Large-scale Limit Research Articles

Sharp Convergence of Nonlinear Functionals of a Class of Gaussian Random Fields

We present a self-contained proof of a uniform bound on multi-point correlations of trigonometric functions of a class of Gaussian random fields. It corresponds to a special case of the general situation considered in Hairer and Xu (large-scale limit of interface fluctuation models. ArXiv e-prints arXiv:1802.08192, 2018), but with improved estimates. As a consequence, we establish convergence of a class of Gaussian fields composite with more general functions. These bounds and convergences are useful ingredients to establish weak universalities of several singular stochastic PDEs.

Temporal Attention-Augmented Bilinear Network for Financial Time-Series Data Analysis.

Financial time-series forecasting has long been a challenging problem because of the inherently noisy and stochastic nature of the market. In the high-frequency trading, forecasting for trading purposes is even a more challenging task, since an automated inference system is required to be both accurate and fast. In this paper, we propose a neural network layer architecture that incorporates the idea of bilinear projection as well as an attention mechanism that enables the layer to detect and focus on crucial temporal information. The resulting network is highly interpretable, given its ability to highlight the importance and contribution of each temporal instance, thus allowing further analysis on the time instances of interest. Our experiments in a large-scale limit order book data set show that a two-hidden-layer network utilizing our proposed layer outperforms by a large margin all existing state-of-the-art results coming from much deeper architectures while requiring far fewer computations.

Imprints of Oscillatory Bispectra on Galaxy Clustering

Long-short mode coupling during inflation, encoded in the squeezed bispectrum of curvature perturbations, induces a dependence of the local, small-scale power spectrum on long-wavelength perturbations, leading to a scale-dependent halo bias. While this scale dependence is absent in the large-scale limit for single-field inflation models that satisfy the consistency relation, certain models such as resonant non-Gaussianity show a peculiar behavior on intermediate scales. We reconsider the predictions for the halo bias in this model by working in Conformal Fermi Coordinates, which isolate the physical effects of long-wavelength perturbations on short-scale physics. We find that the bias oscillates with scale with an envelope similar to that of equilateral non-Gaussianity. Moreover, the bias shows a peculiar modulation with the halo mass. Unfortunately, we find that upcoming surveys will be unable to detect the signal because of its very small amplitude. We also discuss non-Gaussianity due to interactions between the inflaton and massive fields: our results for the bias agree with those in the literature.

Entanglement evolution and generalised hydrodynamics: noninteracting systems

The large-scale properties of homogeneous states after quantum quenches in integrable systems have been successfully described by a semiclassical picture of moving quasiparticles. Here we consider the generalisation for the entanglement evolution after an inhomogeneous quench in noninteracting systems in the framework of generalised hydrodynamics. We focus on the protocol where two semi-infinite halves are initially prepared in different states and then joined together, showing that a proper generalisation of the quasiparticle picture leads to exact quantitative predictions. If the system is initially prepared in a quasistationary state, we find that the entanglement entropy is additive and it can be computed by means of generalised hydrodynamics. Conversely, additivity is lost when the initial state is not quasistationary; yet the entanglement entropy in the large-scale limit can be exactly predicted in the quasiparticle picture, provided that the initial state is low entangled.

Cubic halo bias in Eulerian and Lagrangian space

Predictions of the next-to-leading order, i.e. one-loop, halo power spectra, depend on local and non-local bias parameters up to cubic order. The linear bias parameter can be estimated from the large scale limit of the halo-matter power spectrum, and the second order bias parameters from the large scale, tree-level bispectrum. Cubic operators would naturally be quantified using the tree-level trispectrum. As the latter is computationally expensive, we extend the quadratic field method proposed in Schmittfull et al. 2014 to cubic fields, in order to estimate cubic bias parameters. We cross-correlate a basis set of cubic bias operators with the halo field and express the result in terms of the cross-spectra of these operators, in order to cancel cosmic variance. We obtain significant detections of local and non-local cubic bias parameters, which are partially in tension with predictions based on local Lagrangian bias schemes. We directly measure the Lagrangian bias parameters of the protohaloes associated with our halo sample and clearly detect a non-local quadratic term in Lagrangian space. We do not find a clear detection of non-local cubic Lagrangian terms for low mass bins, but there is some mild evidence for their presence for the highest mass bin. While the method presented here focuses on cubic bias parameters, the approach could also be applied to quantifications of cubic primordial non-Gaussianity.

