arXiv subjects

Alexander K. Hartmann

Publications and source records attributed to Alexander K. Hartmann.

At least 55 records · Page 3Linked to original sources

arXiv2019

Replica Symmetry and Replica Symmetry Breaking for the Traveling Salesperson Problem

We study the energy landscape of the Traveling Salesperson problem (TSP) using exact ground states and a novel linear programming approach to generate excited states with closely defined properties. We look at four different ensembles, notably the classic finite dimensional Euclidean TSP and the mean-field-like (1,2)-TSP, which has its origin directly in the mapping of the Hamiltonian circuit problem on the TSP. Our data supports previous conjectures that the Euclidean TSP does not show signatures of replica symmetry breaking neither in two nor in higher dimension. On the other hand the (1,2)-TSP exhibits some signature which does not exclude broken replica symmetry, making it a candidate for further studies in the future.

cond-mat.dis-nn↗

arXiv2019

Percolation of Fortuin-Kasteleyn clusters for the random-bond Ising model

We apply generalisations of the Swendson-Wang and Wolff cluster algorithms, which are based on the construction of Fortuin-Kasteleyn clusters, to the three-dimensional $\pm 1$ random-bond Ising model. The behaviour of the model is determined by the temperature $T$ and the concentration $p$ of negative (anti-ferromagnetic) bonds. The ground state is ferromagnetic for $0 \le p 0$, our data suggest that the percolation transition is universal, irrespective of whether the ground state exhibits ferromagnetic or spin-glass order, and is in the universality class of standard percolation. This shows that correlations in the bond occupancy of the Fortuin-Kasteleyn clusters are irrelevant, except for $p=0$ where the clusters are tied to Ising correlations so the percolation transition is in the Ising universality class.

cond-mat.dis-nn↗

arXiv2019

Large deviations of the length of the longest increasing subsequence of random permutations and random walks

We study numerically the distributions of the length $L$ of the longest increasing subsequence (LIS) for the two cases of random permutations and of one-dimensional random walks. Using sophisticated large-deviation algorithms, we are able to obtain very large parts of the distribution, especially also covering probabilities smaller than $P(L) = 10^{-1000}$. This enables us to verify for the length of the LIS of random permutations the analytically known asymptotics of the rate function and even the whole Tracy-Widom distribution, to which we observe a rather fast convergence in the larger than typical part. For the length $L$ of LIS of random walks, where no analytical results are known to us, we test a proposed scaling law and observe convergence of the tails into a collapse for increasing system size. Further, we obtain estimates for the leading order behavior of the rate functions of both tails.

cond-mat.dis-nn↗

arXiv2018

Large-deviation properties of the largest biconnected component for random graphs

We study the size of the largest biconnected components in sparse Erdős-Rényi graphs with finite connectivity and Barabási-Albert graphs with non-integer mean degree. Using a statistical-mechanics inspired Monte Carlo approach we obtain numerically the distributions for different sets of parameters over almost their whole support, especially down to the rare-event tails with probabilities far less than $10^{-100}$. This enables us to observe a qualitative difference in the behavior of the size of the largest biconnected component and the largest $2$-core in the region of very small components, which is unreachable using simple sampling methods. Also, we observe a convergence to a rate function even for small sizes, which is a hint that the large deviation principle holds for these distributions.

cond-mat.dis-nn↗

arXiv2018

Large-deviation properties of the extended Moran model

The distributions of the times to the first common ancestor t_mrca is numerically studied for an ecological population model, the extended Moran model. This model has a fixed population size N. The number of descendants is drawn from a beta distribution Beta(alpha, 2-alpha) for various choices of alpha. This includes also the classical Moran model (alpha->0) as well as the uniform distribution (alpha=1). Using a statistical mechanics-based large-deviation approach, the distributions can be studied over an extended range of the support, down to probabilities like 10^{-70}, which allowed us to study the change of the tails of the distribution when varying the value of alpha in [0,2]. We find exponential distributions p(t_mrca)~ delta^{t_mrca} in all cases, with systematically varying values for the base delta. Only for the cases alpha=0 and alpha=1, analytical results are known, i.e., delta=\exp(-2/N^2) and delta=2/3, respectively. We recover these values, confirming the validity of our approach. Finally, we also study the correlations between t_mrca and the number of descendants.

