Showing papers on "Randomness published in 1993"

Common randomness in information theory and cryptography. I. Secret sharing

Abstract: As the first part of a study of problems involving common randomness at distance locations, information-theoretic models of secret sharing (generating a common random key at two terminals, without letting an eavesdropper obtain information about this key) are considered. The concept of key-capacity is defined. Single-letter formulas of key-capacity are obtained for several models, and bounds to key-capacity are derived for other models. >

Approximation theory of output statistics

Abstract: Given a channel and an input process we study the minimum randomness of those input processes whose output statistics approximate the original output statistics with arbitrary accuracy. We introduce the notion of resolvability of a channel, defined as the number of random bits required per channel use in order to generate an input that achieves arbitrarily accurate approximation of the output statistics for any given input process. We obtain a general formula for resolvability which holds regardless of the channel memory structure. We show that, for most channels, resolvability is equal to Shannon capacity. By-products of our analysis are a general formula for the minimum achievable (fixed-length) source coding rate of any finite-alphabet source, and a strong converse of the identification coding theorem, which holds for any channel that satisfies the strong converse of the channel coding theorem.

Chernoff-Hoeffding bounds for applications with limited independence

Abstract: Chernoff-Hoeffding bounds are fundamental tools used in bounding the tail probabilities of the sums of bounded and independent random variables. We present a simple technique which gives slightly better bounds than these and which, more importantly, requires only limited independence among the random variables, thereby importing a variety of standard results to the case of limited independence for free. Additional methods are also presented, and the aggregate results are very sharp and provide a better understanding of the proof techniques behind these bounds. They also yield improved bounds for various tail probability distributions and enable improved approximation algorithms for jobshop scheduling. The ``limited independence'''' result implies that weaker sources of randomness are sufficient for randomized algorithms whose analyses use the Chernoff-Hoeffding bounds; further, it leads to algorithms that require a reduced amount of randomness for any analysis which uses the Chernoff-Hoeffding bounds, e.g., the analysis of randomized algorithms for random sampling and oblivious packet routing.

Network model evaluation of permeability and spatial correlation in a real random sphere packing

Abstract: In principle, network models can replicate exactly the microstructure of porous media. In practice, however, network models have been constructed using various assumptions concerning pore structure. This paper presents a network model of a real, disordered porous medium that invokes no assumptions regarding pore structure. The calculated permeability of the model agrees well with measured permeabilities, providing a new and more rigorous confirmation of the validity of the network approach. Several assumptions commonly used in constructing network models are found to be invalid for a random packing of equal spheres. In addition, the model permits quantification of the effect of pore-scale correlation (departure from randomness) upon permeability. The effect is comparable to reported discrepancies between measured permeabilities and predictions of other network models. The implications of this finding are twofold. First, a key assumption of several theories of transport in porous media, namely that pore dimensions are randomly distributed upon a network, may be invalid for real porous systems. Second, efforts both to model and to measure pore-scale correlations could yield more accurate predictions of permeability.

Nonlinear analysis of phase relationships in quasi-optical oscillator arrays

Abstract: A dynamic theory of coupled oscillators is developed and applied to the class of loosely coupled quasi-optical oscillator arrays. This theory permits the calculation of stable, steady-state phase relationships between the oscillators. The distribution of free-running frequencies and the coupling parameters are most important in determining the behavior of the arrays. It is found that free-running frequencies of the peripheral elements have the strongest influence on the steady-state phase relationships. The influence of randomness in the frequency distribution is considered for the case of broadside beamforming, establishing a critical value for the coupling strength in order to maintain mutual synchronization with a specified maximum beam deviation. Techniques for simplifying the calculation of phase relationships for some common coupling parameters are also developed. >

Statistical inevitability of Horton's laws and the apparent randomness of stream channel networks

Abstract: The remarkably regular geometric relations observed in stream networks have been widely interpreted as evidence of a distinctive structure that reflects particular geomorphic processes. These relations have also been interpreted as evidence that stream networks are topologically random, formed by the laws of chance. Neither of these inferences is justified. The oft-cited geometric properties are not specific to particular kinds of stream networks or to topologically random networks; instead, they describe virtually all possible networks. They therefore compel no particular conclusion about the origin or structure of stream networks.

