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版本更新 · v4arXiv:2609.02948

FrOGS: Discrete Neural Sampler for Independent Alloy Configurations Across Chemical Conditions

Kyucheol Min, Elyssa Hofgard, Tess Smidt

原文摘要

Predicting the thermodynamic properties of an alloy requires sampling its configurations across many chemical conditions and recovering free energies on a common absolute scale. Markov chain Monte Carlo (MCMC) is the standard tool, but it requires separate simulations at different conditions, and auxiliary free-energy methods such as thermodynamic integration are used to place results on a common absolute scale. Modern discrete neural samplers typically use reverse KL divergence as the objective and can be mode-seeking or biased. We present Free energy Offering Generative Sampler (FrOGS), a hybrid discrete neural sampler that couples an autoregressive model to a continuous-time Markov chain (CTMC) to be trained jointly under a single shared loss. FrOGS draws i.i.d. configurations, returns an unbiased estimate of the partition function, and gives consistent estimates of thermodynamic observables. We train a single model across a wide range of chemical conditions to produce estimates on a common absolute free-energy scale. FrOGS matches exact finite-size results on the 2D Ising model and reference phase diagrams for AgPd and CuAu, without mode collapse. We additionally compare to SEGAL, a published autoregressive baseline, and find that only FrOGS recovers the stability range of the CuAu$_3$ phase.

cond-mat.mtrl-sci · cond-mat.stat-mech · cs.LG

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2606.28105

Scaling limit of the Random Language Model

Eric De Giuli

原文摘要

We develop a quantitative theory of the Random Language Model (RLM), an ensemble of stochastic context-free grammars, in a scaling limit where the number of hidden symbols $N \to \infty$ while the grammar temperature $\tildeε_d \to 0$ at fixed $x = \tildeε_d \log N$. In this limit, the model admits a controlled description based on a large-deviation principle over rule-usage patterns. A semi-annealed approximation maps the problem to a class of Random Energy Models with nontrivial combinatorics. We show that the RLM exhibits a freezing transition at a critical value $x_c=1/8$, below which rule usage condenses on a subset of preferred rules and language statistics acquire a nontrivial dependence on corpus length. A second characteristic scale at $x=1/2$, associated with the breakdown of Gaussian fluctuations, separates transition and babbling regimes. Across regimes, we derive explicit scaling laws for the number of distinct rules, entropy, and energy, identifying distinct frozen and critical regimes controlled by the interplay of grammar size, corpus length, and temperature. The theory resolves previous ambiguities regarding the existence of a thermodynamic transition and explains the slow approach to the large-$N$ limit as a consequence of the dependence on $\log N$. It further provides a unified framework in which universal statistical properties of language emerge from typical realizations of generative grammars, with implications for both natural language statistics and the behavior of large language models. Heaps' law and Zipf's law are both derived, and the dependence of predictive entropy on context length is rationalized. Comparison with empirical measurements of Heaps' and Zipf's laws yields parameter estimates for real languages.

cond-mat.dis-nn · cond-mat.stat-mech · cs.CL

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2610.16733

TheoPhysEval: Assessing AI agents at autonomous theoretical physics problem solving

A. Ramesh Chandra, Aaghaz Mahajan, Ainesh Sanyal, Mianqi Wang, Shubhadeep Chakraborty, Sudip Chakraborty, Shailesh Chandrasekharan, Rama Govindarajan 等 35 位作者

原文摘要

We present a suite of 50 unpublished research-level theoretical physics problems with verifiable answers. Without AI assistance, a typical problem in our collection could be solved by a world expert in about a day or less but would take considerably longer for experts in adjacent specialties. We test the ability of off-the-shelf AI agents to tackle these problems autonomously, without human hints. When given access to tools available in a realistic setting, including the ability to search the literature and write scripts for numerical and symbolic work, the best-performing agent, GPT-6-Astra harnessed in Codex, obtains the final answers with a near-perfect accuracy of 98% at an average cost under USD 3 per problem. On average, agents show only small changes in accuracy when denied web search and access to background research papers, suggesting significant memorization of the literature. These results suggest that state-of-the-art agents can accurately perform concrete calculations required across theoretical physics research. We comment on the broader implications of these findings and discuss possible explanations for the current lack of agent-led breakthroughs in theoretical physics despite their outstanding problem-solving abilities.

