield the highest productivity but with limited shape accuracy. The use of mechanical micro-milling, while less productive than EDM and etching, produces a superior quality and geometric accuracy.

20302 インクリメンタル成形による微小円管端部成形装置の開発(GS-10【成形・加工(3)】)

In recent years, products manufactured from minute tube are widely used in electrical and medical field. Conventionally, the press forming of tube or sheet metal is used to make these products. The press forming is suitable for quantity production. However, it can only form limited shape, and its forming limit is low especially in that case of minute products. On the other side, rotary incremental forming with a cylindrical bar tool can make various shapes, and it is possible to improve forming limit because contact area between tool and tube is narrower than previous method. In this study, new forming equipment was developed to forming minute tapered tube with rotary incremental forming at first. Then, experiments with 2mm diameter tube were carried out for the purpose of establishment of basic forming technology required for more minute tube. In these experiments, it was confirmed that the possibility of forming minute tube with rotary incremental forming and that the effect of new tools. Furthermore, effect of each forming parameter on shape and forming rate was examined by experimental trial.

Sensitivity of the limit shape of sample clouds from meta densities

The paper focuses on a class of light-tailed multivariate probability distributions. These are obtained via a transformation of the margins from a heavy-tailed original distribution. This class was introduced in Balkema et al. (J. Multivariate Anal. 101 (2010) 1738-1754). As shown there, for the light-tailed meta distribution the sample clouds, properly scaled, converge onto a deterministic set. The shape of the limit set gives a good description of the relation between extreme observations in different directions. This paper investigates how sensitive the limit shape is to changes in the underlying heavy-tailed distribution. Copulas fit in well with multivariate extremes. By Galambos's theorem, existence of directional derivatives in the upper endpoint of the copula is necessary and sufficient for convergence of the multivariate extremes provided the marginal maxima converge. The copula of the max-stable limit distribution does not depend on the margins. So margins seem to play a subsidiary role in multivariate extremes. The theory and examples presented in this paper cast a different light on the significance of margins. For light-tailed meta distributions, the asymptotic behaviour is very sensitive to perturbations of the underlying heavy-tailed original distribution, it may change drastically even when the asymptotic behaviour of the heavy-tailed density is not affected.

Niche incumbency, dispersal limitation and climate shape geographical distributions in a species‐rich island adaptive radiation

AbstractAimTo test the role of climate, dispersal limitation and biotic interactions in limiting species' distributions within an island adaptive radiation by integrating species traits, phylogeny and estimates of dispersal cost into climate‐based species distribution models.LocationHispaniola.MethodsFocusing on 26 species of Anolis lizards, we used multivariate adaptive regression splines to evaluate the contribution of climate, species interactions, phylogenetic history and dispersal limitation to species distributional limits. For each species, we mapped the morphological similarity of congenerics using traits of known ecological import and predicted that species would be less likely to occur in climatically suitable areas if they were inhabited by ecologically similar species. Dispersal limitation was incorporated by generating spatially explicit estimates of dispersal cost, based on inferred habitat suitability. We compared models including morphological similarity, dispersal cost, phylogeny and climate with climate‐only models. Results were evaluated against a null model that conserved the spatial structure of species occurrences.ResultsClimate had a dominant role in shaping species distributions. However, for over one‐third of species we also found evidence consistent with supplemental effects of species interactions, i.e. ecological niche incumbency. These species were less likely to occur in climatically suitable areas inhabited by a morphologically similar species. Dispersal limitation also supplemented climatic limits in most species. These results were robust to co‐variation with phylogeny and to comparison with our null model.ConclusionsThese results suggest that, rather than act as mutually exclusive alternatives, multiple dimensions of the ecological niche, including climatic limits, biotic interactions and dispersal capacity, interact to shape species distributions and that local interactions can influence the broad‐scale geography of species in a predictable way.

Limit Behavior of Reachable Sets of Linear Time-Invariant Systems with Integral Bounds on Control

In this paper, a linear dynamic system is considered under Lp-constraint on control. We establish the existence of the limit shape of reachable sets as time goes to infinity. Asymptotic formulas are obtained for reachable sets and their shapes. The results throw a bridge between the cases of geometric bounds on control and constraints on the total impulse of control, and create a unified picture of the structure of the limit shapes of reachable sets.

Block characters of the symmetric groups

A block character of a finite symmetric group is a positive definite function which depends only on the number of cycles in a permutation. We describe the cone of block characters by identifying its extreme rays, and find relations of the characters to descent representations and the coinvariant algebra of ${\mathfrak{S}}_{n}$ . The decomposition of extreme block characters into the sum of characters of irreducible representations gives rise to certain limit shape theorems for random Young diagrams. We also study counterparts of the block characters for the infinite symmetric group ${\mathfrak{S}}_{\infty}$ , along with their connection to the Thoma characters of the infinite linear group GL ?(q) over a Galois field.

Gaussian Free Field in an Interlacing Particle System with Two Jump Rates

AbstractWe study the fluctuations of a random surface in a stochastic growth model on a system of interlacing particles placed on a two‐dimensional lattice. There are two different types of particles, one with a low jump rate and the other with a high jump rate. In the large time limit, the random surface has a deterministic shape. Due to the different jump rates, the limit shape and the domain on which it is defined are not smooth. The main result is that the fluctuations of the random surface are governed by the Gaussian free field. © 2012 Wiley Periodicals, Inc.