Single Probability Distribution Research Articles

The total factor characteristics evaluation of photovoltaic power by coarse-fine-grained method

The study of photovoltaic(PV) power uncertainty is of great significance to promote the collaborative optimization of source-grid-load-storage. On the one hand, using a single probability distribution model to describe the fluctuation, intermittency and randomness of PV power is limited and controversial; On the other hand, qualitative analysis oriented to only one or two of them is characterized with poor interpretability and one-sidedness. So that, a evaluation framework for the total factor characteristics named "dual-DMU" is established corresponding to the discrete, mutation and unordered phenomenon of PV power, and a coarse-fine-grained evaluation method is proposed, which can realize the comprehensive description and all-sided evaluation of PV power. Combined with the historical data of a rooftop PV power plant, the analysis reveals the variability of total factor characteristics in different seasons, weather types, and typical time periods, and the methodology can provide a theoretical basis and decision-making support for the construction of PV power forecasting models, energy storage capacity optimization, and adjustable load resource allocation.

Technology Focus: Reserves Management (December 2023)

Time and time again, I meet industry specialists who should have a detailed knowledge of the Petroleum Resources Management System (PRMS) and handling uncertainty in the inputs for resource assessment or to the rest of the evaluation work flow. Often, however, the understanding of PRMS basics, application, or reporting seems to be lacking. This year’s Reserves Management papers are chosen to revisit some basics around both reserves delineation and asset management. Paper SPE 210358 reminds evaluators to use multiple approaches (triangulation?) in the presence of uncertainty to derive the best estimates for Proved Reserves, but this should apply equally to all resource assessment. It also stresses the requirement, sometimes overlooked, to ensure that the approach/technology is proven and calibrated to work in the exact situation, both subsurface and surface. Uncertainty is inherent in our business, though difficult to communicate, and we should do all we can to incorporate it correctly. Also looking at the effect of uncertainty, paper SPE 213385 focuses on Monte Carlo simulation for commercialization around well performance in unconventional settings, but its application can be to any technical or commercial uncertainty. Too many people overly focus on precision (the decimal places in the answer) but lose focus on accuracy or usefulness of the work. Uncertainty can lead to errors in the first, and certainly the second, significant figure, thus invalidating all the misleading precision. Analysis should model the parameters to include the true state of knowledge. It seems ever more difficult to avoid discussions of carbon capture and storage (CCS), a major challenge facing the industry, especially when trying to monetize massive high-CO2 fields. Paper SPE 214359 discusses how the Malaysian authorities and Petronas seek to manage and incentivize CCS in their domain and achieve socially desirable goals with improved efficiency—specifically, centralizing the disposal of CO2 through an organization taking the separated gas from the producers. It creates efficiencies through scale and targeted regulatory incentives that do not need to be applied to the direct producers. It reminds me of how liquefied natural gas facilities often are commercially separated from the upstream to manage project financing and improve project feasibility. The recommended additional reading papers delve further into PRMS disclosure details and implications; how allocating volumes (and costs) for incremental decisions can change the optimal outcome; some more around how volumes and costs flow through to economics and how this can be an issue in complex fields; and, lastly, an unusual proposal of uncertainty around the 1P, 2P, and 3P values. Certainly, they remind the reader to consider the effect of uncertainty on volumetrics and categorization. Recommended additional reading at OnePetro: www.onepetro.org. SPE 209695 Can Reasonable Certainty Be Assessed From Disclosed Proved Reserves Revisions? by Enrique Morales, ISVA Oil and Gas Consultancy, et al. SPE 211458 An Algorithm for Calculating Incremental Production in an Integrated Asset Model by Peter Richard Paul Cunningham, Serafim, et al. SPE 212826 Impact of Financial Assumptions on Economic Viability of Incremental Projects by M.A. Mian, O&G Knowledge Sharing Platform SPE 209217 Probabilistic Reserves Categorization Using Percentiles of a Single Probability Distribution by M.A. Mian, O&G Knowledge Sharing Platform

Temporal Evolution of Trust in Artificial Intelligence-Supported Decision-Making

AI is set to take over some tasks within the decision space that have traditionally been reserved for humans. In return, human decision-makers interacting with AI systems entails rationalization of AI outputs by humans, who may have difficulty forming trust around such AI-generated information. Although a variety of analytical methods have provided some insights into human trust in AI, a more comprehensive understanding of trust may be augmented by generative theories that capture the temporal evolution of trust. Therefore, an open system modeling approach, representing trust as a function of time with a single probability distribution, can potentially improve modeling human trust in an AI system. Results of this study could improve machine behaviors that may help steer a human’s preference to a more Bayesian optimal rationality which is useful in stressful decision-making scenarios.

