In this article, we present an event-driven algorithm that generalizes the recent hard-sphere event-chain Monte Carlo method without introducing discretizations in time or in space. A factorization of the Metropolis filter and the concept of infinitesimal Monte Carlo moves are used to design a rejection-free Markov-chain Monte Carlo algorithm for particle systems with arbitrary pairwise interactions. The algorithm breaks detailed balance, but satisfies maximal global balance and performs better than the classic, local Metropolis algorithm in large systems. The new algorithm generates a continuum of samples of the stationary probability density. This allows us to compute the pressure and stress tensor as a byproduct of the simulation without any additional computations.

We apply the event-chain Monte Carlo algorithm to classical continuum spin models on a lattice and clarify the condition for its validity. In the two-dimensional XY model, it outperforms the local Monte Carlo algorithm by two orders of magnitude, although it remains slower than the Wolff cluster algorithm. In the three-dimensional XY spin glass model at low temperature, the event-chain algorithm is far superior to the other algorithms.

We apply the event-chain Monte Carlo algorithm to the three-dimensional ferromagnetic Heisenberg model. The algorithm is rejection-free and also realizes an irreversible Markov chain that satisfies global balance. The autocorrelation functions of the magnetic susceptibility and the energy indicate a dynamical critical exponent z≈1 at the critical temperature, while that of the magnetization does not measure the performance of the algorithm. We show that the event-chain Monte Carlo algorithm substantially reduces the dynamical critical exponent from the conventional value of z≃2.

PACS 05.10.Ln -Monte Carlo methods PACS 02.70.Tt -Justifications or modifications of Monte Carlo methods PACS 64.70.Md -Transitions in liquid crystalsAbstract -We generalize the rejection-free event-chain Monte Carlo algorithm from manyparticle systems with pairwise interactions to systems with arbitrary three-or many-particle interactions. We introduce generalized lifting probabilities between particles and obtain a general set of equations for lifting probabilities, the solution of which guarantees maximal global balance. We validate the resulting three-particle event-chain Monte Carlo algorithms on three different systems by comparison with conventional local Monte Carlo simulations: i) a test system of three particles with a three-particle interaction that depends on the enclosed triangle area; ii) a hardneedle system in two dimensions, where needle interactions constitute three-particle interactions of the needle end points; iii) a semiflexible polymer chain with a bending energy, which constitutes a three-particle interaction of neighboring chain beads. The examples demonstrate that the generalization to many-particle interactions broadens the applicability of event-chain algorithms considerably.

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Irreversible and rejection-free Monte Carlo methods, recently developed in Physics under the name Event-Chain and known in Statistics as Piecewise Deterministic Monte Carlo (PDMC), have proven to produce clear acceleration over standard Monte Carlo methods, thanks to the reduction of their random-walk behavior. However, while applying such schemes to standard statistical models, one generally needs to introduce an additional randomization for sake of correctness. We propose here a new class of Event-Chain Monte Carlo methods that reduces this extra-randomization to a bare minimum. We compare the efficiency of this new methodology to standard PDMC and Monte Carlo methods. Accelerations up to several magnitudes and reduced dimensional scalings are exhibited.The Hamiltonian dynamics used in Hybrid/Hamiltonian Monte Carlo algorithms [14,27] provides an example of such alternative frameworks [18,34,21]. These methods require however a fine tuning of several parameters, alleviated recently by the development of the statistical software Stan [9]. Also, while aiming at introducing persistency in the successive steps of the Markov chain, these methods still rely on reversible chains with an acceptance-reject scheme.In Physics, recent advances were made in the field of irreversible and rejection-free MCMC simulation methods. These new schemes, referred to as Event-Chain Monte Carlo [2,25], generalize the concept of lifting developed by [12], while drawing on the lines of the recent rejection-free Monte Carlo scheme of [30]. The lifting concept is indeed based upon the transformation of rejections into direction changes. Their successes in different applications [1,22] have motivated the research community to pursue a general framework for implementing irreversible MCMC algorithms beyond Physics. In the statistical community, these methods have been cast into the framework of Piecewise Deterministic Markov Processes (PDMP), see [8,5,4]. In particular, the Bouncy Particle Sampler (BPS) have shown a promising acceleration in comparison to the Hamiltonian MC, as reported in [8]. However, when considering common target distributions encountered in statistics, the PDMC methods can still suffer from some random-walk behavior, partly because they still rely on an additional randomization step to ensure ergodicity.In this paper, we introduce a generalized PDMC framework, the Forward Event-Chain Monte Carlo. This method allows for a fast and global exploration of the sampling space, thanks to a new lifting implementation which leads to a minimal randomization. In this framework, the successive directions are picked according to a full probability distribution conditional on the local potential gradient, contrary to previous PDMC schemes. In addition of being rejection-free, the Forward Event-Chain Monte Carlo algorithm does not require any critical parameter tuning. This paper is organized as follows. Section 2 first recalls and describes the standard MCMC sampling methodologies, as well as classical PDMC sampling schemes. Then, it ...

We propose the clock Monte Carlo technique for sampling each successive chain step in constant time. It is built on a recently proposed factorized transition filter and its core features include its O(1) computational complexity and its generality. We elaborate how it leads to the clock factorized Metropolis (clock FMet) method, and discuss its application in other update schemes. By grouping interaction terms into boxes of tunable sizes, we further formulate a variant of the clock FMet algorithm, with the limiting case of a single box reducing to the standard Metropolis method. A theoretical analysis shows that an overall acceleration of O(N κ ) (0 ≤ κ ≤ 1) can be achieved compared to the Metropolis method, where N is the system size and the κ value depends on the nature of the energy extensivity. As a systematic test, we simulate long-range O(n) spin models in a wide parameter regime: for n = 1, 2, 3, with disordered algebraically decaying or oscillatory Ruderman-Kittel-Kasuya-Yoshida-type interactions and with and without external fields, and in spatial dimensions from d = 1, 2, 3 to mean-field. The O(1) computational complexity is demonstrated, and the expected acceleration is confirmed. Its flexibility and its independence from the interaction range guarantee that the clock method would find decisive applications in systems with many interaction terms.

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