Mean field theory for a simple model of evolution.

Abstract: A simple dynamical model for Darwinian evolution on its slowest time scale is analyzed. Its mean field theory is formulated and solved. A random neighbor version of the model is simulated, as is a one-dimensional version. In one dimension, the dynamics can be described in terms of a ``repetitious random walker'' and anomalous diffusion with exponent 0.4. In all cases the model self-organizes to a robust critical attractor.

Randomness And Undecidability In Physics

Abstract: Part 1 Algorithmic physics: algorithmics automata coding and representation automator worlds algorithmic information and other resources measures. Part 2 Undecidability: true does not equal provable Cantor's diagonaliation method halting problem Godel's incompleteness theorem true > provable intrinsic indeterminism weak physical chaos. Part 3 Randomness: conceptual developments "Lawlessness" = "Algorithmic Incompressibility" "Computational Irreducibility" von Mises collectives statistical based randomness equivalences chaotic systems are optimal analogues of themselves quantum chaos algorithmic probability and information theory entropy origins of entropy increase zeno squeezing definition of chaos via equidecomposibility of attractors metaphysics.

Random square-triangle tilings : a model for twelvefold-symmetric quasicrystals

Abstract: Random tilings that comprise squares and equilateral triangles can model quasicrystals with twelvefold symmetry. A (phason) elastic theory for such tilings is constructed, whose order parameter is the phason field, and whose entropy density includes terms up to third order in the phason strain. Due to an unusual constraint, the phason field of any square-triangle tiling is irrotational and, as a result, the form of the entropy density is simpler than the general form that is required by twelvefold symmetry alone. Using an update move, which rearranges a closed, nonlocal, one-dimensional chain of squares and triangles, the unknown parameters of the elastic theory are estimated via Monte Carlo simulations: (i) One of the two second-order elastic constants and the third-order elastic constant are found by measuring phason fluctuations; athermal systems (maximally random ensembles) with the same background phason strain but different sizes of unit cell are simulated to distinguish the effects of a finite background phason strain from the effects of finite unit-cell size. (ii) The entropy per unit area at zero phason strain and the other second-order elastic constant are found from the entropies that thermal systems (canonical ensembles) gain between zero and infinite temperature, which are estimated using Ferrenberg and Swendsen's histogram method. A way to set up transfer-matrix calculations for random square-triangle tilings is also presented.

A theory of macrodispersion for the scale-up problem

Abstract: Dispersion is the result, observable on large length scales, of events which are random on small length scales. When the length scale on which the randomness operates is not small, relative to the observations, then classical dispersion theory fails. The scale up problem refers to situations in which randomness occurs on all length scales, and for which classical dispersion theory necessarily fails. The purpose of this article is to present non-Fickian, theories of dispersion, which do not assume a scale separation between the randomness and the observed consequences, and which do not assume a single length scale. Porous media flow properties are heterogeneous on all length scales. The geological variation on length scales below the observational length scale can be regarded as unknown and unknowable, and thus as a random variable. We develop a systematic theory relating scaling behavior of the geological heterogeneity to the scaling behavior of the fluid dispersivity. Three qualitatively distinct regimes (Fickian, non-Fickian and nonrenormalizable) are found. The theory gives consistent answers within several distinct analytic approximations, and with numerical simulation of the equations of porous media flow. Comparison to field data is made. The use of Kriging to generate constrained ensembles for conditional simulation is discussed.

Communication complexity and quasi randomness

Abstract: The multiparty communication complexity concerns the least number of bits that must be exchanged among a number of players to collaboratively compute a Boolean function $f ( x_1 , \ldots ,x_k )$, while each player knows at most t inputs for some fixed $t < k$. The relation of the multiparty communication complexity to various hypergraph properties is investigated. Many of these properties are satisfied by random hypergraphs and can be classified by the framework of quasi randomness. Namely, many disparate properties of hypergraphs are shown to be mutually equivalent, and, furthermore, various equivalence classes form a natural hierarchy. In this paper, it is proved that the multiparty communication complexity problems are equivalent to certain hypergraph properties and thereby establish the connections among a large number of combinatorial and computational aspects of hypergraphs or Boolean functions.

