A subset solution to the sign problem in random matrix simulations

We present a solution to the sign problem in dynamical random matrix simulations of a two-matrix model at nonzero chemical potential. The sign problem, caused by the complex fermion determinants, is solved by gathering the matrices into subsets, whose sums of determinants are real and positive even though their cardinality only grows linearly with the matrix size. A detailed proof of this positivity theorem is given for an arbitrary number of fermion flavors. We performed importance sampling Monte Carlo simulations to compute the chiral condensate and the quark number density for varying chemical potential and volume. The statistical errors on the results only show a mild dependence on the matrix size and chemical potential, which confirms the absence of sign problem in the subset method. This strongly contrasts with the exponential growth of the statistical error in standard reweighting methods, which was also analyzed quantitatively using the subset method. Finally, we show how the method elegantly resolves the Silver Blaze puzzle in the microscopic limit of the matrix model, where it is equivalent to QCD.Comment: 18 pages, 11 figures, as published in Phys. Rev. D; added references; in Sec. VB: added discussion of model satisfying the Silver Blaze for all N (proof in Appendix E

Evading the sign problem in random matrix simulations

We show how the sign problem occurring in dynamical simulations of random matrices at nonzero chemical potential can be avoided by judiciously combining matrices into subsets. For each subset the sum of fermionic determinants is real and positive such that importance sampling can be used in Monte Carlo simulations. The number of matrices per subset is proportional to the matrix dimension. We measure the chiral condensate and observe that the statistical error is independent of the chemical potential and grows linearly with the matrix dimension, which contrasts strongly with its exponential growth in reweighting methods.Comment: 4 pages, 3 figures, minor corrections, as published in Phys. Rev. Let

Phase Transitions and Symmetry Breaking in Genetic Algorithms with Crossover

In this paper, we consider the role of the crossover operator in genetic algorithms. Specifically, we study optimisation problems that exhibit many local optima and consider how crossover affects the rate at which the population breaks the symmetry of the problem. As an example of such a problem, we consider the subset sum problem. In so doing, we demonstrate a previously unobserved phenomenon, whereby the genetic algorithm with crossover exhibits a critical mutation rate, at which its performance sharply diverges from that of the genetic algorithm without crossover. At this critical mutation rate, the genetic algorithm with crossover exhibits a rapid increase in population diversity. We calculate the details of this phenomenon on a simple instance of the subset sum problem and show that it is a classic phase transition between ordered and disordered populations. Finally, we show that this critical mutation rate corresponds to the transition between the genetic algorithm accelerating or preventing symmetry breaking and that the critical mutation rate represents an optimum in terms of the balance of exploration and exploitation within the algorithm

Anonymous Networking amidst Eavesdroppers

The problem of security against timing based traffic analysis in wireless networks is considered in this work. An analytical measure of anonymity in eavesdropped networks is proposed using the information theoretic concept of equivocation. For a physical layer with orthogonal transmitter directed signaling, scheduling and relaying techniques are designed to maximize achievable network performance for any given level of anonymity. The network performance is measured by the achievable relay rates from the sources to destinations under latency and medium access constraints. In particular, analytical results are presented for two scenarios: For a two-hop network with maximum anonymity, achievable rate regions for a general m x 1 relay are characterized when nodes generate independent Poisson transmission schedules. The rate regions are presented for both strict and average delay constraints on traffic flow through the relay. For a multihop network with an arbitrary anonymity requirement, the problem of maximizing the sum-rate of flows (network throughput) is considered. A selective independent scheduling strategy is designed for this purpose, and using the analytical results for the two-hop network, the achievable throughput is characterized as a function of the anonymity level. The throughput-anonymity relation for the proposed strategy is shown to be equivalent to an information theoretic rate-distortion function

Capacity Analysis for Continuous Alphabet Channels with Side Information, Part I: A General Framework

Capacity analysis for channels with side information at the receiver has been an active area of interest. This problem is well investigated for the case of finite alphabet channels. However, the results are not easily generalizable to the case of continuous alphabet channels due to analytic difficulties inherent with continuous alphabets. In the first part of this two-part paper, we address an analytical framework for capacity analysis of continuous alphabet channels with side information at the receiver. For this purpose, we establish novel necessary and sufficient conditions for weak* continuity and strict concavity of the mutual information. These conditions are used in investigating the existence and uniqueness of the capacity-achieving measures. Furthermore, we derive necessary and sufficient conditions that characterize the capacity value and the capacity-achieving measure for continuous alphabet channels with side information at the receiver.Comment: Submitted to IEEE Trans. Inform. Theor

A physicist's approach to number partitioning

The statistical physics approach to the number partioning problem, a classical NP-hard problem, is both simple and rewarding. Very basic notions and methods from statistical mechanics are enough to obtain analytical results for the phase boundary that separates the ``easy-to-solve'' from the ``hard-to-solve'' phase of the NPP as well as for the probability distributions of the optimal and sub-optimal solutions. In addition, it can be shown that solving a number partioning problem of size $N$ to some extent corresponds to locating the minimum in an unsorted list of \bigo{2^N} numbers. Considering this correspondence it is not surprising that known heuristics for the partitioning problem are not significantly better than simple random search.Comment: 35 pages, to appear in J. Theor. Comp. Science, typo corrected in eq.1