On Ontological Alternatives to Bohmian Mechanics.

The article describes an interpretation of the mathematical formalism of standard quantum mechanics in terms of relations. In particular, the wave function is interpreted as a complex-valued relation between an entity (often called “particle”) and a second entity x (often called “spatial point”). Such complex-valued relations can also be formulated for classical physical systems. Entanglement is interpreted as a relation between two entities (particles or properties of particles). Such relations define the concept of “being next to each other”, which implies that entangled entities are close to each other, even though they might appear to be far away with respect to a classical background space. However, when space is also considered to be a network of relations (of which the classical background space is a large-scale continuum limit), such nearest neighbor configurations are possible. The measurement problem is discussed from the perspective of this interpretation. It should be emphasized that this interpretation is not meant to be a serious attempt to describe the ontology of our world, but its purpose is to make it obvious that, besides Bohmian mechanics, presumably many other ontological interpretations of quantum theory exist.

Absorber Model: the Halo-like model for the Lyman-α forest

We present a semi-analytic model for the Lyman-α forest that is inspired by the Halo Model. This model is built on the absorption line decomposition of the forest. Flux correlations are decomposed into those within each absorption line (the 1-absorber term) and those between separate lines (the 2-absorber term), treating the lines as biased tracers of the underlying matter fluctuations. While the nonlinear exponential mapping between optical depth and flux requires an infinite series of moments to calculate any statistic, we show that this series can be re-summed (truncating at the desired order in the linear matter overdensity). We focus on the z=2–3 line-of-sight power spectrum. Our model finds that 1-absorber term dominates the power on all scales, with most of its contribution coming from H I columns of 1014–1015 cm−2, while the smaller 2-absorber contribution comes from lower columns that trace overdensities of a few. The prominence of the 1-absorber correlations indicates that the line-of-sight power spectrum is shaped principally by the lines' number densities and their absorption profiles, with correlations between lines contributing to a lesser extent. We present intuitive formulae for the effective optical depth as well as the large-scale limits of 1-absorber and 2-absorber terms, which simplify to integrals over the H I column density distribution with different equivalent-width weightings. With minimalist models for the bias of absorption systems and their peculiar velocity broadening, our model predicts values for the density bias and velocity gradient bias that are consistent with those found in simulations.

The Scaling Limit of the KPZ Equation in Space Dimension 3 and Higher

We study in the present article the Kardar–Parisi–Zhang (KPZ) equation $$\begin{aligned} \partial _t h(t,x)=\nu \Delta h(t,x)+\lambda |\nabla h(t,x)|^2 +\sqrt{D}\, \eta (t,x), \qquad (t,x)\in \mathbb {R}_+\times \mathbb {R}^d \end{aligned}$$ in $$d\ge 3$$ dimensions in the perturbative regime, i.e. for $$\lambda >0$$ small enough and a smooth, bounded, integrable initial condition $$h_0=h(t=0,\cdot )$$ . The forcing term $$\eta $$ in the right-hand side is a regularized space-time white noise. The exponential of h—its so-called Cole-Hopf transform—is known to satisfy a linear PDE with multiplicative noise. We prove a large-scale diffusive limit for the solution, in particular a time-integrated heat-kernel behavior for the covariance in a parabolic scaling. The proof is based on a rigorous implementation of K. Wilson’s renormalization group scheme. A double cluster/momentum-decoupling expansion allows for perturbative estimates of the bare resolvent of the Cole-Hopf linear PDE in the small-field region where the noise is not too large, following the broad lines of Iagolnitzer and Magnen (Commun Math Phys 162(1):85–121, 1994). Standard large deviation estimates for $$\eta $$ make it possible to extend the above estimates to the large-field region. Finally, we show, by resumming all the by-products of the expansion, that the solution h may be written in the large-scale limit (after a suitable Galilei transformation) as a small perturbation of the solution of the underlying linear Edwards–Wilkinson model ( $$\lambda =0$$ ) with renormalized coefficients $$\nu _{eff}=\nu +O(\lambda ^2),D_{eff}=D+O(\lambda ^2)$$ .