Enhanced backscattering of light from weakly rough, random metal surfaces

Abstract: The enhanced backscattering of p-polarized light from a small-rms-height, small-rms-slope, one-dimensional, random metal surface was first predicted on the basis of an infinite-order perturbation calculation that used the techniques of many-body theory and the concepts of weak localization theory. We present an elementary calculation of the contribution to the mean differential reflection coefficient from the incoherent component of the scattered light that is based on the first two nonzero terms in the expansion of this function in powers of the surface-profile function. We show that this approach not only accounts for enhanced backscattering but also gives the correct order of magnitude of the effect predicted by infinite-order perturbation theory.

Recursive sampling of random walks: self-avoiding walks in disordered media

Abstract: The author presents a new algorithm for simulating random walks which is simple, versatile and efficient. It uses recursive function calls and can be used to obtain unbiased samples with any given length distribution. This makes it particularly useful in disordered geometries where the effective connectivity constant is not known a priori. When applying it to self-avoiding random walks on two-dimensional media with quenched randomness, he finds evidence for at least two new renormalization group fixed points.

Ground state of random copolymers and the discrete random energy model

Abstract: We propose and investigate a modification of the random energy model where the energies of different states are still independent random values but may take only discrete values. This model appears naturally in studies of random heteropolymers with monomers of two types. We calculate the probability that the ground state of such a polymer is nondegenerate and test this result against a lattice model of a heteropolymer with exhaustively enumerated conformations. The theory is in excellent agreement with numerical experiment. Our results imply that the lower the energy of the ground state the less probable that it is degenerate. The probability of degeneracy decays exponentially as ground state energy decreases.

A random activity network generator

Abstract: Exact and heuristic procedures are often developed to obtain optimal and near-optimal solutions to decision problems modeled as activity networks. Testing the accuracy and efficiency of these procedures requires the use of activity networks with various sizes, structures, and parameters. The size of the network is determined by its number of nodes and arcs, where the structure is chosen from the set of all structures for the specified network size. The network parameters depend on the nature of the decision problem. Often, it is desirable for test problems to be generated at random from the space of all feasible networks. This paper deals with the problem of generating the size and structure of the network at random from the space of all feasible networks. It develops a theory which guarantees the randomness of the network structure. The theory is the basis for two methods. One can be used to generate dense networks, where the other is used to generate nondense networks. The methods, which are practical a...

Exact integer algorithm for the two-dimensional +/-J Ising spin glass.

Abstract: We describe an exact integer algorithm to compute the partition function of a two-dimensional \ifmmode\pm\else\textpm\fi{}J Ising spin glass. Given a set of quenched random bonds, the algorithm returns the density of states as a function of energy. The computation time is polynomial in the lattice size. We investigate defects, low-lying excitations, and zeros of the partition function in the complex plane. We also discuss the potential to examine other types of quenched randomness.

Critical behavior induced by quenched disorder

Abstract: Domain arguments and renormalization-group calculations indicate that all temperature-driven symmetry-breaking first-order phase transitions are converted to second order by quenched bond randomness. This occurs for infinitesimal randomness in d ⩽2 or d ⩽4 respectively for discrete or continuous ( n = 1 or n ⩾2 component) microscopic degrees of freedom. Even strongly first-order transitions undergo this conversion to second order! Above these dimensions this conversion still occurs but requires a threshold bond randomness, presumably with an intervening new tricritical point. For example, q -state Potts transitions can be made second order for any q in any d , via bond randomness. Non-symmetry-breaking “temperature-driven” first-order transitions are eliminated under the above conditions. These quenched-fluctuation-induced second-order phase transitions suggest the possibility of new universality classes of criticality and tricriticality.