Assessment of Multidimensional Health Care Parameters Among Adults in Japan for Developing a Virtual Human Generative Model: Protocol for a Cross-sectional Study

Background Human health status can be measured on the basis of many different parameters. Statistical relationships among these different health parameters will enable several possible health care applications and an approximation of the current health status of individuals, which will allow for more personalized and preventive health care by informing the potential risks and developing personalized interventions. Furthermore, a better understanding of the modifiable risk factors related to lifestyle, diet, and physical activity will facilitate the design of optimal treatment approaches for individuals. Objective This study aims to provide a high-dimensional, cross-sectional data set of comprehensive health care information to construct a combined statistical model as a single joint probability distribution and enable further studies on individual relationships among the multidimensional data obtained. Methods In this cross-sectional observational study, data were collected from a population of 1000 adult men and women (aged ≥20 years) matching the age ratio of the typical adult Japanese population. Data include biochemical and metabolic profiles from blood, urine, saliva, and oral glucose tolerance tests; bacterial profiles from feces, facial skin, scalp skin, and saliva; messenger RNA, proteome, and metabolite analyses of facial and scalp skin surface lipids; lifestyle surveys and questionnaires; physical, motor, cognitive, and vascular function analyses; alopecia analysis; and comprehensive analyses of body odor components. Statistical analyses will be performed in 2 modes: one to train a joint probability distribution by combining a commercially available health care data set containing large amounts of relatively low-dimensional data with the cross-sectional data set described in this paper and another to individually investigate the relationships among the variables obtained in this study. Results Recruitment for this study started in October 2021 and ended in February 2022, with a total of 997 participants enrolled. The collected data will be used to build a joint probability distribution called a Virtual Human Generative Model. Both the model and the collected data are expected to provide information on the relationships between various health statuses. Conclusions As different degrees of health status correlations are expected to differentially affect individual health status, this study will contribute to the development of empirically justified interventions based on the population. International Registered Report Identifier (IRRID) DERR1-10.2196/47024

PROBABILITY MODELING OF LOW FLOWS AT DIFFERENT SITES OF INDUS BASIN IN PAKISTAN USING L-MOMENTS AND TL-MOMENTS

At-site Frequency Analysis (ASFA) of low flow was carried out for nine sites ofIndus basin in Pakistan. In the present study, 10-day annual low flow series were analyzed by robustestimation methods such as Method of L-moment (ML) and TL-moment (MTL) to identify best fitprobability distributions for each site. Best distribution for each site was identified using differentgoodness-of-fit Tests (GFT). No single probability distribution was declared as the best-fit distributionfor all sites included in the plan. The GFT results indicated GPA was the most appropriate distributionfor most of the sites followed by GLO and GEV distributions. On comparison, it was found that formost of the sites ML was best estimation method and for others MTL. For ASFA, the quantiles of bestfit distribution were also estimated. It was found that estimated low flows based on fitted distributionwere in close agreement with observed flows.

PROBABILITY MODELING OF LOW FLOWS AT DIFFERENT SITES OF INDUS BASIN IN PAKISTAN USING L-MOMENTS AND TL-MOMENTS

At-site Frequency Analysis (ASFA) of low flow was carried out for nine sites ofIndus basin in Pakistan. In the present study, 10-day annual low flow series were analyzed by robustestimation methods such as Method of L-moment (ML) and TL-moment (MTL) to identify best fitprobability distributions for each site. Best distribution for each site was identified using differentgoodness-of-fit Tests (GFT). No single probability distribution was declared as the best-fit distributionfor all sites included in the plan. The GFT results indicated GPA was the most appropriate distributionfor most of the sites followed by GLO and GEV distributions. On comparison, it was found that formost of the sites ML was best estimation method and for others MTL. For ASFA, the quantiles of bestfit distribution were also estimated. It was found that estimated low flows based on fitted distributionwere in close agreement with observed flows

Assessment of the Uncertainty Associated with Statistical Modeling of Precipitation Extremes for Hydrologic Engineering Applications in Amman, Jordan

