arXiv subjects

Werner Krauth

Publications and source records attributed to Werner Krauth.

At least 19 recordsLinked to original sources

arXiv2026

Markov chains at the onset of non-reversibility

For a one-dimensional path graph and a lifted path graph constructed from a duplication of each of its sites, we study how a reversible Markov chain can be perturbed and gradually driven into non-reversibility. The reversible Markov chain has a transition matrix that is diagonalizable and features real-valued eigenvalues and eigenvectors. The left and right eigenvectors form a biorthogonal system. We discuss in concrete examples how the transition matrix of a non-reversible Markov chain may be diagonalizable or non-diagonalizable, and it may have real eigenvalues and complex-conjugate pairs. For a number of steady states (flat, square-wave, wedge, V-shape), we compute eigenvalue spectra on both graphs and discuss the speedup that can be achieved through lifting. We develop a Green's matrix formalism, which we use to compute Kemeny times and mean first-passage times, and which provides valuable information and allows us to interpret the results for the characteristic times.

cond-mat.stat-mech↗

arXiv2026

Path convergence in diffusion models

We discuss diffusion-model paths interpolating between a target distribution known only through p patterns and a reference distribution that can be sampled. These interpolating paths can be constructed symmetrically or else in forward direction (often referred to as a "noising") from the target patterns to the reference distribution or in backward direction (as a "denoising") from the reference distribution to the patterns. For backward paths with identical diffusion noise, we consider the path convergence in number of patterns p towards the path for infinitely many patterns. In a one-dimensional test case, we show that this convergence is on a scale 1/sqrt(p), but with infinite mean square deviation. We demonstrate that the path convergence allows for extrapolation towards the p=infinity path which samples the target distribution. We provide a proof-of-concept extrapolation algorithm and propose the convergence and extrapolation of paths as a possible strategy for density estimation and generalization. We illustrate all our algorithms through pseudo-codes and provide Python implementations.

cond-mat.stat-mech↗

arXiv2026

Mixed-dimensional quantum Monte Carlo studies of M-point moir\'e materials

A new moir\'e-material platform has recently been proposed based on twisting two-dimensional triangular-lattice monolayers whose low-energy states lie at the three M points of the Brillouin zone. Continuum models derived from extensive ab initio simulations suggest that electrons in the conduction bands of one such M-point moir\'e material, twisted AA-stacked SnSe$_2$, realize a three-orbital Hubbard model with orbitally-selective, quasi-one-dimensional (quasi-1D) hopping, protected by a projective mirror symmetry. Here, we show that the resulting "mixed-dimensional" limit -- in which the hopping is exactly quasi-1D in each valley, while the valleys are coupled by interactions into a fully two-dimensional network -- can be sampled with Stochastic Series Expansion (SSE) quantum Monte Carlo (QMC) without a sign problem at any filling. We develop an efficient new SSE QMC algorithm that combines custom global updates with parallel tempering to overcome the equilibration challenges posed by the mixed-dimensional setting. We then use this algorithm to explore the phase diagram of M-point twisted AA-stacked SnSe$_2$. Over extended and realistic ranges of twist angles and interaction strengths, we find that at integer fillings the system supports correlated insulators whose nature and strength depend strongly on angle. At certain commensurate fractional fillings, we further find evidence for Wigner-Mott insulators. We analytically account for the main features observed numerically using a strong-coupling description. Finally, we discuss perturbations away from the mixed-dimensional limit and the possibility of applying our method to other realizations of mixed-dimensional Hubbard models.

cond-mat.str-el↗

arXiv2026

Lifting the fog - a case for non-reversible "lifted" Markov chains

Phase transitions appear all over science, and are familiar from everyday life, as water boiling, sugar melting into caramel or as nematic molecules turning smectic in liquid-crystal displays. The dynamics of phase transitions can be extremely slow, as for example when fog in winter does not lift, that is when the coarsening takes much time from many tiny water droplets to fewer but larger rain drops that feel the pull of gravity. The dynamics of phase transitions is relevant also for the performance of computer algorithms. In the ubiquitous Metropolis Monte Carlo algorithm, the mixing dynamics towards equilibrium leads towards the solution of a sampling problem. It is governed by the same reversibility and detailed-balance principles as the overdamped physical dynamics of fog. For the phase-separated Lennard-Jones system, we describe here how the coarsening dynamics of non-reversible "lifted" variants of the Metropolis algorithm proceeds on much faster time scales, with the microscopic non-reversibility translating into large-scale relative motion of droplets that is impossible under the Ostwald-ripening condition of reversibility. A density-displacement coupling moves droplets relative to each other through a lensing effect. Efficient implementations of the long-range Metropolis algorithm and its non-reversible lifting (event-chain Monte Carlo) allow us to show that, in consequence, the coarsening growth exponent is larger under lifting. For large system sizes, the computing problem is thus solved infinitely faster than before, with the outcome strictly unchanged with respect to the Metropolis algorithm. We also discuss the larger setting of our findings, namely that "lifted" non-reversible algorithms can be set up for generic reversible sampling methods, with applications going much beyond our example of lifting fog.

