arXiv subjects

Hendrik Schawe

Publications and source records attributed to Hendrik Schawe.

26 records · Page 2Linked to original sources

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

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

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↗

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↗

arXiv2015

Phase Transitions of Traveling Salesperson Problems solved with Linear Programming and Cutting Planes

The Traveling Salesperson problem asks for the shortest cyclic tour visiting a set of cities given their pairwise distances and belongs to the NP-hard complexity class, which means that with all known algorithms in the worst case instances are not solveable in polynomial time, i.e., the problem is hard. Though that does not mean, that there are not subsets of the problem which are easy to solve. To examine numerically transitions from an easy to a hard phase, a random ensemble of cities in the Euclidean plane given a parameter σ, which governs the hardness, is introduced. Here, a linear programming approach together with suitable cutting planes is applied. Such algorithms operate outside the space of feasible solutions and are often used in practical application but rarely studied in physics so far. We observe several transitions. To characterize these transitions, scaling assumptions from continuous phase transitions are applied

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