Estimates of extreme precipitation are commonly associated with different sources of uncertainty. One of the primary sources of uncertainty in the statistical modeling of precipitation extremes comes from extreme data series (i.e., sampling uncertainty). Therefore, this research aimed to quantify the sampling uncertainty in terms of confidence intervals. In addition, this article examined how the data record length affects predicted extreme precipitation estimates and data set statistics. A nonparametric bootstrap resample was utilized to quantify the precipitation quantile sampling distribution at a particular non exceedance probability. This sampling distribution can provide a point estimation of the precipitation quantile and the confidence interval at a particular non exceedance probability. It has been shown that the different types of probability distributions fit the extreme precipitation data series of various weather stations. Therefore, the uncertainty analysis should be conducted using the best-fit probability distribution for extreme precipitation data series rather than a predefined single probability distribution for all stations based on modern extreme value theory. According to the 95% confidence intervals, precipitation quantiles are subject to significant uncertainty and the band of the uncertainty intervals increases with the return period. These uncertainty bounds need to be integrated into any frequency analysis from historical data. The average, standard deviation, skewness and kurtosis are highly affected by the data record length. Thus, a longer record length is desirable to decrease the sampling uncertainty and, therefore, decrease the error in the predicted quantile values. Moreover, the results suggest that a series of at least 40 years of data records is needed to obtain reasonably accurate estimates of the distribution parameters and the precipitation quantiles for 100 years return periods and higher. Using only 20 to 25 years of data to obtain estimates of the higher return period quantile is risky, since it created high sampling variability relative to the full data length.

A tool for the ages: The Probabilistic Cosmogenic Age Analysis Tool (P-CAAT)

While revolutionary to the geomorphic community, the application of terrestrial cosmogenic nuclide (TCN) dating is complicated by geological uncertainties, which often lead to skewed or poorly clustered TCN age distributions. Although a range of statistical approaches are typically used to detect and remove outliers, few are optimized for analysis of TCN datasets. Many are mean- or median-based and therefore explicitly assume a single probability distribution (e.g., Mean Squared Weighted Deviates, Chauvenet's Criterion, etc.). Given the ubiquity of pre- and post-depositional modification of rock surfaces, which occur at different rates in different geomorphic settings, these approaches struggle with multimodal distributions which often characterize TCN datasets. In addition, most statistical approaches do not propagate measurement or production rate uncertainties, which become increasingly important as dataset size or clustering increases. Finally, most approaches provide arithmetic single solutions, irrespective of geologic context. To address these limitations, we present the Probabilistic Cosmogenic Age Analysis Tool (P-CAAT), a new approach for outlier detection and landform age analysis. This tool incorporates both sample age and geologic uncertainties and uses Monte Carlo simulations to eliminate dataset skewness by isolating component normal distributions from a cumulative probability density estimate for datasets with three or more samples. This approach allows geologic context to inform post-analysis interpretations, as researchers can assign landform ages based upon statistically distinct subpopulations, informed by the characteristics of geomorphic systems (e.g., exhumation of boulders as moraines degrade through time). To evaluate the effectiveness of P-CAAT, we analyzed a range of synthetic TCN datasets and compared the results to commonly used statistical approaches for outlier detection. Irrespective of dataset size or clustering, P-CAAT outperformed other approaches and returned accurate solutions that improve in precision as sample size increases. To enable more comprehensive utilization of our approach, P-CAAT is packaged with a GUI interface and is available for download at kgs. uky.edu/anorthite/PCAAT.

The application of the distributed-order time fractional Bloch model to magnetic resonance imaging

It is now well known that the magnetic resonance imaging (MRI) signal decay deviates from the classical mono-exponential relaxation. This deviation is referred to in the literature as anomalous relaxation. The modelling of this anomalous relaxation can provide a better understanding of MRI magnetization. The purpose of this work is to investigate the utility of the distributed-order time fractional Bloch equations to describe anomalous relaxation processes in human brain MRI data. Two choices of continuous distribution weight functions, which are parameterised by their mean μ and standard deviation σ, are studied to investigate their impact on the model solution behaviour. An implicit numerical method implemented on a graded mesh is proposed to solve the model and the stability and convergence analysis are presented. We also derive semi-analytical solutions of the fully coupled Bloch equations using the Laplace transform technique to assess the accuracy of the numerical scheme. Furthermore, three different voxel models of continuous distribution weight functions, namely a single continuous probability distribution (model 1), two distinct continuous probability distributions (model 2) and a mixture of two continuous probability distributions (model 3), are applied to the in vivo human brain MRI data, and a feasible and reliable parameter estimation method based on a modified hybrid Nelder-Mead simplex search and particle swarm optimization is presented to perform the voxel-level temporal fitting of the MRI data. The application of these distributed-order time fractional Bloch models highlights the validity of the proposed models, and based on the mean square error we conclude that models 2 and 3 might be more suitable than model 1 to characterize anomalous relaxation processes in human brain MRI data.