hep-th · cond-mat.stat-mech · cond-mat.str-el · cs.AI · gr-qc

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.16336

Visibility graphs of irregularly sampled two-dimensional fields via Voronoi tessellations

Francisco J. Muñoz, Juan Carlos Nuño

原文摘要

Visibility graphs map a time series into a network by linking two data whenever the straight line between them is not obstructed by intermediate data. Their extension to two-dimensional fields has so far been restricted to regular lattices, where visibility is checked along a finite number of lattice directions. We propose a definition for scalar data sampled at arbitrary, irregularly scattered points of the plane. Each point is assigned its Voronoi cell, so that the data become a landscape of prisms, and two points are linked when the line of sight between them clears the cells crossed by the connecting segment. This yields horizontal ($\HVGV$) and natural ($\NVGV$) Voronoi visibility graphs that reduce exactly to the classical horizontal and natural visibility graphs when the points are collinear, contain the Gabriel graph of the point set, and can be built by an exact walk through the Delaunay triangulation. For uncorrelated heights, the horizontal graph depends only on the ranks of the data, so its statistics are independent of the marginal distribution. The probability that two points see each other is $2/[(M+1)(M+2)]$, where $M$ is the number of Voronoi cells crossed, which reduces the problem to the geometry of the tessellation. For Poisson points this gives a degree distribution with a scale-free tail $P(k)\simeq(π^3/4)\,k^{-2}$ and a mean degree that grows as $(π^3/8)\ln N$, in contrast with the finite value $\langle k\rangle=4$ of one-dimensional series. Two dimensions are therefore the marginal case in which isotropic horizontal visibility ceases to be local. Numerical simulations with up to $N=1.28\times10^5$ points support these predictions.

math.DS · cond-mat.stat-mech · cs.CG · physics.data-an

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2611.06059

Exact Transport-Information Correspondence for Distribution-Adaptive Discretization

Koretaka Yuge

原文摘要

Discretization of continuous probability distributions is an operation on the sample space, and the structure induced by this operation cannot generally be determined solely from information geometry. In our recent work on unavoidable canonical nonlinearity (UCN), we established a non-arbitrary connection between the transport cost associated with discretization and the information-geometric indistinguishability of continuous probability distributions. Through this transport-information correspondence, UCN quantifies the information-theoretic cost associated with discretization in terms of the expected Kullback-Leibler divergence between nearby continuous distributions characterized by the discretization geometry. In the original UCN construction, discretization is implemented by transporting the probability mass within each cell to its geometric centroid. The resulting transport-information correspondence, however, is derived under specific conditions involving the cell geometry, spatial coverage, and the vanishing-cell-scale limit. In this work, we generalize the local-map discretization framework by allowing an arbitrary representative point for each cell and establish an exact decomposition of the associated transport cost into contributions from the conditional covariance of the probability distribution and the displacement of the representative point from its probability-weighted centroid. Consequently, the resulting transport cost is exactly represented by the expectation of the associated Fisher-induced quadratic form, without requiring uniform cell geometry, complete coverage of the sample space, or an infinitesimal cell-scale limit. The proposed construction thus establishes an exact transport-information correspondence for finite-scale discretization using arbitrary collections of non-overlapping bounded sets.

