arXiv subjects

Alexander K. Hartmann

Publications and source records attributed to Alexander K. Hartmann.

At least 37 records · Page 2Linked to original sources

arXiv2023

Rare Events of Host Switching for Diseases using a SIR Model with Mutations

We numerically study disease dynamics that lead to the disease switching from one host species to another, resulting in diseases gaining the ability to infect, e.g., humans. Unlike previous studies that focused on branching processes starting with the first infected humans, we begin by considering a disease pathogen that initially cannot infect humans. We model the entire process, starting from an infection in the animal population, including mutations that eventually enable the disease to cause an epidemic outbreak in the human population. We use an SIR model on a network consisting of 132 dog and 1320 human nodes, with a single parameter representing the gene of the pathogen. We use numerical large-deviation techniques, specifically the $1/t$ Wang-Landau algorithm, to calculate the potentially very small probability of the host switching event. With this approach we are able to resolve probabilities as small as $10^{-120}$. Additionally the $1/t$ Wang-Landau algorithm allows us to obtain the complete probability density function $P(C)$ of the cumulative fraction $C$ of infected humans, which is an indicator for the severity of the disease in the human population. We also calculate correlations of $C$ with selected quantities $q$ that characterize the outbreak. Due to the application of the rare-event algorithm, this is possible for the entire range of $C$ values.

physics.soc-ph↗

arXiv2023

Energy landscapes of some matching-problem ensembles

The maximum-weight matching problem and the behavior of its energy landscape is numerically investigated. We apply a perturbation method adapted from the analysis of spin glasses. This gives inside into the complexity of the energy landscape of different ensembles. Erdös-Renyi graphs and ring graphs with randomly added edges are considered and two types of distributions for the random edge weighs are used. For maximum-weight matching, fast and scalable algorithms exist, such that we can study large graphs of more than $10^5$ nodes. Our results show that the structure of the energy landscape for standard ensembles of matching is simple, comparable to the energy landscape of a ferromagnet. Nonetheless, for some of the here presented ensembles our results allow for the presence of complex energy landscapes in the spirit of Replica-Symmetry Breaking.

cond-mat.dis-nn↗

arXiv2023

Current fluctuations in stochastically resetting particle systems

We consider a system of non-interacting particles on a line with initial positions distributed uniformly with density $ρ$ on the negative half-line. We consider two different models: (i) each particle performs independent Brownian motion with stochastic resetting to its initial position with rate $r$ and (ii) each particle performs run and tumble motion, and with rate $r$ its position gets reset to its initial value and simultaneously its velocity gets randomised. We study the effects of resetting on the distribution $P(Q,t)$ of the integrated particle current $Q$ up to time $t$ through the origin (from left to right). We study both the annealed and the quenched current distributions and in both cases, we find that resetting induces a stationary limiting distribution of the current at long times. However, we show that the approach to the stationary state of the current distribution in the annealed and the quenched cases are drastically different for both models. In the annealed case, the whole distribution $P_{\rm an}(Q,t)$ approaches its stationary limit uniformly for all $Q$. In contrast, the quenched distribution $P_{\rm qu}(Q,t)$ attains its stationary form for $Q Q_{\rm crit}(t)$. We show that $Q_{\rm crit}(t)$ increases linearly with $t$ for large $t$. On the scale where $Q \sim Q_{\rm crit}(t)$, we show that $P_{\rm qu}(Q,t)$ has an unusual large deviation form with a rate function that has a third-order phase transition at the critical point. We have computed the associated rate functions analytically for both models. Using an importance sampling method that allows to probe probabilities as tiny as $10^{-14000}$, we were able to compute numerically this non-analytic rate function for the resetting Brownian dynamics and found excellent agreement with our analytical prediction.

cond-mat.stat-mech↗

arXiv2023

Simulated annealing, optimization, searching for ground states

The chapter starts with a historical summary of first attempts to optimize the spin glass Hamiltonian, comparing it to recent results on searching largest cliques in random graphs. Exact algorithms to find ground states in generic spin glass models are then explored in Section 1.2, while Section 1.3 is dedicated to the bidimensional case where polynomial algorithms exist and allow for the study of much larger systems. Finally Section 1.4 presents a summary of results for the assignment problem where the finite size corrections for the ground state can be studied in great detail.