Fractal characteristics of turbulent–non-turbulent interface in supersonic turbulent boundary layers

The turbulent–non-turbulent interface (TNTI) of supersonic turbulent boundary layers is a fundamental but relatively unexplored physics problem. In this study, we present experimental results from fractal analysis on the TNTI of supersonic turbulent boundary layers, and test the applicability of the additive law for these flows. By applying the nanoparticle-tracer planar laser scattering (NPLS) technique in a supersonic wind tunnel, we obtain data covering nearly three decades in scale. The box-counting results indicate that the TNTI of supersonic turbulent boundary layers is a self-similar fractal with a fractal dimension of 2.31. By comparing data sets acquired from two orthogonal planes, we find that the scaling exponent does not depend on direction, consistent with the validity of the additive law for the TNTI of turbulent boundary layers in a scale range with the large-scale limit not exceeding approximately $0.05\unicode[STIX]{x1D6FF}$.

Impact of hardware impairments on large-scale MIMO systems over composite RG fading channels

In this paper, we analyze the impact of receiver hardware impairments on the achievable sum rate of multiple-input multiple-output (MIMO) systems, where the channel follows composite Rician-Gamma (RG) distribution and may be correlated at the transmitters. More specifically, we derive the analytical expressions on the lower bound for the achievable sum rate of regular and large-scale MIMO systems with zero-forcing (ZF) receivers. In order to obtain deeper insights, the asymptotic analysis for the achievable sum rate of regular MIMO systems at high signal-to-noise ratio (SNR) regime is explored. It explicitly reveals that there is a finite rate ceiling on the achievable sum rate for regular MIMO systems at high SNRs, which is irrespective of the transmit power. Additionally, for large antenna configuration, three representative cases are considered insightfully by deriving in closed-form expressions for the asymptotic achievable sum rate. It demonstrates that the finite rate ceiling vanishes for large-scale MIMO limits, which means that large-scale MIMO systems are robust to hardware impairments.

Massive MIMO Antenna Selection: Switching Architectures, Capacity Bounds, and Optimal Antenna Selection Algorithms

Antenna selection is a multiple-input multiple-output (MIMO) technology, which uses radio frequency (RF) switches to select a good subset of antennas. Antenna selection can alleviate the requirement on the number of RF transceivers, thus being attractive for massive MIMO systems. In massive MIMO antenna selection systems, RF switching architectures need to be carefully considered. In this paper, we examine two switching architectures, i.e., full-array and sub-array. By assuming independent and identically distributed Rayleigh flat fading channels, we use asymptotic theory on order statistics to derive the asymptotic upper capacity bounds of massive MIMO channels with antenna selection for the both switching architectures in the large-scale limit. We also use the derived bounds to further derive the upper bounds of the ergodic achievable spectral efficiency considering the channel state information (CSI) acquisition. It is also showed that the ergodic capacity of sub-array antenna selection system scales no faster than double logarithmic rate. In addition, optimal antenna selection algorithms based on branch-and-bound are proposed for both switching architectures. Our results show that the derived asymptotic bounds are effective and also apply to the finite-dimensional MIMO. The CSI acquisition is one of the main limits for the massive MIMO antenna selection systems in the time-variant channels. The proposed optimal antenna selection algorithms are much faster than the exhaustive-search-based antenna selection, e.g., 1000 × speedup observed in the large-scale system. Interestingly, the full-array and sub-array systems have very close performance, which is validated by their exact capacities and their close upper bounds on capacity.