q-bio.PE↗

arXiv2018

Phase Transitions of the Typical Algorithmic Complexity of the Random Satisfiability Problem Studied with Linear Programming

Here we study the NP-complete $K$-SAT problem. Although the worst-case complexity of NP-complete problems is conjectured to be exponential, there exist parametrized random ensembles of problems where solutions can typically be found in polynomial time for suitable ranges of the parameter. In fact, random $K$-SAT, with $α=M/N $ as control parameter, can be solved quickly for small enough values of $α$. It shows a phase transition between a satisfiable phase and an unsatisfiable phase. For branch and bound algorithms, which operate in the space of feasible Boolean configurations, the empirically hardest problems are located only close to this phase transition. Here we study $K$-SAT ($K=3,4$) and the related optimization problem MAX-SAT by a linear programming approach, which is widely used for practical problems and allows for polynomial run time. In contrast to branch and bound it operates outside the space of feasible configurations. On the other hand, finding a solution within polynomial time is not guaranteed. We investigated several variants like including artificial objective functions, so called cutting-plane approaches, and a mapping to the NP-complete vertex-cover problem. We observed several easy-hard transitions, from where the problems are typically solvable (in polynomial time) using the given algorithms, respectively, to where they are not solvable in polynomial time. For the related vertex-cover problem on random graphs these easy-hard transitions can be identified with structural properties of the graphs, like percolation transitions. For the present random $K$-SAT problem we have investigated numerous structural properties also exhibiting clear transitions, but they appear not be correlated to the here observed easy-hard transitions. This renders the behaviour of random $K$-SAT more complex than, e.g., the vertex-cover problem.

cond-mat.dis-nn↗

arXiv2018

Large Deviations of Convex Hulls of the "True" Self-Avoiding Random Walk

We study the distribution of the area and perimeter of the convex hull of the "true" self-avoiding random walk in a plane. Using a Markov chain Monte Carlo sampling method, we obtain the distributions also in their far tails, down to probabilities like $10^{-800}$. This enables us to test previous conjectures regarding the scaling of the distribution and the large-deviation rate function $Φ$. In previous studies, e.g., for standard random walks, the whole distribution was governed by the Flory exponent $ν$. We confirm this in the present study by considering expected logarithmic corrections. On the other hand, the behavior of the rate function deviates from the expected form. For this exception we give a qualitative reasoning.

cond-mat.stat-mech↗

arXiv2018

Ground state energy of noninteracting fermions with a random energy spectrum

We derive analytically the full distribution of the ground-state energy of $K$ non-interacting fermions in a disordered environment, modelled by a Hamiltonian whose spectrum consists of $N$ i.i.d.~random energy levels with distribution $p(\varepsilon)$ (with $\varepsilon \geq 0$), in the same spirit as the `Random Energy Model'. We show that for each fixed $K$, the distribution $P_{K,N}(E_0)$ of the ground-state energy $E_0$ has a universal scaling form in the limit of large $N$. We compute this universal scaling function and show that it depends only on $K$ and the exponent $α$ characterizing the small $\varepsilon$ behaviour of $p(\varepsilon) \sim \varepsilon^α$. We compared the analytical predictions with results from numerical simulations. For this purpose we employed a sophisticated importance-sampling algorithm that allowed us to obtain the distributions over a large range of the support down to probabilities as small as $10^{-160}$. We found asymptotically a very good agreement between analytical predictions and numerical results.

cond-mat.dis-nn↗

arXiv2018

The distribution of shortest path lengths in subcritical Erdős-Rényi networks

Networks that are fragmented into small disconnected components are prevalent in a large variety of systems. These include the secure communication networks of commercial enterprises, government agencies and illicit organizations, as well as networks that suffered multiple failures, attacks or epidemics. The properties of such networks resemble those of subcritical random networks, which consist of finite components, whose sizes are non-extensive. Surprisingly, such networks do not exhibit the small-world property that is typical in supercritical random networks, where the mean distance between pairs of nodes scales logarithmically with the network size. Unlike supercritical networks whose structure has been studied extensively, subcritical networks have attracted little attention. A special feature of these networks is that the statistical and geometric properties vary between different components and depend on their sizes and topologies. The overall statistics of the network can be obtained by a summation over all the components with suitable weights. We use a topological expansion to perform a systematic analysis of the degree distribution and the distribution of shortest path lengths (DSPL) on components of given sizes and topologies in subcritical Erdos-Renyi (ER) networks. From this expansion we obtain an exact analytical expression for the DSPL of the entire subcritical network, in the asymptotic limit. The DSPL, which accounts for all the pairs of nodes that reside on the same finite component (FC), is found to follow a geometric distribution of the form $P_{\rm FC}(L=\ell|L<\infty)=(1-c)c^{\ell-1}$, where $c<1$ is the mean degree. We confirm the convergence to this asymptotic result using computer simulations. Using the duality relations between subcritical and supercritical ER networks, we obtain the DSPL on the non-giant components above the percolation transition.