cs.IT · cond-mat.stat-mech

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.14039

Thermodynamics of words

Klaus M. Frahm, Dima L. Shepelyansky

原文摘要

We introduce the word thermalization hypothesis according to which in a literary work there are effective interactions between words that drive them to thermalization where their word frequency distribution is the same as for the phenomenon of Rayleigh-Jeans (RJ) thermalization and condensation. In this concept, the different words play a role of classical oscillators (particles) and their word frequencies, or related tokens, are analogous to certain filling factors over energy state levels. The thermalization process is characterized by the conservation of total system energy, or total number of words/tokens, and total probability norm, or total number of different words. We apply this approach to analyze 16 literary works, including Bible, complete works of Shakespeare, and data from American National Corpus. We show that for these cases the effective system energy is very low and thus there is the RJ condensation phenomenon with a big fraction of total probability concentrated at the lowest energy states corresponding to a big fraction of rare words appearing only once or twice in the whole text. Such a RJ condensation is observed for light propagation in multimode optical fibers and other physical systems. We relate the thermalization of words to the recently proposed wealth thermalization for households in world countries. The economy concepts of Lorenz and Pareto curves, and also Zipf curves, are applied to the word frequency distributions of the selected 16 cases and it is shown that the thermalization theory gives a strikingly good description of real data.

cond-mat.stat-mech · cs.CL · physics.soc-ph

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.13904

Thermodynamic Formalism for Non-Invariant Probabilities: Entropy, Divergence, Channels

Artur O. Lopes, Elismar R. Oliveira

原文摘要

Let $\mathcal{M}$ and $\mathcal{M}_σ$ denote, respectively, the sets of Borel and $σ$-invariant probabilities on the uncountable set $Ω=\{1,2,...,d\}^\mathbb{N}$. Given a Hölder Jacobian $J$, the dual Ruelle operator $μ\in \mathcal{M} \mapsto \mathcal{L}_{J}^* (μ)$ will play the role of a dynamical channel; generalizes the action of a stochastic matrix on probability vectors. Since $\mathcal{L}_{J}^*$ generally breaks $σ$-invariance, we introduce the \emph{ni-entropy} $h_{ni}(μ)$ for non-invariant measures, employing the concept of the expected value function on cylinders of size one. $h_{ni}(μ)$ differs from Shannon entropy but extends the Kolmogorov-Sinai entropy from $\mathcal{M}_σ$ to $\mathcal{M}$; $h_{ni}$ is concave and u.s.c. This leads us to the concept of ni-pressure $\mathcal{P}(A)$ of a potential $A:Ω\to \mathbb{R}$. $μ\in \mathcal{M}$ is not a density with respect to a dominating measure. We define an {\it ni-Kullback-Leibler divergence} $D_{ni}(ρ|| ν)$ not assuming absolute continuity. We investigate concepts of Information Theory such as Minimal Output Entropy, Donsker-Varadhan variational principle, a variational formula for $h_{ni}$, bounds for divergence, information projections, and their connections to the ni-pressure $\mathcal{P}$.

math.DS · cond-mat.stat-mech · cs.IT · math.PR

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.13174

How wide must an eavesdropper be? Leakage of a toric-code logical bit to a strip and the Nishimori point

Konstantinos Karagiannis

原文摘要

Fault-tolerant quantum computers hand pieces of each logical qubit to separate decoder modules and shared readout lines. How much does a piece reveal? We prove that for any region and any Pauli noise, including correlated noise, an observer's optimal advantage in learning a logical bit is exactly 1 - 2 P_fail, where P_fail is the failure probability of maximum-likelihood decoding confined to that region. Two lessons follow. Geometry, not noise, provides the protection: a region that misses a single column of the code learns exactly nothing at any noise level, while a region that spans the code is protected by noise only at impractical distances (d > 350 for one row at a 10^-3 flip rate). And width is a poor guide to risk: under i.i.d. noise a two-row strip learns exactly as little as one row, yet below threshold an eavesdropper needs a width growing only as ln d. Through the identity, the strip's operational threshold is the Nishimori point of the random-bond Ising model, and a Walsh-Hadamard transfer matrix up to width 16 locates it to within a few thousandths. On two IBM Heron processors (d = 3, 5), an emulated strip adversary shows the predicted jump at width three, and non-uniform qubit errors let a trained attacker profit from the second row that uniform noise makes useless. We turn these results into design rules for partitioned decoding and multiplexed readout. Code and data are included.