The third AI summer: AAAI Robert S. Engelmore Memorial Lecture

The third AI summer: AAAI Robert S. Engelmore Memorial Lecture

Probabilistic Reserves Categorization Using Percentiles of a Single Probability Distribution

Summary Petroleum reserves assessment is a routine exercise performed by every oil and gas company. The reserves estimation could be achieved using deterministic or probabilistic methods. Depending on the type of data available, different assumptions and data are used at various stages of the assessment throughout the E&P life cycle (i.e., exploration, appraisal, development, depletion, and so on). The accuracy of the reserves estimate is highly dependent on the amount and quality of data available and experience of the evaluator. The estimates become more accurate and refined as more and more data become available. The objective of this paper is to urge the industry to rethink the terminology used for categorizing probabilistic reserves (P90, P50, and P10) using a single probability distribution. As a routine practice, we consider 1P = P90 = proved reserves, 2P = P50 = proved + probable reserves, and 3P = P10 = proved + probable + possible reserves. The reserves categories depend on the range of uncertainty as shown in the Petroleum Resources Classification Framework of the Petroleum Resources Management System (PRMS). It is shown here that P50 (or mean/expected value) of each reserve category is a more robust and logical value to be used for categorizing reserves. The P90 and P10 demonstrate confidence levels of a certain probability distribution, which is owing to the standard deviation (proxy for uncertainty) of the specific distribution and not the range of uncertainty. The range in uncertainty is assessed from varying standard deviations from more than one probability distribution. Therefore, the use of P90 and P10 for reserves categorization contradicts the x-axis title of the Petroleum Resources Classification Framework in the PRMS. The P90, P50, and P10 from a single probability distribution with a single standard deviation will have the same uncertainty (not range of uncertainty). The difference between P90 and P10 is also owing to this single standard deviation. Some examples are presented in this paper to show this and to show that P50 (or mean) with its corresponding standard deviation (indication of uncertainty) could be more appropriately used to book and categorize reserves. The concept of expected value (EV) is also revisited as this may not be correctly reported in some publications.

Accurate Modeling of Grazing Transits Using Umbrella Sampling

Abstract Grazing transits present a special problem for statistical studies of exoplanets. Even though grazing planetary orbits are rare (due to geometric selection effects), for many low to moderate signal-to-noise ratio cases, a significant fraction of the posterior distribution is nonetheless consistent with a grazing geometry. A failure to accurately model grazing transits can therefore lead to biased inferences even for cases where the planet is not actually on a grazing trajectory. With recent advances in stellar characterization, the limiting factor for many scientific applications is now the quality of available transit fits themselves, and so the time is ripe to revisit the transit fitting problem. In this paper, we model exoplanet transits using a novel application of umbrella sampling and a geometry-dependent parameter basis that minimizes covariances between transit parameters. Our technique splits the transit fitting problem into independent Monte Carlo sampling runs for the grazing, nongrazing, and transition regions of the parameter space, which we then recombine into a single joint posterior probability distribution using a robust weighting scheme. Our method can be trivially parallelized and so requires no increase in the wall clock time needed for computations. Most importantly, our method produces accurate estimates of exoplanet properties for both grazing and nongrazing orbits, yielding more robust results than standard methods for many common star–planet configurations.

Smoothing approximations for piecewise smooth functions: A probabilistic approach

<p style='text-indent:20px;'>In this article, we present a new approach to construct smoothing approximations for piecewise smooth functions. This approach proposes to formulate any piecewise smooth function as the expectation of a random variable. Based on this formulation, we show that smoothing all elements of a defined space of piecewise smooth functions is equivalent to smooth a single probability distribution. Furthermore, we propose to use the Boltzmann distribution as a smoothing approximation for this probability distribution. Moreover, we present the theoretical results, error estimates, and some numerical examples for this new smoothing method in both one-dimensional and multiple-dimensional cases.</p>

Molecular dynamics lattice gas equilibrium distribution function for Lennard-Jones particles.