cond-mat.dis-nn↗

arXiv2022

Large-deviations of the SIR model around the epidemic threshold

We numerically study the dynamics of the SIR disease model on small-world networks by using a large-deviation approach. This allows us to obtain the probability density function of the total fraction of infected nodes and of the maximum fraction of simultaneously infected nodes down to very small probability densities like $10^{-2500}$. We analyze the structure of the disease dynamics and observed three regimes in all probability density functions, which correspond to quick mild, quick extremely severe and sustained severe dynamical evolutions, respectively. Furthermore, the mathematical rate functions of the densities are investigated. The results indicate that the so called large-deviation property hold for the SIR model. Finally, we measured correlations with other quantities like the duration of an outbreak or the peak position of the fraction of infections, also in the rare regions which are not accessible by standard simulation techniques.

physics.soc-ph↗

arXiv2021

Phase transition in the bipartite z-matching

We study numerically the maximum $z$-matching problems on ensembles of bipartite random graphs. The $z$-matching problems describes the matching between two types of nodes, users and servers, where each server may serve up to $z$ users at the same time. By using a mapping to standard maximum-cardinality matching, and because for the latter there exists a polynomial-time exact algorithm, we can study large system sizes of up to $10^6$ nodes. We measure the capacity and the energy of the resulting optimum matchings. First, we confirm previous analytical results for bipartite regular graphs. Next, we study the finite-size behaviour of the matching capacity and find the same scaling behaviour as before for standard matching, which indicates the universality of the problem. Finally, we investigate for bipartite Erdős-Rényi random graphs the saturability as a function of the average degree, i.e., whether the network allows as many customers as possible to be served, i.e. exploiting the servers in an optimal way. We find phase transitions between unsaturable and saturable phases. These coincide with a strong change of the running time of the exact matching algorithm, as well with the point where a minimum-degree heuristic algorithm starts to fail.

cond-mat.dis-nn↗

arXiv2021

Critical behavior of the Anderson model on the Bethe lattice via a large-deviation approach

We present a new large-deviation approach to investigate the critical properties of the Anderson model on the Bethe lattice close to the localization transition in the thermodynamic limit. Our method allows us to study accurately the distribution of the local density of states (LDoS) down to very small probability tails as small as $10^{-50}$ which are completely out of reach for standard numerical techniques. We perform a thorough analysis of the functional form and of the tails of the probability distribution of the LDoS which yields for the first time a direct, transparent, and precise estimation of the correlation volume close to the Anderson transition. Such correlation volume is found to diverge exponentially when the localization is approached from the delocalized regime, in a singular way that is in agreement with the analytic predictions of the supersymmetric treatment.

cond-mat.dis-nn↗

arXiv2021

Observing symmetry-broken optimal paths of stationary Kardar-Parisi-Zhang interface via a large-deviation sampling of directed polymers in random media

Consider the short-time probability distribution $\mathcal{P}(H,t)$ of the one-point interface height difference $h(x=0,τ=t)-h(x=0,τ=0)=H$ of the stationary interface $h(x,τ)$ described by the Kardar-Parisi-Zhang equation. It was previously shown that the optimal path -- the most probable history of the interface $h(x,τ)$ which dominates the upper tail of $\mathcal{P}(H,t)$ -- is described by any of \emph{two} ramp-like structures of $h(x,τ)$ traveling either to the left, or to the right. These two solutions emerge, at a critical value of $H$, via a spontaneous breaking of the mirror symmetry $x\leftrightarrow -x$ of the optimal path, and this symmetry breaking is responsible for a second-order dynamical phase transition in the system. We simulate the interface configurations numerically by employing a large-deviation Monte Carlo sampling algorithm in conjunction with the mapping between the KPZ interface and the directed polymer in a random potential at high temperature. This allows us to observe the optimal paths, which determine each of the two tails of $\mathcal{P}(H,t)$, down to probability densities as small as $10^{-500}$. At short times we observe mirror-symmetry-broken traveling optimal paths for the upper tail, and a single mirror-symmetric path for the lower tail, in good quantitative agreement with analytical predictions. At long times, even at moderate values of $H$, where the optimal fluctuation method is \emph{not} supposed to apply, we still observe two well-defined dominating paths. Each of them violates the mirror symmetry $x\leftrightarrow -x$ and is a mirror image of the other.