cond-mat.dis-nn↗

arXiv2018

Large Deviations of Convex Hulls of Self-Avoiding Random Walks

A global picture of a random particle movement is given by the convex hull of the visited points. We obtained numerically the probability distributions of the volume and surface of the convex hulls of a selection of three types of self-avoiding random walks, namely the classical Self-Avoiding Walk, the Smart-Kinetic Self-Avoiding Walk, and the Loop-Erased Random Walk. To obtain a comprehensive description of the measured random quantities, we applied sophisticated large-deviation techniques, which allowed us to obtain the distributions over a large range of the support down to probabilities far smaller than $P = 10^{-100}$ . We give an approximate closed form of the so-called large-deviation rate function $Φ$ which generalizes above the upper critical dimension to the previously studied case of the standard random walk. Further we show correlations between the two observables also in the limits of atypical large or small values.

cond-mat.stat-mech↗

arXiv2018

Large-deviation Properties of Linear-programming Computational Hardness of the Vertex Cover Problem

The distribution of the computational cost of linear-programming (LP) relaxation for vertex cover problems on Erdos-Renyi random graphs is evaluated by using the rare-event sampling method. As a large-deviation property, differences of the distribution for "easy" and "hard" problems are found reflecting the hardness of approximation by LP relaxation. In particular, by evaluating the total variation distance between conditional distributions with respect to the hardness, it is suggested that those distributions are almost indistinguishable in the replica symmetric (RS) phase while they asymptotically differ in the replica symmetry breaking (RSB) phase. In addition, we seek for a relation to graph structure by investigating a similarity to bipartite graphs, which exhibits a quantitative difference between the RS and RSB phase. These results indicate the nontrivial relation of the typical computational cost of LP relaxation to the RS-RSB phase transition as present in the spin-glass theory of models on the corresponding random graph structure.

cond-mat.dis-nn↗

arXiv2018

High-precision simulation of the height distribution for the KPZ equation

The one-point distribution of the height for the continuum Kardar-Parisi-Zhang (KPZ) equation is determined numerically using the mapping to the directed polymer in a random potential at high temperature. Using an importance sampling approach, the distribution is obtained over a large range of values, down to a probability density as small as 10^{-1000} in the tails. Both short and long times are investigated and compared with recent analytical predictions for the large-deviation forms of the probability of rare fluctuations. At short times the agreement with the analytical expression is spectacular. We observe that the far left and right tails, with exponents 5/2 and 3/2 respectively, are preserved until large time. We present some evidence for the predicted non-trivial crossover in the left tail from the 5/2 tail exponent to the cubic tail of Tracy-Widom, although the details of the full scaling form remains beyond reach.

cond-mat.dis-nn↗

arXiv2017

Distribution of diameters for Erdös-Rényi random graphs

We study the distribution of diameters d of Erdös-Rényi random graphs with average connectivity c. The diameter d is the maximum among all shortest distances between pairs of nodes in a graph and an important quantity for all dynamic processes taking place on graphs. Here we study the distribution P(d) numerically for various values of c, in the non-percolating and the percolating regime. Using large-deviations techniques, we are able to reach small probabilities like 10^{-100} which allow us to obtain the distribution over basically the full range of the support, for graphs up to N=1000 nodes. For values c<1, our results are in good agreement with analytical results, proving the reliability of our numerical approach. For c>1 the distribution is more complex and no complete analytical results are available. For this parameter range, P(d) exhibits an inflection point, which we found to be related to a structural change of the graphs. For all values of c, we determined the finite-size rate function Phi(d/N) and were able to extrapolate numerically to N->infinity, indicating that the large deviation principle holds.

