Stephan Mertens

Exhaustive search for low-autocorrelation binary sequences

Binary sequences with low autocorrelations are important in communication engineering and in statistical mechanics as ground states of the Bernasconi model. Computer searches are the main tool in the construction of such sequences. Owing to the exponential size of the configuration space, exhaustive searches are limited to short sequences. We discuss an exhaustive search algorithm with run-time characteristic and apply it to compile a table of exact ground states of the Bernasconi model up to N = 48. The data suggest F > 9 for the optimal merit factor in the limit .

Phase Transition in the Number Partitioning Problem

Universit¨at Magdeburg, Institut fu¨r Theoretische Physik, Universit¨atsplatz 2, D-39106 Magdeburg, Germany(February 1, 2008)Number partitioning is an NP-complete problem of com-binatorial optimization. A statistical mechanics analysis re-veals the existence of a phase transition that separates theeasy from the hard to solve instances and that reflects thepseudo-polynomiality of number partitioning. The phase di-agram and the value of the typical ground state energy arecalculated.64.60.Cn, 02.60.Pn, 02.70.Lq, 89.80+h

Continuum percolation thresholds in two dimensions.

A wide variety of methods have been used to compute percolation thresholds. In lattice percolation, the most powerful of these methods consists of microcanonical simulations using the union-find algorithm to efficiently determine the connected clusters, and (in two dimensions) using exact values from conformal field theory for the probability, at the phase transition, that various kinds of wrapping clusters exist on the torus. We apply this approach to percolation in continuum models, finding overlaps between objects with real-valued positions and orientations. In particular, we find precise values of the percolation transition for disks, squares, rotated squares, and rotated sticks in two dimensions and confirm that these transitions behave as conformal field theory predicts. The running time and memory use of our algorithm are essentially linear as a function of the number of objects at criticality.

Random costs in combinatorial optimization

The random cost problem is the problem of indentifying the minimum in a list of random numbers. By definition, this problem cannot be solved faster than by exhaustive search. It is shown that a classical NP-hard optimization problem, number partitioning, is essentially equivalent to the random cost problem. On the one hand this explains the bad performance of heuristic approaches to the number partitioning problem, but on the other hand it allows one to calculate the probability distributions of the optimum and suboptimum costs.

Random numbers for large-scale distributed Monte Carlo simulations.

Monte Carlo simulations are one of the major tools in statistical physics, complex system science, and other fields, and an increasing number of these simulations is run on distributed systems like clusters or grids. This raises the issue of generating random numbers in a parallel, distributed environment. In this contribution we demonstrate that multiple linear recurrences in finite fields are an ideal method to produce high quality pseudorandom numbers in sequential and parallel algorithms. Their known weakness (failure of sampling points in high dimensions) can be overcome by an appropriate delinearization that preserves all desirable properties of the underlying linear sequence.

Computational complexity for physicists

The theory of computational complexity has some interesting links to physics, in particular to quantum computing and statistical mechanics. The article contains an informal introduction to this theory and its links to physics.

Lattice animals: A fast enumeration algorithm and new perimeter polynomials

A fast computer algorithm for enumerating isolated connected clusters on a regular lattice and its Fortran implementation are presented. New perimeter polynomials are calculated for the square, the triangular, the simple cubic, and the square lattice with next nearest neighbors.

On the ground states of the Bernasconi model

The ground states of the Bernasconi model are binary sequences of length N with low autocorrelations. We introduce the notion of perfect sequences, binary sequences with one-valued off-peak correlations of minimum amount. If they exist, they are ground states. Using results from the mathematical theory of cyclic difference sets, we specify all values of N for which perfect sequences exist and how to construct them. For other values of N, we investigate almost perfect sequences, i.e. sequences with two-valued off-peak correlations of minimum amount. Numerical and analytical results support the conjecture that almost perfect sequences exist for all values of N, but that they are not always ground states. We present a construction for low-energy configurations that works if N is the product of two odd primes.

Learning grey-toned patterns in neural networks

The problem of learning multi-state patterns in neural networks is investigated. An analysis of the space of couplings (Gardner approach) yields the distribution of local fields, the critical storage capacity alpha c and the minimum number of errors for an overloaded network. For noisy local fields the classification error is minimized if the local fields of the patterns are allowed to lie in intervals of finite width. A fast converging, adaptive learning algorithm is presented, which finds the coupling matrix of optimal stability.

Universality in the level statistics of disordered systems

Energy spectra of disordered systems share a common feature: If the entropy of the quenched disorder is larger than the entropy of the dynamical variables, the spectrum is locally that of a random energy model and the correlation between energy and configuration is lost. We demonstrate this effect for the Edwards-Anderson model, but we also discuss its universality.