quant-ph · cond-mat.stat-mech · cs.CR

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2610.39568

Diffusion models for the $O(3)$ nonlinear sigma model

Daniel Schubring

原文摘要

A score-based diffusion model is trained on the two-dimensional classical Heisenberg model, the lattice representation of the $O(3)$ nonlinear sigma model. Neural networks are trained on three different coarse-grainings of the same training set to implement a multiscale sampling scheme that modestly improves both the accuracy and computational cost of direct sampling on the finest lattice. Stein (or Schwinger-Dyson) diagnostics for the two-point correlation function are introduced to assess the quality of the learned score without relying on reverse diffusion sampling. These diagnostics are also incorporated into an additional training loss. We discuss several technical considerations relevant to implementing diffusion models on $S^2$. These include a consideration of lifting the network input to $S^3$, as in the $CP^1$ representation, and developing a boundary-layer approximation to the short-time spherical heat kernel that extends its usefulness to larger diffusion times.

cond-mat.stat-mech · cs.LG · hep-lat

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.12485

Resetting-based designated information distribution

Xiaojuan Ma, Yan Wang, Indranil Gupta

原文摘要

For a network of $n$ nodes under random gossip dynamics, we show that by introducing the additional resetting mechanism (with rate $β$) to a designated subset of $r$ nodes, the expected sum of all node values is asymptotically equal to $\tfrac{n}{r}$ times the sum of the initial values of the $r$ designated nodes, in the small-$β$ and long-time limit. We also analyze the convergence speed to this asymptotic state and provide an approximate high-probability confidence interval for the expected sum at finite time $t$ and finite $β$. Operationally, we propose a protocol that distributes the initial node values over the designated subset evenly (up to a prescribed precision), by appropriately turning on and off the resetting dynamics. Our work not only offers insights into classical problems such as distributed computing and network control, but also serves as a potential foundation for designing quantum algorithms that prepare desired states in quantum networks using only local operations.

cs.DC · cond-mat.stat-mech · cs.SI

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arXiv ↗PDF ↗
版本更新 · v4arXiv:2607.17469

Local Search with Correlated Randomness

Yunbei Xu

原文摘要

The running-time distribution of a randomized algorithm is computed under independent randomness, but independence is rarely what a source guarantees. We study sources $ν$ with $ν[w]\le DP[w]^s$ for every finite prefix $w$ of the tape, where $P$ is an independent reference law, $0<s\le1$, and $D\ge1$; individual choices may be predictable, even determined by the past. The worst-case probability of any closed tape event $E$ is exactly $\min\{1,DC_s(E)\}$, where $C_s(E)$ is the infimum cost of covering $E$ by prefixes of varying depth. This tree max-flow/min-cut identity makes worst-case probabilities exact and optimizable. Three applications follow. In scheduling, $n$ tasks on one machine are each retried until the next random symbol lies in its success set; every schedule has the same makespan distribution under independent randomness, yet under our constraint two fixed orders miss a linear deadline with worst-case probabilities $1$ and $\exp[-Θ(n)]$, and a static priority rule is optimal at every deadline. For the standard local walk on a $k$-CNF with an assignment satisfying a majority of each clause, we find the sharp source threshold $s_*$: below it, on some instances one admissible source forces nontermination for every clause selection; at it, with fixed $D$, the expected number of flips is at most cubic in the initial distance, against linear under independence, with matching instances at distances up to a constant fraction of the number of variables; a fixed margin above it restores linear dependence. For noisy predecessor search with prefix-bounded comparison errors, verified backtracking attains any prescribed error probability at or above a source threshold whose sharpness we leave open, with the same cubic-to-linear query bounds. Restarting on the shared tape restores exponential tails exactly when the nontermination event has a prefix cover of cost below one.