The molecular dynamics lattice gas (MDLG) method maps a molecular dynamics (MD) simulation onto a lattice gas using a coarse-graining procedure. This is a novel fundamental approach to derive the lattice Boltzmann method (LBM) by taking a Boltzmann average over the MDLG. A key property of the LBM is the equilibrium distribution function, which was originally derived by assuming that the particle displacements in the MD simulation are Boltzmann distributed. However, we recently discovered that a single Gaussian distribution function is not sufficient to describe the particle displacements in a broad transition regime between free particles and particles undergoing many collisions in one time step. In a recent publication, we proposed a Poisson weighted sum of Gaussians which shows better agreement with the MD data. We derive a lattice Boltzmann equilibrium distribution function from the Poisson weighted sum of Gaussians model and compare it to a measured equilibrium distribution function from MD data and to an analytical approximation of the equilibrium distribution function from a single Gaussian probability distribution function. This article is part of the theme issue 'Progress in mesoscale methods for fluid dynamics simulation'.

A nearby galaxy perspective on dust evolution

Context. The efficiency of the different processes responsible for the evolution of interstellar dust on the scale of a galaxy are, to date, very uncertain, spanning several orders of magnitude in the literature. Yet, precise knowledge of the grain properties is key to addressing numerous open questions about the physics of the interstellar medium and galaxy evolution. Aims. This article presents an empirical statistical study, aimed at quantifying the timescales of the main cosmic dust evolution processes as a function of the global properties of a galaxy. Methods. We modeled a sample of ≃800 nearby galaxies, spanning a wide range of metallicities, gas fractions, specific star formation rates, and Hubble stages. We derived the dust properties of each object from its spectral energy distribution. Through an additional level of analysis, we inferred the timescales of dust condensation in core-collapse supernova ejecta, grain growth in cold clouds, and dust destruction by shock waves. Throughout this paper, we have adopted a hierarchical Bayesian approach, resulting in a single large probability distribution of all the parameters of all the galaxies, to ensure the most rigorous interpretation of our data. Results. We confirm the drastic evolution with metallicity of the dust-to-metal mass ratio (by two orders of magnitude), found by previous studies. We show that dust production by core-collapse supernovae is efficient only at very low metallicity, a single supernova producing on average less than ≃0.03 M⊙/SN of dust. Our data indicate that grain growth is the dominant formation mechanism at metallicity above ≃1/5 solar, with a grain growth timescale shorter than ≃50 Myr at solar metallicity. Shock destruction is relatively efficient, a single supernova clearing dust on average in at least ≃1200 M⊙/SN of gas. These results are robust when assuming different stellar initial mass functions. In addition, we show that early-type galaxies are outliers in several scaling relations. This feature could result from grain thermal sputtering in hot X-ray emitting gas, which is a hypothesis supported by a negative correlation between the dust-to-stellar mass ratio and the X-ray photon rate per grain. Finally, we confirm the well-known evolution of the aromatic-feature-emitting grain mass fraction as a function of metallicity and interstellar radiation field intensity. Our data indicate that the relation with metallicity is significantly stronger. Conclusions. Our results provide valuable constraints for simulations of galaxies. They imply that grain growth is the likely dust production mechanism in dusty high-redshift objects. We also emphasize the determinant role of local, low metallicity systems in order to address these questions.

Euclid preparation

Forthcoming large photometric surveys for cosmology require precise and accurate photometric redshift (photo-z) measurements for the success of their main science objectives. However, to date, no method has been able to produce photo-zs at the required accuracy using only the broad-band photometry that those surveys will provide. An assessment of the strengths and weaknesses of current methods is a crucial step in the eventual development of an approach to meet this challenge. We report on the performance of 13 photometric redshift code single value redshift estimates and redshift probability distributions (PDZs) on a common set of data, focusing particularly on the 0.2 − 2.6 redshift range that the Euclid mission will probe. We designed a challenge using emulated Euclid data drawn from three photometric surveys of the COSMOS field. The data was divided into two samples: one calibration sample for which photometry and redshifts were provided to the participants; and the validation sample, containing only the photometry to ensure a blinded test of the methods. Participants were invited to provide a redshift single value estimate and a PDZ for each source in the validation sample, along with a rejection flag that indicates the sources they consider unfit for use in cosmological analyses. The performance of each method was assessed through a set of informative metrics, using cross-matched spectroscopic and highly-accurate photometric redshifts as the ground truth. We show that the rejection criteria set by participants are efficient in removing strong outliers, that is to say sources for which the photo-z deviates by more than 0.15(1 + z) from the spectroscopic-redshift (spec-z). We also show that, while all methods are able to provide reliable single value estimates, several machine-learning methods do not manage to produce useful PDZs. We find that no machine-learning method provides good results in the regions of galaxy color-space that are sparsely populated by spectroscopic-redshifts, for example z > 1. However they generally perform better than template-fitting methods at low redshift (z < 0.7), indicating that template-fitting methods do not use all of the information contained in the photometry. We introduce metrics that quantify both photo-z precision and completeness of the samples (post-rejection), since both contribute to the final figure of merit of the science goals of the survey (e.g., cosmic shear from Euclid). Template-fitting methods provide the best results in these metrics, but we show that a combination of template-fitting results and machine-learning results with rejection criteria can outperform any individual method. On this basis, we argue that further work in identifying how to best select between machine-learning and template-fitting approaches for each individual galaxy should be pursued as a priority.