cond-mat.stat-mech↗

arXiv2025

Thermodynamic integration, fermion sign problem, and real-space renormalization

We reconsider real-space renormalization for the two-dimensional Ising model, following the path traced out by Wilson in Sect. VI of his 1975 Reviews of Modern Physics. In that reference, Wilson considerably extended the Kadanoff decimation procedure towards a possibly rigorous construction of a real-space scale-invariant hamiltonian. Wilson's construction has, to the best of our knowledge, never been fully understood and thus neither reproduced nor generalized. In the present work, we use Monte Carlo sampling in combination with thermodynamic integration in order to retrace Wilson's computation for a real-space renormalization with a number of terms in the hamiltonian. We elaborate on the connection of real-space renormalization with the fermion sign problem and discuss to which extent our Monte Carlo procedure actually implements Wilson's program from half a century ago.

cond-mat.stat-mech↗

arXiv2025

Velocity trapping in the lifted TASEP and the true self-avoiding random walk

We discuss non-reversible Markov-chain Monte Carlo algorithms that, for particle systems, rigorously sample the positional Boltzmann distribution and that have faster than physical dynamics. These algorithms all feature a non-thermal velocity distribution. They are exemplified by the lifted TASEP (totally asymmetric simple exclusion process), a one-dimensional lattice reduction of event-chain Monte Carlo. We analyze its dynamics in terms of a velocity trapping that arises from correlations between the local density and the particle velocities. This allows us to formulate a conjecture for its out-of-equilibrium mixing time scale, and to rationalize its equilibrium superdiffusive time scale. Both scales are faster than for the (unlifted) TASEP. They are further justified by our analysis of the lifted TASEP in terms of many-particle realizations of true self-avoiding random walks. We discuss velocity trapping beyond the case of one-dimensional lattice models and in more than one physical dimensions. Possible applications beyond physics are pointed out.

cond-mat.stat-mech↗

arXiv2025

Lifted TASEP: long-time dynamics,generalizations, and continuum limit

We investigate the lifted TASEP and its generalization, the GL-TASEP. We analyze the spectral properties of the transition matrix of the lifted TASEP using its Bethe ansatz solution, and use them to determine the scaling of the relaxation time (the inverse spectral gap) with particle number. The observed scaling with particle number was previously found to disagree with Monte Carlo simulations of the equilibrium autocorrelation times of the structure factor and of other large-scale density correlators for a particular value of the pullback $\alpha_{\rm crit}$. We explain this discrepancy. We then construct the continuum limit of the lifted TASEP, which remains integrable, and connect it to the event-chain Monte Carlo algorithm. The critical pullback $\alpha_{\rm crit}$ then equals the system pressure. We generalize the lifted TASEP to a large class of nearest-neighbor interactions, which lead to stationary states characterized by non-trivial Boltzmann distributions. By tuning the pullback parameter in the GL-TASEP to a particular value we can again achieve a polynomial speedup in the time required to converge to the steady state. We comment on the possible integrability of the GL-TASEP.

cond-mat.stat-mech↗

arXiv2024

Hamiltonian Monte Carlo vs. event-chain Monte Carlo: an appraisal of sampling strategies beyond the diffusive regime

We discuss Hamiltonian Monte Carlo (HMC) and event-chain Monte Carlo (ECMC) for the one-dimensional chain of particles with harmonic interactions and benchmark them against local reversible Metropolis algorithms. While HMC achieves considerable speedup with respect to local reversible Monte Carlo algorithms, its autocorrelation functions of global observables such as the structure factor have slower scaling with system size than for ECMC, a lifted non-reversible Markov chain. This can be traced to the dependence of ECMC on a parameter of the harmonic energy, the equilibrium distance, which drops out when energy differences or gradients are evaluated. We review the recent literature and provide pseudocodes and Python programs. We finally discuss related models and generalizations beyond one-dimensional particle systems.