cs.CC · cond-mat.stat-mech · cs.LG · math.CO · math.PR

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.11826

How Path Integral Diffusion Decides a Finite Spin Glass

Michael Chertkov

原文摘要

When an AI diffusion-type generator samples a frustrated spin glass, which macroscopic alternatives does it decide first, and when? We study this for bridge-diffusion models of the Path Integral Diffusion (PID) type started from a point source. We show that every isotropic PID is a time-changed Gaussian stochastic localization, and conversely every signal-to-noise schedule is realized by a PID protocol. What a continuous-time generator decides therefore does not depend on the protocol, which only sets when and at what cost. We describe the generator's state by its partial commitments: the probabilities it assigns, at each moment, to nested classes of configurations fixed by the target. The partial commitments are martingales that move only along class contrasts. Their ensemble average gives the information about each level at an exact rate set by squared magnetization contrasts. Each split is decided, approximately, like a binary Gaussian channel at signal-to-noise ratio 1.04/λ_P, with λ_P the leading eigenvalue of the parent's correlation matrix. The theory is applied to Bethe-lattice and Sherrington--Kirkpatrick (SK) glasses with up to 64 spins. It is observed generically that a trajectory first shifts its weight to one of the two mirror halves of the zero-field measure and then to the glassy classes inside it. These classes appear near the glass transition and keep their identity on cooling. Nested glassy levels have similar λ_P and are decided together, so a trajectory commits in one or two distinct events---the mirror choice and a compound glassy decision---which separate further with coupling and size, as the spectra predict. For a generator that takes finitely many steps, the protocol matters through the late, local stage rather than the macroscopic decisions.

cond-mat.dis-nn · cond-mat.stat-mech · cs.AI · stat.ML

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2604.00858

Statistical Physics of Coding for the Integers

Neri Merhav

原文摘要

We study a paradigm of coding for compression of the natural numbers via the zeta distribution and develop a statistical-mechanical interpretation, both in terms of Hagedorn systems and a Bose gas with energy levels given by logarithms of prime numbers. We also propose a simple coding scheme for the zeta distribution that nearly achieves the ideal code length. For block coding of vectors of natural numbers, we derive the micro-canonical entropy function and demonstrate its asymptotic linearity implying that its behavior is analogous to that of a Hagedorn system. We also derive the large deviations rate function, and provide a formula for the best coding parameter in the large deviations sense. We show that due the Hagedorn-type phase transition there is only partial equivalence of ensembles, due to the degeneration of the domain of the partition function.

cond-mat.stat-mech · cs.IT

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2602.12449

Computationally sufficient statistics for Ising models

Abhijith Jayakumar, Shreya Shukla, Marc Vuffray, Andrey Y. Lokhov, Sidhant Misra

原文摘要

Learning Gibbs distributions using only sufficient statistics has long been recognized as a computationally hard problem. On the other hand, computationally efficient algorithms for learning Gibbs distributions rely on access to full sample configurations generated from the model. For many systems of interest that arise in physical contexts, expecting a full sample to be observed is not practical, and hence it is important to look for computationally efficient methods that solve the learning problem with access to only a limited set of statistics. We examine the trade-offs between the power of computation and observation within this scenario, employing the Ising model as a paradigmatic example. We show that the couplings of the model can be learned to a desired accuracy from statistics whose order grows linearly with the $\ell_1$ width of the model and only logarithmically with the inverse of the accuracy. In particular, for a fixed accuracy, the required order is proportional to the width. This allows us to infer the model's structure, using statistics whose order grows logarithmically with the inverse of the weakest coupling, and also to learn its magnetic fields. When the width and the accuracy are fixed, the running time is polynomial in the number of variables. We also discuss a setting where prior information about structure of the model is available and show that the learning problem can be solved efficiently with even more limited observational power.

cs.LG · cond-mat.stat-mech · stat.ML

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.10338

On the universal structure in deterministic generative diffusion

Defang Chen, Zhenyu Zhou, Can Wang, Zeyu Zheng, Siwei Lyu

原文摘要

Diffusion-based generative modeling has emerged as a general computational framework for simulating high-dimensional, structured data across various domains. Understanding its generative mechanisms is paramount for advancing model training and inference in a principled way. Here, we demonstrate that deterministic sampling trajectories share a universal geometric structure invariant to data modalities and diffusion model implementations. This structure can be largely explained by an exactly solvable model based on the optimal linear denoiser.

cs.LG · cond-mat.stat-mech · cs.CV · stat.ML

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2605.10642

Composing diffusion priors with explicit physical context via generative Gibbs sampling

