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arXiv:2608.31114

Overcoming critical slowing down in frustrated spin systems by learned multiscale sampling

Gabriele Bandini, Giulio Biroli, Patrick Charbonneau, Andrea Gambassi

Published Aug 31, 2026• 0 citations
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Abstract

Cluster algorithms, such as the Swendsen--Wang and Wolff methods, are among the most successful MCMC methods for mitigating critical slowing down in statistical systems. These constructive cluster algorithms, however, fail in the presence of even extremely weak frustration. Here, we sidestep this fundamental limitation by learning rather than constructing the relevant clusters. Specifically, we use the wavelet conditional renormalization group (WCRG) sampling method to learn the probability distribution of collective fluctuations of a frustrated two-dimensional soft-spin model. Configurations are then generated recursively from coarse to fine scales by sampling conditional wavelet distributions. The WCRG method reproduces the main statistical properties of the system across different phases, including the local-field distribution and the structure factor. At an Ising-like critical point, the conditional dynamics remains decorrelated within $\mathcal{O}(1)$ sweeps at each scale, yielding an overall sampling complexity of $\mathcal{O}(\log_2 L)$, thus making WCRG much more efficient than standard local MCMC methods. These results show that learned multiscale sampling can overcome critical slowing down in frustrated systems for which conventional cluster algorithms fail. By assessing the sampling accuracy of different observables, we also clarify the main tradeoff of the WCRG method: the accuracy of the fast sampling scheme depends on the expressiveness of the energy-based model used to estimate the wavelet conditional distributions.