cond-mat.stat-mech↗

arXiv2024

Damage spreading and coupling in spin glasses and hard spheres

We study the connection between damage spreading, a phenomenon long discussed in the physics literature, and the coupling of Markov chains, a technique used to bound the mixing time. We discuss in parallel the Edwards-Anderson spin glass model and the hard-disk system, focusing on how coupling provides insights into the region of fast coupling within the paramagnetic and liquid phases. We also work out the connection between path coupling and damage spreading. Numerically, the scaling analysis of the mean coupling time determines a critical point between fast and slow couplings. The exact relationship between fast coupling and disordered phases has not been established rigorously, but we suggest that it will ultimately enhance our understanding of phase behavior in disordered systems.

cond-mat.dis-nn↗

arXiv2024

Markov-chain sampling for long-range systems without evaluating the energy

In past decades, enormous effort has been expended to develop algorithms and even to construct special-purpose computers in order to efficiently evaluate total energies and forces for long-range-interacting particle systems, with the particle-mesh Ewald and the fast multipole methods as well as the 'Anton' series of supercomputers serving as examples for biomolecular simulations. Cutoffs in the range of the interaction have also been used for large systems. All these methods require extrapolations. Within Markov-chain Monte Carlo, in thermal equilibrium, the Boltzmann distribution can however be sampled natively without evaluating the total interaction potential. Using as an example the Lennard-Jones interaction, we review past attempts in this direction, and then discuss in detail the class of cell-veto algorithms which make possible fast, native sampling of the Boltzmann distribution without any approximation, extrapolation, or cutoff even for the slowly decaying Coulomb interaction. The computing effort per move remains constant with increasing system size, as we show explicitly. We provide worked-out illustrations and pseudocode representations of the discussed algorithms. Python implementations are made available in an associated open-source software repository.

cond-mat.stat-mech↗

arXiv2023

Diameters of symmetric and lifted simple exclusion models

We determine diameters of Markov chains describing one-dimensional N -particle models with an exclusion interaction, namely the Ssep (symmetric simple exclusion process) and one of its non-reversible liftings, the lifted Tasep (totally asymmetric simple exclusion process). The diameters provide lower bounds for the mixing times, and we discuss the implications of our findings for the analysis of these models.

cond-mat.stat-mech↗

arXiv2023

Concepts in Monte Carlo sampling

We discuss modern ideas in Monte Carlo algorithms in the simplified setting of the one-dimensional anharmonic oscillator. After reviewing the connection between molecular dynamics and Monte Carlo, we introduce to the Metropolis and the factorized Metropolis algorithms and to lifted non-reversible Markov chains. We furthermore illustrate the concept of thinning, where moves are accepted by simple bounding potentials rather than, in our case, the harmonic and quartic constituents of the anharmonic oscillator. We point out the multiple connections of our example algorithms with real-world sampling problems. The paper is fully self-contained and Python implementations are provided.

cond-mat.stat-mech↗

arXiv2023

Molecular simulation from modern statistics: Continuous-time, continuous-space, exact

In a world made of atoms, the computer simulation of molecular systems, such as proteins in water, plays an enormous role in science. Software packages that perform these computations have been developed for decades. In molecular simulation, Newton's equations of motion are discretized and long-range potentials are treated through cutoffs or spacial discretization, which all introduce approximations and artifacts that must be controlled algorithmically. Here, we introduce a paradigm for molecular simulation that is based on modern concepts in statistics and is rigorously free of discretizations, approximations, and cutoffs. Our demonstration software reaches a break-even point with traditional molecular simulation at high precision. We stress the promise of our paradigm as a gold standard for critical applications and as a future competitive approach to molecular simulation.

physics.chem-ph↗

arXiv2023

Lifted TASEP: a Bethe ansatz integrable paradigm for non-reversible Markov chains