Weizhou Wang, Jonathan Weare, Aaron R. Dinner

原文摘要

Pretrained diffusion models provide powerful learned priors, but in scientific sampling the target distribution often depends on physical context that a single generative model can neither fully represent nor learn from scarce joint-system data. We introduce Generative Gibbs for Physics-Aware Sampling (GG-PA), a training-free framework that composes learned partial priors with explicit physical context through Gibbs sampling. We prove that this sampler is asymptotically exact as the diffusion time vanishes and remains exact at finite diffusion time for quadratic interactions. As a special case, the finite-time result corrects a finite-noise bias in plug-and-play split Gibbs samplers for linear inverse problems at no additional cost. We further introduce replica exchange over diffusion time to accelerate mixing. In experiments, a one-dimensional double-well prior recovers a context-induced distribution shift when coupled to an environment, and copies of such a prior composed on a lattice reproduce the emergent collective behavior of the $φ^4$ model. In atomistic peptide systems, GG-PA captures a context-induced distribution shift and the spontaneous formation of hydrogen-bonded dimers from two copies of a prior trained only on the monomer.

cs.LG · cond-mat.stat-mech

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.10139

The satisfiability conjecture for random k-SAT formulas. Human-friendly exposition of the recent OpenAI resolution

David Gamarnik

原文摘要

The satisfiability conjecture was one of the most important open problems in discrete probability. It was resolved recently first by Carenini~\cite{GCarenini}, and shortly after by OpenAI~\cite{k-sat-OpenAI}, among other 370 problems solved by its internal model. The conjecture concerned the satisfiability property of random instances of the k-SAT problem. It posits the existence of a fixed value for the clause-to-variables ratio around which the probability that the formula is satisfiable drops sharply from nearly one to nearly zero as the ratio increases. While the case of large constant $k$ was resolved a decade ago by Ding, Sly and Sun~\cite{ding2015proof,ding2022proof} using the heavy machinery of the second moment method, the case of general $k$ eluded proof until ~\cite{GCarenini} and~\cite{k-sat-OpenAI}. In this short note we provide a ''human-friendly'' and detailed exposition of the OpenAI's resolution of the conjecture.

math.PR · cond-mat.stat-mech · cs.DM

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2604.12151

Distinct mechanisms underlying in-context learning in transformers

Cole Gibson, Wenping Cui, Gautam Reddy

原文摘要

Modern distributed networks, notably transformers, acquire a remarkable ability (termed `in-context learning') to adapt their computation to input statistics, such that a fixed network can be applied to data from a broad range of systems. Here, we provide a complete mechanistic characterization of this behavior in transformers trained on a finite set $S$ of discrete Markov chains. The transformer displays four algorithmic phases, characterized by whether the network memorizes and generalizes, and whether it uses 1-point or 2-point statistics. We show that the four phases are implemented by multi-layer subcircuits that exemplify two qualitatively distinct mechanisms for implementing context-adaptive computations. Minimal models isolate the key features of both motifs. Memorization and generalization phases are delineated by two boundaries that depend on data diversity, $K = |S|$. The first ($K_1^\ast$) is set by a kinetic competition between subcircuits and the second ($K_2^\ast$) is set by a representational bottleneck. A symmetry-constrained theory of a transformer's training dynamics explains the sharp transition from 1-point to 2-point generalization and identifies key features of the loss landscape that allow the network to generalize. Put together, we show that transformers develop distinct subcircuits to implement in-context learning and identify conditions that favor certain mechanisms over others.