Overhanging rock slope by design: An integrated approach using rock mass strength characterisation, large-scale numerical modelling and limit equilibrium methods

Overhanging rock slopes (steeper than 90°) are typically avoided in rock engineering design, particularly where the scale of the slope exceeds the scale of fracturing present in the rock mass. This paper highlights an integrated approach of designing overhanging rock slopes where the relative dimensions of the slope exceed the scale of fracturing and the rock mass failure needs to be considered rather than kinematic release of individual blocks. The key to the method is a simplified limit equilibrium (LE) tool that was used for the support design and analysis of a multi-faceted overhanging rock slope. The overhanging slopes required complex geometries with constantly changing orientations. The overhanging rock varied in height from 30 m to 66 m. Geomechanical modelling combined with discrete fracture network (DFN) representation of the rock mass was used to validate the rock mass strength assumptions and the failure mechanism assumed in the LE model. The advantage of the simplified LE method is that buttress and support design iterations (along with sensitivity analysis of design parameters) can be completed for various cross-sections along the proposed overhanging rock sections in an efficient manner, compared to the more time-intensive, sophisticated methods that were used for the initial validation. The method described presents the development of this design tool and assumptions made for a specific overhanging rock slope design. Other locations will have different geological conditions that can control the potential behaviour of rock slopes, however, the approach presented can be applied as a general guiding design principle for overhanging rock cut slope.

Synchronization of reinforced stochastic processes with a network-based interaction

Randomly evolving systems composed by elements which interact among each other have always been of great interest in several scientific fields. This work deals with the synchronization phenomenon, that could be roughly defined as the tendency of different components to adopt a common behavior. We continue the study of a model of interacting stochastic processes with reinforcement, that recently has been introduced in Crimaldi et al. (2016, arXiv:1602.06217). Generally speaking, by reinforcement we mean any mechanism for which the probability that a given event occurs has an increasing dependence on the number of times that events of the same type occurred in the past. The particularity of systems of such stochastic processes is that synchronization is induced along time by the reinforcement mechanism itself and does not require a large-scale limit. We focus on the relationship between the topology of the network of the interactions and the long-time synchronization phenomenon. After proving the almost sure synchronization, we provide some CLTs in the sense of stable convergence that establish the convergence rates and the asymptotic distributions for both convergence to the common limit and synchronization. The obtained results lead to the construction of asymptotic confidence intervals for the limit random variable and of statistical tests to make inference on the topology of the network given the observation of the reinforced stochastic processes positioned at the vertices.

Quantum coherence of cosmological perturbations

In this paper, the degrees of quantum coherence of cosmological perturbations of different spins are computed in the large-scale limit and compared with the standard results holding for a single mode of the electromagnetic field in an optical cavity. The degree of second-order coherence of curvature inhomogeneities (and, more generally, of the scalar modes of the geometry) reproduces faithfully the optical limit. For the vector and tensor fluctuations, the numerical values of the normalized degrees of second-order coherence in the zero time-delay limit are always larger than unity (which is the Poisson benchmark value) but differ from the corresponding expressions obtainable in the framework of the single-mode approximation. General lessons are drawn on the quantum coherence of large-scale cosmological fluctuations.

Simplified and still meaningful: assessing butterfly habitat quality in grasslands with data collected by pupils

Public participation in scientific research, now commonly referred to as citizen science, is increasingly promoted as a possibility to overcome the large-scale data limitations related to biodiversity and conservation research. Furthermore, public data-collection projects can stimulate public engagement and provide transformative learning situations. However, biodiversity monitoring depends on sound data collection and warranted data quality. Therefore, we investigated if and how trained and supervised pupils are able to systematically collect data about the occurrence of diurnal butterflies, and how this data could contribute to a permanent butterfly monitoring system. We developed a specific assessment scheme suitable for laypeople and applied it at 35 sampling sites in Tyrol, Austria. Data quality and its explanatory power to predict butterfly habitat quality was investigated comparing data collected by pupils with independent assessments of professional butterfly experts. Despite substantial identification uncertainties for some species or species groups, the data collected by pupils was successfully used to predict the general habitat quality for butterflies using a linear regression model (r² = 0.73, p <0.001). Applying the proposed method in a citizen science context with laypeople could support both the long term monitoring of butterfly habitat quality, as well as the efficient selection of sites for professional in-depth assessments.

