On the Efficient Calculation of a Linear Combination of Chi-Square Random Variables with an Application in Counting String Vacua

Linear combinations of chi square random variables occur in a wide range of fields. Unfortunately, a closed, analytic expression for the pdf is not yet known. As a first result of this work, an explicit analytic expression for the density of the sum of two gamma random variables is derived. Then a computationally efficient algorithm to numerically calculate the linear combination of chi square random variables is developed. An explicit expression for the error bound is obtained. The proposed technique is shown to be computationally efficient, i.e. only polynomial in growth in the number of terms compared to the exponential growth of most other methods. It provides a vast improvement in accuracy and shows only logarithmic growth in the required precision. In addition, it is applicable to a much greater number of terms and currently the only way of computing the distribution for hundreds of terms. As an application, the exponential dependence of the eigenvalue fluctuation probability of a random matrix model for 4d supergravity with N scalar fields is found to be of the asymptotic form exp(-0.35N).Comment: 21 pages, 19 figures. 3rd versio

Applying Rule Ensembles to the Search for Super-Symmetry at the Large Hadron Collider

In this note we give an example application of a recently presented predictive learning method called Rule Ensembles. The application we present is the search for super-symmetric particles at the Large Hadron Collider. In particular, we consider the problem of separating the background coming from top quark production from the signal of super-symmetric particles. The method is based on an expansion of base learners, each learner being a rule, i.e. a combination of cuts in the variable space describing signal and background. These rules are generated from an ensemble of decision trees. One of the results of the method is a set of rules (cuts) ordered according to their importance, which gives useful tools for diagnosis of the model. We also compare the method to a number of other multivariate methods, in particular Artificial Neural Networks, the likelihood method and the recently presented boosted decision tree method. We find better performance of Rule Ensembles in all cases. For example for a given significance the amount of data needed to claim SUSY discovery could be reduced by 15 % using Rule Ensembles as compared to using a likelihood method.Comment: 24 pages, 7 figures, replaced to match version accepted for publication in JHE

An efficient stochastic approach to uncertainty quantification in 3-D FDTD magnetized cold plasma

pre-printAn efficient stochastic finite-difference time-domain (S-FDTD) method is developed to analyze electromagnetic field variability in three dimensional anisotropic magnetized plasma. The new S-FDTD plasma model provides a full understanding of the true physics due to the associated uncertainties and has broad potential applicability. This new algorithm efficiently calculates in a single simulation not only the mean electromagnetic field values, but also their variance as caused by the variability or uncertainty of the content of the ionosphere. This ability will, for example, provide the capability of determining the confidence level that a communications / remote sensing / radar system will operate as expected under abnormal ionospheric conditions

Statistical mechanics of budget-constrained auctions

Finding the optimal assignment in budget-constrained auctions is a combinatorial optimization problem with many important applications, a notable example being the sale of advertisement space by search engines (in this context the problem is often referred to as the off-line AdWords problem). Based on the cavity method of statistical mechanics, we introduce a message passing algorithm that is capable of solving efficiently random instances of the problem extracted from a natural distribution, and we derive from its properties the phase diagram of the problem. As the control parameter (average value of the budgets) is varied, we find two phase transitions delimiting a region in which long-range correlations arise.Comment: Minor revisio