cond-mat.dis-nn↗

arXiv2017

Exact Ground States of the Kaya-Berker Model

Here we study the two-dimensional Kaya-Berker model, with a site occupancy p of one sub lattice, by using a polynomial-time exact ground-state algorithm. Thus, we were able to obtain T=0 results in exact equilibrium for rather large system sizes up to 777^2 lattice sites. We obtained sub-lattice magnetization and the corresponding Binder parameter. We found a critical point p_c=0.6423(3) beyond which the sub-lattice magnetization vanishes. This is clearly smaller than previous results which were obtained by using non-exact approaches for much smaller systems. We also created for each realization minimum-energy domain walls from two ground-state calculations for periodic and anti-periodic boundary conditions, respectively. The analysis of the mean and the variance of the domain-wall distribution shows that there is no thermodynamic stable spin-glass phase, in contrast to previous claims about this model.

cond-mat.dis-nn↗

arXiv2017

Convex Hulls of Random Walks in Higher Dimensions: A Large Deviation Study

The distribution of the hypervolume $V$ and surface $\partial V$ of convex hulls of (multiple) random walks in higher dimensions are determined numerically, especially containing probabilities far smaller than $P = 10^{-1000}$ to estimate large deviation properties. For arbitrary dimensions and large walk lengths $T$, we suggest a scaling behavior of the distribution with the length of the walk $T$ similar to the two-dimensional case, and behavior of the distributions in the tails. We underpin both with numerical data in $d=3$ and $d=4$ dimensions. Further, we confirm the analytically known means of those distributions and calculate their variances for large $T$.

cond-mat.stat-mech↗

arXiv2017

Directed negative-weight percolation

We consider a directed variant of the negative-weight percolation model in a two-dimensional, periodic, square lattice. The problem exhibits edge weights which are taken from a distribution that allows for both positive and negative values. Additionally, in this model variant all edges are directed. For a given realization of the disorder, a minimally weighted loop/path configuration is determined by performing a non-trivial transformation of the original lattice into a minimum weight perfect matching problem. For this problem, fast polynomial-time algorithms are available, thus we could study large systems with high accuracy. Depending on the fraction of negatively and positively weighted edges in the lattice, a continuous phase transition can be identified, whose characterizing critical exponents we have estimated by a finite-size scaling analyses of the numerically obtained data. We observe a strong change of the universality class with respect to standard directed percolation, as well as with respect to undirected negative-weight percolation. Furthermore, the relation to directed polymers in random media is illustrated.

cond-mat.dis-nn↗

arXiv2016

Parameter Optimisation of a Virtual Synchronous Machine in a Microgrid

Parameters of a virtual synchronous machine in a small microgrid are optimised. The dynamical behaviour of the system is simulated after a perturbation, where the system needs to return to its steady state. The cost functional evaluates the system behaviour for different parameters. This functional is minimised by Parallel Tempering. Two perturbation scenarios are investigated and the resulting optimal parameters agree with analytical predictions. Dependent on the focus of the optimisation different optima are obtained for each perturbation scenario. During the transient the system leaves the allowed voltage and frequency bands only for a short time if the perturbation is within a certain range.

math.OC↗

arXiv2016

Convex Hulls of Multiple Random Walks: A Large-Deviation Study

We study the polygons governing the convex hull of a point set created by the steps of $n$ independent two-dimensional random walkers. Each such walk consists of $T$ discrete time steps, where $x$ and $y$ increments are i.i.d. Gaussian. We analyze area $A$ and perimeter $L$ of the convex hulls. We obtain probability densities for these two quantities over a large range of the support by using a large-deviation approach allowing us to study densities below $10^{-900}$. We find that the densities exhibit a universal scaling behavior as a function of $A/T$ and $L/\sqrt{T}$, respectively. As in the case of one walker ($n=1$), the densities follow Gaussian distributions for $L$ and $\sqrt{A}$, respectively. We also obtained the rate functions for the area and perimeter, rescaled with the scaling behavior of their maximum possible values, and found limiting functions for $T \rightarrow \infty$, revealing that the densities follow the large-deviation principle. These rate functions can be described by a power law for $n \rightarrow \infty$ as found in the $n=1$ case. We also investigated the behavior of the averages as a function of the number of walks $n$ and found good agreement with the predicted behavior.

cond-mat.stat-mech↗