Randomness in interactive proofs

Abstract: This paper initiates a study of the quantitative aspects of randomness in interactive proofs. Our main result, which applies to the equivalent form of IP known as Arthur-Merlin (AM) games, is a randomness-efficient technique for decreasing the error probability. Given an AM proof system forL which achieves error probability 1/3 at the cost of Arthur sendingl(n) random bits per round, and given a polynomialk=k(n), we show how to construct an AM proof system forL which, in the same number of rounds as the original proof system, achieves error 2 −k(n) at the cost of Arthur sending onlyO(l+k) random bits per round. Underlying the transformation is a novel sampling method for approximating the average value of an arbitrary functionf:{0,1} l → [0,1]. The method evaluates the function onO(∈−2 log γ−1) sample points generated using onlyO(l + log γ−1) coin tosses to get an estimate which with probability at least 1-δ is within ∈ of the average value of the function.

Limits of infinite interaction radius, dimensionality and the number of components for random operators with off-diagonal randomness

Abstract: We consider random one-body operators that are analogs of the statistical mechanics Hamiltonians with a varying interaction radiusR, the dimensionality of spaced and the number of the field components (orbitals)n. We prove that all the moments of the Green functions for nonreal energies of these operators converge asR, d, n→∞ to the products of the average Green functions, just as in the mean field approximation of statistical mechanics. We find in particular the selfconsistent equation for the limiting integrated density of states and the limiting form of the conductivity, which is nonzero on the whole support of the integrated density of states.

Exact integer algorithm for the two-dimensional ±J ising spin glass

Abstract: We describe an exact integer algorithm to compute the partition function of a two-dimensional ±J Ising spin glass. Given a set of quenched random bonds, the algorithm returns the density of states as a function of energy. The computation time is polynomial in the lattice size. We investigate defects, low-lying excitations, and zeros of the partition function in the complex plane. We also discuss the potential to examine other types of quenched randomness

Phase diagram of gauge glasses

Abstract: The authors introduce a general class of random spin systems which are symmetric under local gauge transformations. Their model is a generalization of the usual Ising spin glass and includes the Zq, XY, and SU (2) gauge glasses. For this general class of systems, the internal energy and an upper bound on the specific heat are calculated explicitly in any dimensions on a special line in the phase diagram. Although the line intersects a phase boundary at a multicritical point, the internal energy and the bound on the specific heat are found to be written in terms of a simple function. They also show that the boundary between the ferromagnetic and nonferromagnetic phases is parallel to the temperature axis in the low-temperature region of the phase diagram. This means the absence of re-entrant transitions. All these properties are derived by simple applications of gauge transformations of spin and randomness degrees of freedom.

The influence of random delays on parallel execution times

Abstract: Stochastic models are widely used for the performance evaluation of parallel programs and systems. The stochastic assumptions in such models exe intended to represent non-deterministic processing requirements as well as random delays due to inter-process communication end resource contention. In this paper, we provide compelling analytical and experimental evidence that in current and foreseeable shared-memory programs, communication delays introduce negligible variance into the execution time between synchronization points. Furthermore, we show using direct measurements of variance that other sources of randomness, particularly non-deterministic computational requirements, also do not introduce significant variance in many programs. We then use two examples to demonstrate the implications of these results for parallel program performance prediction models, as well as for general stochastic models of parallel systems.

Borel Normality and Algorithmic Randomness.

Abstract: We prove that all random sequences (in Chaitin-Martin-Löf sense) and almost all random strings (in both Kolmogorov-Chaitin and Chaitin senses) satisfy various conditions of normality (first introduced by Borel). All proofs are constructive.

Onset of chaos and its signature in the spectral autocorrelation function

Abstract: The signature of chaos in the spectral autocorrelation function and in its Fourier transform, the survival probability, is shown to be in good agreement with the predictions of random matrix theory. An expression is proposed for the survival probability of an experimentally prepared nonstationary state when the dynamics are intermediate between chaotic and regular. Its validity is tested through the study of a model Hamiltonian . Two parameters can be extracted from the above observable, one which characterizes the level statistics and one which characterizes the distribution of transition intensities.