Metal-insulator transition in a disordered nanotube

According to the Anderson localization theory, the wavefunctions of a sufficiently strong disordered system are localized. We show that shifting hopping energy between nearest neighbors would induce an anomalous localization-delocalization transition in a disordered square lattice nanotube modelled by tight-binding. For this purpose, the consecutive level spacing statistics and the singularity spectrum analyses were performed. The quantum analysis of singularity spectrum reveals distinctive multifractality structures of the wavefunctions associated with localized and delocalized phases. We find that while in finite-size limit the system has a sudden metal-insulator transition, in large-scale limit the system experiences a rapid but continuous crossover. Interestingly, we report a critical value of hopping energy for which the system behavior is fairly close to metallic phase and especially independent of the system size. Passing this critical value, a great difference in the electronic transport properties of the system occurs. It follows that in the large-scale size, the system tends to follow semi-metallic behavior, while in finite size behaves more like to an insulator. The localization-delocalization transition is also reflected in the electrical current. In accordance with the indicators studied, we find that in delocalized regime there is a spreading electrical current throughout the whole system with an azimuthal symmetric characteristic.

Large population and size scale limit of a stochastic particle model

The aim of this work is to study a stochastic individual-based model, structured with respect to age (progression within the cell cycle) and space (radial distance from the oocyte). We prove the existence of solutions and the convergence in large population and size scale limit to a solution of a partial differential equation.

Large-scale heterogeneous service systems with general packing constraints

Abstract A service system with multiple types of customers, arriving according to Poisson processes, is considered. The system is heterogeneous in that the servers can also be of multiple types. Each customer has an independent, exponentially distributed service time, with the mean determined by its type. Multiple customers (possibly of different types) can be placed for service into one server, subject to `packing' constraints, which depend on the server type. Service times of different customers are independent, even if served simultaneously by the same server. The large-scale asymptotic regime is considered such that the customer arrival rates grow to ∞. We consider two variants of the model. For the infinite-server model, we prove asymptotic optimality of the greedy random (GRAND) algorithm in the sense of minimizing the weighted (by type) number of occupied servers in steady state. (This version of GRAND generalizes that introduced by Stolyar and Zhong (2015) for homogeneous systems, with all servers of the same type.) We then introduce a natural extension of the GRAND algorithm for finite-server systems with blocking. Assuming subcritical system load, we prove existence, uniqueness, and local stability of the large-scale system equilibrium point such that no blocking occurs. This result strongly suggests a conjecture that the steady-state blocking probability under the algorithm vanishes in the large-scale limit.

Large-scale assembly bias of dark matter halos

We present precise measurements of the assembly bias of dark matter halos, i.e. the dependence of halo bias on other properties than the mass, using curved “separate universe” N-body simulations which effectively incorporate an infinite-wavelength matter overdensity into the background density. This method measures the LIMD (local-in-matter-density) bias parameters bn in the large-scale limit. We focus on the dependence of the first two Eulerian biases bE1 and bE2 on four halo properties: the concentration, spin, mass accretion rate, and ellipticity. We quantitatively compare our results with previous works in which assembly bias was measured on fairly small scales. Despite this difference, our findings are in good agreement with previous results. We also look at the joint dependence of bias on two halo properties in addition to the mass. Finally, using the excursion set peaks model, we attempt to shed new insights on how assembly bias arises in this analytical model.

Three-dimensional automated reporter quantification (3D-ARQ) technology enables quantitative screening in retinal organoids.

The advent of stem cell-derived retinal organoids has brought forth unprecedented opportunities for developmental and physiological studies, while presenting new therapeutic promise for retinal degenerative diseases. From a translational perspective, organoid systems provide exciting new prospects for drug discovery, offering the possibility to perform compound screening in a three-dimensional (3D) human tissue context that resembles the native histoarchitecture and to some extent recapitulates cellular interactions. However, inherent variability issues and a general lack of robust quantitative technologies for analyzing organoids on a large scale pose severe limitations for their use in translational applications. To address this need, we have developed a screening platform that enables accurate quantification of fluorescent reporters in complex human iPSC-derived retinal organoids. This platform incorporates a fluorescence microplate reader that allows xyz-dimensional detection and fine-tuned wavelength selection. We have established optimal parameters for fluorescent reporter signal detection, devised methods to compensate for organoid size variability, evaluated performance and sensitivity parameters, and validated this technology for functional applications.