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Alexander K. Hartmann

Publications and source records attributed to Alexander K. Hartmann.

126 records · Page 7Linked to original sources
arXiv2001

Minimal vertex covers on finite-connectivity random graphs - a hard-sphere lattice-gas picture

The minimal vertex-cover (or maximal independent-set) problem is studied on random graphs of finite connectivity. Analytical results are obtained by a mapping to a lattice gas of hard spheres of (chemical) radius one, and they are found to be in excellent agreement with numerical simulations. We give a detailed description of the replica-symmetric phase, including the size and the entropy of the minimal vertex covers, and the structure of the unfrozen component which is found to percolate at connectivity $c\simeq 1.43$. The replica-symmetric solution breaks down at $c=e\simeq 2.72$. We give a simple one-step replica symmetry broken solution, and discuss the problems in interpretation and generalization of this solution.

cond-mat.dis-nn↗
arXiv2000

Typical solution time for a vertex-covering algorithm on finite-connectivity random graphs

In this letter, we analytically describe the typical solution time needed by a backtracking algorithm to solve the vertex-cover problem on finite-connectivity random graphs. We find two different transitions: The first one is algorithm-dependent and marks the dynamical transition from linear to exponential solution times. The second one gives the maximum computational complexity, and is found exactly at the threshold where the system undergoes an algorithm-independent phase transition in its solvability. Analytical results are corroborated by numerical simulations.

cond-mat.dis-nn↗
arXiv2000

Stress Relaxation of Near-Critical Gels

The time-dependent stress relaxation for a Rouse model of a crosslinked polymer melt is completely determined by the spectrum of eigenvalues of the connectivity matrix. The latter has been computed analytically for a mean-field distribution of crosslinks. It shows a Lifshitz tail for small eigenvalues and all concentrations below the percolation threshold, giving rise to a stretched exponential decay of the stress relaxation function in the sol phase. At the critical point the density of states is finite for small eigenvalues, resulting in a logarithmic divergence of the viscosity and an algebraic decay of the stress relaxation function. Numerical diagonalization of the connectivity matrix supports the analytical findings and has furthermore been applied to cluster statistics corresponding to random bond percolation in two and three dimensions.

cond-mat.soft↗
arXiv2000

Ground-state clusters of two-, three- and four-dimensional +-J Ising spin glasses

A huge number of independent true ground-state configurations is calculated for two-, three- and four-dimensional +- J spin-glass models. Using the genetic cluster-exact approximation method, system sizes up to N=20^2,8^3,6^4 spins are treated. A ``ballistic-search'' algorithm is applied which allows even for large system sizes to identify clusters of ground states which are connected by chains of zero-energy flips of spins. The number of clusters n_C diverges with N going to infinity. For all dimensions considered here, an exponential increase of n_C appears to be more likely than a growth with a power of N. The number of different ground states is found to grow clearly exponentially with N. A zero-temperature entropy per spin of s_0=0.078(5)k_B (2d), s_0=0.051(3)k_B (3d) respectively s_0=0.027(5)k_B (4d) is obtained.

cond-mat.dis-nn↗
arXiv2000

Statistical mechanics perspective on the phase transition in vertex covering finite-connectivity random graphs

The vertex-cover problem is studied for random graphs $G_{N,cN}$ having $N$ vertices and $cN$ edges. Exact numerical results are obtained by a branch-and-bound algorithm. It is found that a transition in the coverability at a $c$-dependent threshold $x=x_c(c)$ appears, where $xN$ is the cardinality of the vertex cover. This transition coincides with a sharp peak of the typical numerical effort, which is needed to decide whether there exists a cover with $xN$ vertices or not. For small edge concentrations $c\ll 0.5$, a cluster expansion is performed, giving very accurate results in this regime. These results are extended using methods developed in statistical physics. The so called annealed approximation reproduces a rigorous bound on $x_c(c)$ which was known previously. The main part of the paper contains an application of the replica method. Within the replica symmetric ansatz the threshold $x_c(c)$ and the critical backbone size $b_c(c)$ can be calculated. For $c<e/2$ the results show an excellent agreement with the numerical findings. At average vertex degree $2c=e$, an instability of the simple replica symmetric solution occurs.