Markov-chain Monte Carlo (MCMC), the field of stochastic algorithms built on the concept of sampling, has countless applications in science and technology. The overwhelming majority of MCMC algorithms are time-reversible and satisfy the detailed-balance condition, just like physical systems in thermal equilibrium. The underlying Markov chains typically display diffusive dynamics, which leads to a slow exploration of sample space. Significant speed-ups can be achieved by non-reversible MCMC algorithms exhibiting non-equilibrium dynamics, whose steady states exactly reproduce the target equilibrium states of reversible Markov chains. Such algorithms have had successes in applications but are generally difficult to analyze, resulting in a scarcity of exact results. Here, we introduce the "lifted" TASEP (totally asymmetric simple exclusion process) as a paradigm for lifted non-reversible Markov chains. Our model can be viewed as a second-generation lifting of the reversible Metropolis algorithm on a one-dimensional lattice and is exactly solvable by an unusual kind of coordinate Bethe ansatz. We establish the integrability of the model and present strong evidence that the lifting leads to faster relaxation than in the KPZ universality class.

cond-mat.stat-mech↗

arXiv2023

The Liquid--Hexatic Transition for Soft Disks

We study the liquid--hexatic transition of soft disks with massively parallel simulations and determine the equation of state as a function of system size. For systems with interactions decaying as the inverse $m$th power of the separation, the liquid--hexatic phase transition is continuous for $m = 12$ and $m=8$, while it is of first order for $m = 24$. The critical power $m$ for the transition between continuous and first-order behavior is larger than previously reported. The continuous transition for $ m=12 $ implies that the two-dimensional Lennard-Jones model has a continuous liquid--hexatic transition at high temperatures. We also study the Weeks--Chandler--Andersen model and find a continuous transition at high temperatures, that is consistent with the soft-disk case for $m=12$. Pressure data as well as our implementation are available from an open-source repository.

cond-mat.soft↗

arXiv2022

Direction-sweep Markov chains

We discuss a non-reversible, lifted Markov-chain Monte Carlo (MCMC) algorithm for particle systems in which the direction of proposed displacements is changed deterministically. This algorithm sweeps through directions analogously to the popular MCMC sweep methods for particle or spin indices. Direction-sweep MCMC can be applied to a wide range of original reversible or non-reversible Markov chains, such as the Metropolis algorithm or the event-chain Monte Carlo algorithm. For a single two-dimensional dipole, we consider direction-sweep MCMC in the limit where restricted equilibrium is reached among the accessible configurations before changing the direction. We show rigorously that direction-sweep MCMC leaves the stationary probability distribution unchanged, and that it profoundly modifies the Markov-chain trajectory. Long excursions, with persistent rotation in one direction, alternate with long sequences of rapid zigzags resulting in persistent rotation in the opposite direction in the limit of small direction increments. The mapping to a Langevin equation then yields the exact scaling of excursions while the zigzags are described through a non-linear differential equation that is solved exactly. We show that the direction-sweep algorithm can have shorter mixing times than the algorithms with random updates of directions. We point out possible applications of direction-sweep MCMC in polymer physics and in molecular simulation.

cond-mat.stat-mech↗

arXiv2022

Hard-disk computer simulations -- a historic perspective

We discuss historic pressure computations for the hard-disk model performed since 1953, and compare them to results that we obtain with a powerful event-chain Monte Carlo and a massively parallel Metropolis algorithm. Like other simple models in the sciences, such as the Drosophila model of biology, the hard-disk model has needed monumental effort to be understood. In particular, we argue that the difficulty of estimating the pressure has not been fully realized in the decades-long controversy over the hard-disk phase-transition scenario. We present the physics of the hard-disk model, the definition of the pressure and its unbiased estimators, several of which are new. We further treat different sampling algorithms and crucial criteria for bounding mixing times in the absence of analytical predictions. Our definite results for the pressure, for up to one million disks, may serve as benchmarks for future sampling algorithms. A synopsis of hard-disk pressure data as well as different versions of the sampling algorithms and pressure estimators are made available in an open-source repository.

cond-mat.stat-mech↗

arXiv2021

Hard-disk dipoles and non-reversible Markov chains

We benchmark event-chain Monte Carlo (ECMC) algorithms for tethered hard-disk dipoles in two dimensions in view of application of ECMC to water models in molecular simulation. We characterize the rotation dynamics of dipoles through the integrated autocorrelation times of the polarization. The non-reversible straight, reflective, forward, and Newtonian ECMC algorithms are all event-driven, and they differ only in their update rules at event times. They realize considerable speedups with respect to the local reversible Metropolis algorithm. We also find significant speed differences among the ECMC variants. Newtonian ECMC appears particularly well-suited for overcoming the dynamical arrest that has plagued straight ECMC for three-dimensional dipolar models with Coulomb interactions.

cond-mat.stat-mech↗