cond-mat.stat-mech↗

arXiv2021

Replica-symmetry breaking for directed polymers

Directed polymers on 1+1 dimensional lattices coupled to a heat bath at temperature $T$ are studied numerically for three ensembles of the site disorder. In particular correlations of the disorder as well as fractal patterning are considered. Configurations are directly sampled in perfect thermal equilibrium for very large system sizes with up to $N=L^2= 32768 \times 32768 \approx 10^{9}$ sites. The phase-space structure is studied via the distribution of overlaps and hierarchical clustering of configurations. One ensemble shows a simple behavior like a ferromagnet. The other two ensembles exhibit indications for complex behavior reminiscent of multiple replica-symmetry breaking. Also results for the ultrametricity of the phase space and the phase transition behavior of $P(q)$ when varying the temperature $T$ are studied. In total, the present model ensembles offer convenient numerical accesses to comprehensively studying complex behavior.

cond-mat.dis-nn↗

arXiv2020

Extremely rare ultra-fast non-equilibrium processes can be close to equilibrium: RNA unfolding and refolding

We study numerically the behavior of RNA secondary structures under influence of a varying external force. This allows to measure the work $W$ during the resulting fast unfolding and refolding processes. Here, we investigate a medium-size hairpin structure. Using a sophisticated large-deviation algorithm, we are able to measure work distributions with high precision down to probabilities as small as $10^{-46}$. Due to this precision and by comparison with exact free-energy calculations we are able to verify the theorems of Crooks and Jarzynski. Furthermore, we analyze force-extension curves and the configurations of the secondary structures during unfolding and refolding for typical equilibrium processes and non-equilibrium processes, conditioned to selected values of the measured work $W$, typical and rare ones. We find that the non-equilibrium processes where the work values are close to those which are most relevant for applying Crooks and Jarzynski theorems, respectively, are most and quite similar to the equilibrium processes. Thus, a similarity of equilibrium and non-equilibrium behavior with respect to a mere scalar variable, which occurs with a very small probability but can be generated in a controlled but non-targeted way, is related to a high similarity for the set of configurations sampled along the full dynamical trajectory.

cond-mat.stat-mech↗

arXiv2020

Large deviations of a random walk model with emerging territories

We study an agent-based model of animals marking their territory and evading adversarial territory in one dimension, with respect to the distribution of the size of the resulting territories. In particular, we use sophisticated sampling methods to determine it over a large part of territory sizes, including atypically small and large configurations, which occur with probability of less than $10^{-30}$. We find hints for the validity of a large deviation principle, the shape of the rate function for the right tail of the distribution and insight into the structure of atypical realizations.

cond-mat.stat-mech↗

arXiv2020

Large deviations of connected components in the stochastic block model

We study the stochastic block model which is often used to model community structures and study community-detection algorithms. We consider the case of two blocks in regard to its largest connected component and largest biconnected component, respectively. We are especially interested in the distributions of their sizes including the tails down to probabilities smaller than $10^{-800}$. For this purpose we use sophisticated Markov chain Monte Carlo simulations to sample graphs from the stochastic block model ensemble. We use this data to study the large-deviation rate function and conjecture that the large-deviation principle holds. Further we compare the distribution to the well known Erdős-Rényi ensemble, where we notice subtle differences at and above the percolation threshold.

physics.soc-ph↗

arXiv2020

How many longest increasing subsequences are there?

We study the entropy $S$ of longest increasing subsequences (LIS), i.e., the logarithm of the number of distinct LIS. We consider two ensembles of sequences, namely random permutations of integers and sequences drawn i.i.d.\ from a limited number of distinct integers. Using sophisticated algorithms, we are able to exactly count the number of LIS for each given sequence. Furthermore, we are not only measuring averages and variances for the considered ensembles of sequences, but we sample very large parts of the probability distribution $p(S)$ with very high precision. Especially, we are able to observe the tails of extremely rare events which occur with probabilities smaller than $10^{-600}$. We show that the distribution of the entropy of the LIS is approximately Gaussian with deviations in the far tails, which might vanish in the limit of long sequences. Further we propose a large-deviation rate function which fits best to our observed data.