cond-mat.stat-mech↗
arXiv2000

The number of guards needed by a museum: A phase transition in vertex covering of random graphs

In this letter we study the NP-complete vertex cover problem on finite connectivity random graphs. When the allowed size of the cover set is decreased, a discontinuous transition in solvability and typical-case complexity occurs. This transition is characterized by means of exact numerical simulations as well as by analytical replica calculations. The replica symmetric phase diagram is in excellent agreement with numerical findings up to average connectivity $e$, where replica symmetry becomes locally unstable.

cond-mat.dis-nn↗
arXiv2000

Comment on `Glassy Transition in a Disordered Model for the RNA Secondary Structure'

In cond-mat/9907125 the low-temperature behavior of a model for RNA secondary structure was studied. It is claimed that the model exhibits a breaking of the replica symmetry, since the width of the distribution P(q) of overlaps may converge to a finite value at T=0. The authors used an exact enumeration method to obtain all ground states for a given RNA sequence. Because of the exponential growing degeneracy, only sequences up to length L=256 could be studied. Here it is shown that, in contrast to the previous results, by going to much larger sizes as L=2000 the variance coverges towards zero, i.e. P(q) is a delta-function in the thermodynamic limit.

cond-mat.dis-nn↗
arXiv1999

How to evaluate ground-state landscapes of disordered systems thermodynamical correctly

Ground states of three-dimensional EA Ising spin glasses are calculated for sizes up to 14^3 using a combination of a genetic algorithm and cluster-exact approximation. For each realization several independent ground states are obtained. Then, by applying ballistic search and T=0 Monte-Carlo simulations, it is ensured that each ground state appears with the same probability. Consequently, the results represent the true T=0 thermodynamic behavior. The distribution P(|q|) of overlaps is evaluated. For increasing size the width of P(|q|) and the fraction of the distribution below q_0=0.5 converge to zero. This indicates that for the infinite system P(|q|) is a delta function, in contrast to previous results. Thus, the ground-state behavior is dominated by few large clusters of similar ground states.

cond-mat.dis-nn↗
arXiv1999

A new method for analyzing ground-state landscapes: ballistic search

A ``ballistic-search'' algorithm is presented which allows the identification of clusters (or funnels) of ground states in Ising spin glasses even for moderate system sizes. The clusters are defined to be sets of states, which are connected in state-space by chains of zero-energy flips of spins. The technique can also be used to estimate the sizes of such clusters. The performance of the method is tested with respect to different system sizes and choices of parameters. As an application the ground-state funnel structure of two-dimensional +or- J spin glasses of systems up to size L=20 is analyzed by calculating a huge number of ground states per realization. A T=0 entropy per spin of s_0=0.086(4)k_B is obtained.

cond-mat.dis-nn↗
arXiv1999

Analysis of the statistical behavior of genetic cluster-exact approximation

The genetic cluster-exact approximation algorithm is an efficient method to calculate ground states of EA spin glasses. The method can be used to study ground-state landscapes by calculating many independent ground states for each realization of the disorder. The algorithm is analyzed with respect to the statistics of the ground states and the valleys of the energy landscape. Furthermore, the distribution inside each valley is evaluated. It is shown that the algorithm does not lead to a true T=0 thermodynamic distribution, i.e. each ground state has not the same frequency of occurrence when performing many runs. An extension of the technique is outlined, which guarantees that each ground states occurs with the same probability.

cond-mat.dis-nn↗
arXiv1999

Calculation of ground states of four-dimensional +or- J Ising spin glasses

Ground states of four-dimensional (d=4) EA Ising spin glasses are calculated for sizes up to 7x7x7x7 using a combination of a genetic algorithm and cluster-exact approximation. The ground-state energy of the infinite system is extrapolated as e_0=-2.095(1). The ground-state stiffness (or domain wall) energy D is calculated. A D~L^Θ behavior with Θ=0.65(4) is found which confirms that the d=4 model has an equilibrium spin-glass-paramagnet transition for non-zero T_c.