Eigensystem Multiscale Analysis for the Anderson Model via the Wegner Estimate

We present a new approach to the eigensystem multiscale analysis (EMSA) for random Schr\"odinger operators that relies on the Wegner estimate. The EMSA treats all energies of the finite volume operator in an energy interval at the same time, simultaneously establishing localization of all eigenfunctions with eigenvalues in the energy interval with high probability. It implies all the usual manifestations of localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization). The new method removes the restrictive level spacing hypothesis used in the previous versions of the EMSA. The method is presented in the context of the Anderson model, allowing for single site probability distributions that are H\"older continuous of order $\alpha \in (0,1]$.

Application of random effects negative binomial model with clustered dataset for vehicle crash frequency analysis

Abstract One of the major challenges with vehicle crash frequency studies is how to deal with the unobserved heterogeneity in crash data. While statistical models of crash frequency analysis based on single probability distributions are constantly improving, several researchers discovered that multiple distribution models may better describe crash frequency data and capture more unobserved heterogeneity. Based on the hypothesis that total crash counts occurring at an intersection are affected by unique sets of factors, this research proposes a two-step approach to studying the contributing factors to crashes at intersections in the Mississippi coastal area. In this study, the data of single crash accidents are first clustered into subgroups using a hierarchical clustering method, and then a Random Effects Negative Binomial model is applied to each subgroup with crash counts at an intersection as observations. A model with no data clustering is also estimated to serve as the comparison benchmark. The analysis results show that this two-step approach can reveal more information about crash contributing factors and improve the predictive power and goodness of fit.

Assessing the sustainability of emerging technologies: A probabilistic LCA method applied to advanced photovoltaics

A key source of uncertainty in the environmental assessment of emerging technologies is the unpredictable manufacturing, use, and end-of-life pathways a technology can take as it progresses from lab to industrial scale. This uncertainty has sometimes been addressed in life cycle assessment (LCA) by performing scenario analysis. However, the scenario-based approach can be misleading if the probabilities of occurrence of each scenario are not incorporated. It also brings about a practical problem; considering all possible pathways, the number of scenarios can quickly become unmanageable. We present a modelling approach in which all possible pathways are modelled as a single product system with uncertain processes. These processes may or may not be selected once the technology reaches industrial scale according to given probabilities. An uncertainty analysis of such a system provides a single probability distribution for each impact score. This distribution accounts for uncertainty about the product system’s final configuration along with other sources of uncertainty. Furthermore, a global sensitivity analysis can identify whether the future selection of certain pathways over others will be of importance for the variance of the impact score. We illustrate the method with a case study of an emerging technology for front-side metallization of photovoltaic cells.

Toppling and height probabilities in sandpiles

We study Abelian sandpiles numerically, using exact sampling. Our method uses a combination of Wilson’s algorithm to generate uniformly distributed spanning trees, and Majumdar and Dhar’s bijection with sandpiles. We study the probability of topplings of individual vertices in avalanches initiated at the centre of large cubic lattices in dimensions and 5. Based on these, we estimate the values of the toppling probability exponent in the infinite volume limit in dimensions , and find good agreement with theoretical results on the mean-field value of the exponent in . We also study the distribution of the number of waves in 2D avalanches. Our simulation method, combined with a variance reduction idea, lends itself well to studying some problems even in very high dimensions. We illustrate this with an estimation of the single site height probability distribution in d = 32, and compare this to the asymptotic behaviour as .