DOI
10.48550/arxiv.2608.31114

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Cluster algorithms, such as the Swendsen--Wang and Wolff methods, are among the most successful MCMC methods for mitigating critical slowing down in statistical systems. These constructive cluster algorithms, however, fail in the presence of even extremely weak frustration. Here, we sidestep this fundamental limitation by learning rather than constructing the relevant clusters. Specifically, we use the wavelet conditional renormalization group (WCRG) sampling method to learn the probability distribution of collective fluctuations of a frustrated two-dimensional soft-spin model. Configurations are then generated recursively from coarse to fine scales by sampling conditional wavelet distributions. The WCRG method reproduces the main statistical properties of the system across different phases, including the local-field distribution and the structure factor. At an Ising-like critical point, the conditional dynamics remains decorrelated within O(1)\mathcal{O}(1)O(1) sweeps at each scale, yielding an overall sampling complexity of O(log⁡2L)\mathcal{O}(\log_2 L)O(log2​L), thus making WCRG much more efficient than standard local MCMC methods. These results show that learned multiscale sampling can overcome critical slowing down in frustrated systems for which conventional cluster algorithms fail. By assessing the sampling accuracy of different observables, we also clarify the main tradeoff of the WCRG method: the accuracy of the fast sampling scheme depends on the expressiveness of the energy-based model used to estimate the wavelet conditional distributions.
We develop a multiscale approach to estimate high-dimensional probability distributions.Our approach applies to cases in which the energy function (or Hamiltonian) is not known from the start.Using data acquired from experiments or simulations we can estimate the energy function.We obtain a representation of the approximate probability distribution based on a multiscale cascade of conditional probabilities.This representation allows for fast sampling of many-body systems in various domains, from statistical physics to cosmology.Our method-the wavelet-conditional renormalization group (WCRG)-proceeds scale by scale, estimating models for the conditional probabilities of "fast degrees of freedom" conditioned by coarse-grained fields.These probability distributions are modeled by energy functions associated with scale interactions, and are represented in an orthogonal wavelet basis.The WCRG decomposes the microscopic energy function as a sum of interaction energies at all scales and can efficiently generate new samples by going from coarse to fine scales.Near phase transitions, this representation of the approximate probability distribution completely avoids the "critical slowing-down" of direct estimation and sampling algorithms.This is explained theoretically by combining results from RG and wavelet theories, and verified numerically for the Gaussian and φ 4 -field theories.We show that multiscale WCRG energy-based models are more general than local potential models and can capture the physics of complex many-body interacting systems at all length scales.This is demonstrated for weak-gravitational-lensing fields reflecting darkmatter distributions in cosmology, which include long-range interactions with long-tail probability distributions.The WCRG has a large number of potential applications in nonequilibrium systems, where the underlying distribution is not known a priori.Finally, we discuss the connection between the WCRG and deep network architectures.
In the theoretical study of phase transitions and critical phenomena, renormalization groups (RG) provide a course-grain representation of a many particle system and aid in the determination of the critical exponents associated with phase transitions. Monte Carlo and Molecular Dynamic simulations are often the simulation tools used for verification. However, the rich structure and multiscale properties of wavelets have found limited use in these applications. To promote exploration here, we apply and investigate wavelets to the solution of the simplest of molecular models, the Ising model and evaluate the most fundamental of objects in statistical mechanics, the partition function, now based on a wavelet-modifed Hamiltonian. As in we iteratively compute the waveletbased coarse grain 2-D finite-size Ising model. Thermodynamic properties are calculated to predict, given the finite size effect, potential for phase transition. Exact calculations are confirmed using Monte Carlo simulations. Based on this investigation and corresponding insight gained, we propose a wavelet-based investigation in three open areas: Wavelet Bases, Critical Exponents and Phase Transitions, Renormalization Groups and Continuous Wavelets.
We simulate single and multiple Ising models coupled to 2–d gravity using both the Swendsen-Wang and Wolff algorithms to update the spins. We study the integrated autocorrelation time and find that there is considerable critical slowing down, particularly in the magnetization. We argue that this is primarily due to the local nature of the dynamical triangulation algorithm and to the generation of a distribution of baby universes which inhibits cluster growth. SU–HEP–93–4241–560 SCCS 568 hep-lat/9311036 Considerable work has been devoted to the study of the performance of cluster algorithms in reducing the critical slowing down of many statistical models. The Swendsen-Wang [1] and the Wolff [2] algorithms have proven very effective in beating the critical slowing down (CSD) exhibited by these models when simulated with a standard Metropolis [3] update. Reviews on cluster algorithms and CSD may be found on refs. [4]. In recent years, the numerical study of Potts models coupled to 2–d gravity has received much attention [5, 6, 7], aided as well by increased analytical understanding of these models [8]. Cluster algorithms have proved useful in saving computational effort on the update of the Potts variables. The present study is motivated by the fact that there is little understanding of the actual extent of the improvement achieved in these simulations. It is worthwhile, therefore, to measure CSD in the case of simple Ising spins coupled to a dynamical lattice in order to quantify the performance of cluster algorithms. We find that there is considerable CSD, especially in the magnetization, and we relate this to the dynamics of cluster formation on a random lattice.
In the present work, we present an extensive Monte Carlo simulation study on the dynamical properties of the two-dimensional random-bond Ising model. The correlation time τ of the Swendsen-Wang and Wolff cluster algorithms is calculated at the critical point. The dynamic critical exponent z of both algorithms is also measured by using the numerical data for several lattice sizes up to L=512. It is found for both algorithms that the autocorrelation time decreases considerably and the critical slowing-down effect reduces upon the introduction of bond disorder. Additionally, simulations with the Metropolis algorithm are performed, and the critical slowing-down effect is observed to be more pronounced in the presence of disorder, confirming the previous findings in the literature. Moreover, the existence of the non-self-averaging property of the model is demonstrated by calculating the scaled form of the standard deviation of autocorrelation times. Finally, the critical exponent ratio of the magnetic susceptibility is estimated by using the average cluster size of the Wolff algorithm.
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Disordered lattice spin systems are crucial in both theoretical and applied physics. However, understanding their properties poses significant challenges to Monte Carlo simulations. In this work, we investigate the two-dimensional random-bond Ising model using the recently proposed tensor network Monte Carlo (TNMC) method. This method generates biased samples from conditional probabilities computed via tensor network contractions and corrects the bias using the Metropolis scheme. Consequently, the proposals provided by tensor networks function as block updates for Monte Carlo simulations. Through extensive numerical experiments, we demonstrate that TNMC simulations can be performed on lattices as large as $1024\ifmmode\times\else\texttimes\fi{}1024$ spins with moderate computational resources, a substantial increase from the previous maximum size of $64\ifmmode\times\else\texttimes\fi{}64$ in Markov-chain Monte Carlo. Notably, we observe an almost complete absence of critical slowing down, enabling the efficient collection of unbiased samples and averaging over a large number of random realizations of bond disorders. We successfully pinpoint the multicritical point along the Nishimori line with significant precision and accurately determined the bulk and surface critical exponents. Our findings suggest that TNMC is a highly efficient algorithm for exploring disordered and frustrated systems in two dimensions.