cond-mat.dis-nn↗

arXiv2020

Phase transition for parameter learning of Hidden Markov Models

We study a phase transition in parameter learning of Hidden Markov Models (HMMs). We do this by generating sequences of observed symbols from given discrete HMMs with uniformly distributed transition probabilities and a noise level encoded in the output probabilities. By using the Baum-Welch (BW) algorithm, an Expectation-Maximization algorithm from the field of Machine Learning, we then try to estimate the parameters of each investigated realization of an HMM. We study HMMs with n=4, 8 and 16 states. By changing the amount of accessible learning data and the noise level, we observe a phase-transition-like change in the performance of the learning algorithm. For bigger HMMs and more learning data, the learning behavior improves tremendously below a certain threshold in the noise strength. For a noise level above the threshold, learning is not possible. Furthermore, we use an overlap parameter applied to the results of a maximum-a-posteriori (Viterbi) algorithm to investigate the accuracy of the hidden state estimation around the phase transition.

cond-mat.stat-mech↗

arXiv2020

Asymptotic behavior of the length of the longest increasing subsequences of random walks

We numerically estimate the leading asymptotic behavior of the length $L_{n}$ of the longest increasing subsequence of random walks with step increments following Student's $t$-distribution with parameter in the range $1/2 \leq ν\leq 5$. We find that the expected value $\mathbb{E}(L_{n}) \sim n^θ\ln{n}$ with $θ$ decreasing from $θ(ν=1/2) \approx 0.70$ to $θ(ν\geq 5/2) \approx 0.50$. For random walks with distribution of step increments of finite variance ($ν> 2$), this confirms previous observation of $\mathbb{E}(L_{n}) \sim \sqrt{n}\ln{n}$ to leading order. We note that this asymptotic behavior (including the subleading term) resembles that of the largest part of random integer partitions under the uniform measure and that, curiously, both random variables seem to follow Gumbel statistics. We also provide more refined estimates for the asymptotic behavior of $\mathbb{E}(L_{n})$ for random walks with step increments of finite variance.

cond-mat.stat-mech↗

arXiv2019

Optimal paths of non-equilibrium stochastic fields: the Kardar-Parisi-Zhang interface as a test case

Atypically large fluctuations in macroscopic non-equilibrium systems continue to attract interest. Their probability can often be determined by the optimal fluctuation method (OFM). The OFM brings about a conditional variational problem, the solution of which describes the "optimal path" of the system which dominates the contribution of different stochastic paths to the desired statistics. The OFM proved efficient in evaluating the probabilities of rare events in a host of systems. However, theoretically predicted optimal paths were observed in stochastic simulations only in diffusive lattice gases, where the predicted optimal density patterns are either stationary, or travel with constant speed. Here we focus on the one-point height distribution of the paradigmatic Kardar-Parisi-Zhang interface. Here the optimal paths, corresponding to the distribution tails at short times, are intrinsically non-stationary and can be predicted analytically. Using the mapping to the directed polymer in a random potential at high temperature, we obtain "snapshots" of the optimal paths in Monte-Carlo simulations which probe the tails with an importance sampling algorithm. For each tail we observe a very narrow "tube" of height profiles around a single optimal path which agrees with the analytical prediction. The agreement holds even at long times, supporting earlier assertions of the validity of the OFM in the tails well beyond the weak-noise limit.

cond-mat.stat-mech↗

arXiv2019

Probing the large deviations of the Kardar-Parisi-Zhang equation at short time with an importance sampling of directed polymers in random media

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. The short time behavior is investigated and compared with recent analytical predictions for the large-deviation forms of the probability of rare fluctuations, showing a spectacular agreement with the analytical expressions. The flat and stationary initial conditions are studied in the full space, together with the droplet initial condition in the half-space.

cond-mat.stat-mech↗

arXiv2019

Rare-Event Properties of the Nagel-Schreckenberg Model

We have studied the distribution of traffic flow $q$ for the Nagel-Schreckenberg model by computer simulations. We applied a large-deviation approach, which allowed us to obtain the distribution $P(q)$ over more than one hundred decades in probability, down to probabilities like $10^{-140}$. This allowed us to characterize the flow distribution over a large range of the support and identify the characteristics of rare and even very rare traffic situations. We observe a change of the distribution shape when increasing the density of cars from the free flow to the congestion phase. Furthermore, we characterize typical and rare traffic situations by measuring correlations of $q$ to other quantities like density of standing cars or number and size of traffic jams.

physics.data-an↗