cond-mat.dis-nn↗
arXiv1998

Ground-state landscape of 2d +-J Ising spin glasses

Large numbers of ground states of two-dimensional Ising spin glasses with periodic boundary conditions in both directions are calculated for sizes up to 40^2. A combination of a genetic algorithm and Cluster-Exact Approximation is used. For each quenched realization of the bonds up to 40 independent ground states are obtained. For the infinite system a ground-state energy of e=-1.4015(3) is extrapolated. The ground-state landscape is investigated using a finite-size scaling analysis of the distribution of overlaps. The mean-field picture assuming a complex landscape describes the situation better than the droplet-scaling model, where for the infinite system mainly two ground states exist. Strong evidence is found that the ground states are not organized in an ultrametric fashion in contrast to previous results for three-dimensional spin glasses.

cond-mat.dis-nn↗
arXiv1998

Ground-state behavior of the 3d +/-J random-bond Ising model

Large numbers of ground states of the three-dimensional $\pm J$ random-bond Ising model are calculated for sizes up to $14^3$ using a combination of a genetic algorithm and Cluster-Exact Approximation. Several quantities are calculated as function of the concentration $p$ of the antiferromagnetic bonds. The critical concentration where the ferromagnetic order disappears is determined using the Binder cumulant of the magnetization. A value of $p_c=0.222\pm 0.005$ is obtained. From the finite-size behavior of the Binder cumulant and the magnetization critical exponents $ν=1.1 \pm 0.3$ and $β=0.2 \pm 0.1$ are calculated.

cond-mat.dis-nn↗
arXiv1998

Scaling of stiffness energy for 3d +/-J Ising spin glasses

Large numbers of ground states of 3d EA Ising spin glasses are calculated for sizes up to 10^3 using a combination of a genetic algorithm and Cluster-Exact Approximation. A detailed analysis shows that true ground states are obtained. The ground state stiffness (or domain wall) energy D is calculated. A D ~ L^t behavior with t=0.19(2) is found which strongly indicates that the 3d model has an equilibrium spin-glass-paramagnet transition for non-zero T_c.

cond-mat.dis-nn↗
arXiv1998

Are Ground States of 3d +/-J Spin Glasses ultrametric ?

Ground states of 3d EA Ising spin glasses are calculated for sizes up to 14^3 using a combination of a genetic algorithm and Cluster-Exact Approximation. Evidence for an ultrametric structure is found by studying triplets of independent ground states where one or two values of the three overlaps are fixed.

cond-mat.dis-nn↗
arXiv1997

Evidence for existence of many pure ground states in 3d $\pm J$ Spin Glasses

Ground states of 3d EA Ising spin glasses are calculated for sizes up to $14^3$ using a combination of genetic algorithms and cluster-exact approximation . The distribution $P(|q|)$ of overlaps is calculated. For increasing size the width of $P(|q|)$ converges to a nonzero value, indicating that many pure ground states exist for short range Ising spin glasses.

cond-mat.dis-nn↗
arXiv1997

Ground state structure of diluted antiferromagnets and random field systems

A method is presented for the calculation of all exact ground states of diluted antiferromagnets and random field systems in an arbitrary range of fields. It works by calculating all jump-fields B,Δwhere the system changes it's ground state. For each field value all degenerated ground states are represented by a set of (anti-) ferromagnetic clusters and a relation between the clusters. So a complete description of the ground state structure of these systems is possible. Systems are investigated up to size 48^3 on the whole field-range and up to 160^3 for some particular fields. The behavior of order parameters is investigated, the number of jumps is analyzed and the degree of degeneracy as functions of size and fields is calculated.

cond-mat.dis-nn↗