cs.LG · cond-mat.dis-nn · cond-mat.stat-mech

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2602.14885

Embedding stochastic dynamics in latent manifolds of recurrent neural networks

Ramón Nartallo-Kaluarachchi, Renaud Lambiotte, Alain Goriely

原文摘要

Recurrent neural networks (RNNs) provide a theoretical framework for understanding computation in biological neural circuits through their ability to support rich time-dependent behaviour. Whilst high-dimensional by construction, they can be used to encode low-dimensional dynamical systems, for example by restricting to low-rank connectivity, hence mirroring the emergence of latent neural manifolds found in empirical neural data. Here, we exploit a relationship between RNNs and a class of universal approximators to train networks that can represent arbitrary, nonlinear stochastic differential equations (SDEs), with given drift and diffusion coefficients, within a low-dimensional latent subspace. We show that RNNs can faithfully embed the drift and diffusion of a given SDE, including nonlinear and nonequilibrium dynamics such as chaotic attractors. We apply this approach by constructing RNN realisations of stochastic systems that transiently explore various attractors through both input-driven switching and autonomous transitions driven by nonequilibrium currents, which we interpret as models of associative and sequential (episodic) memory. To elucidate how these dynamics are encoded in the network, we introduce decompositions of the RNN based on its asymmetric connectivity and its time-irreversibility. Our results show that neural populations can implement a broad class of stochastic dynamics within low-dimensional manifolds, unifying ideas from the dynamics of neural computation, the theory of neural manifolds, associative memory, and nonequilibrium statistical mechanics.

cond-mat.dis-nn · cond-mat.stat-mech · cs.LG · q-bio.NC

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.09138

Deep Learning Theory through the Lens of Statistical Physics: A Topical Review

Wei Huang, Youjin Deng

原文摘要

The recent success of foundation models and deep learning has far outpaced our theoretical understanding of how large-scale neural networks learn and generalize. Statistical physics provides a natural language for describing such high-dimensional, nonlinear, and strongly coupled systems, offering concepts such as mean-field theory, phase transitions, energy landscapes, and stochastic dynamics. In this topical review, we adopt a statistical-physics way of thinking -- from concepts including phases and criticality, and free-energy viewpoints, to methods such as kernel and Gaussian-process limits, mean-field and dynamical mean-field theories, random-matrix tools, and Fokker-Planck formalisms, and to a mindset emphasizing universality, scaling, and tractable limits with explicit validity regimes. We survey recent progress on understanding modern deep learning through this lens, including mean-field analyses of wide networks, spectral and geometric perspectives on loss landscapes, stochastic-gradient-based training viewed as a dynamical process, and phase-transition analogies that organize phenomena such as overparameterization, generalization, and representation change. Our goal is to synthesize these threads into a coherent framework that clarifies what each approximation explains, when it is expected to hold, and how apparently competing narratives relate or diverge. Together, these perspectives form a unified physical framework that views representation learning as a collective phenomenon in high-dimensional systems, bridging deterministic and stochastic regimes of gradient-based optimization.

cs.LG · cond-mat.stat-mech

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.10039

Efficient classical simulation of Fermi-Hubbard models with Kramers symmetry

Chaithanya Rayudu, Jun Takahashi, Ojas Parekh, Andrew Zhao

原文摘要

Quantum Monte Carlo methods are widely used to solve quantum many-body problems. However, rigorous guarantees for their performance are poorly understood, especially for simulating interacting fermions. In this work, we develop a quantum Monte Carlo algorithm for a broad family of fermionic systems exhibiting Kramers time-reversal symmetry, and prove that their partition functions can be estimated in polynomial time. This family, which contains many important classes of Hubbard-type models, is known to both avoid the sign problem and obey a Lee-Yang zero-freeness theorem. Thus our results suggest that these two properties together may enable efficient classical algorithms for quantum many-body systems more broadly.

cs.DS · cond-mat.stat-mech · cond-mat.str-el · quant-ph

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2605.26032

万物皆尺度：尺度不变的连续超分辨率扩散模型

Everything at Every Scale: Scale-Invariant Diffusion with Continuous Super-Resolution

Zixin Jessie Chen, Zhuo Chen, Archer Wang, Jeff Gore, William T. Freeman, Congyue Deng, Marin Soljačić

AI 速读 基于摘要

提出 SKILD，用尺度作为扩散动力学的显式坐标，在单一无条件框架内统一图像生成与连续超分辨率。

cs.CV · cond-mat.stat-mech · cs.AI · cs.LG

arXiv ↗PDF ↗
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Creating images from noise is image generation; reconstructing fine details from coarse inputs is super-resolution. Despite their practical differences, both can be understood as reversing information loss across scales. We introduce $\textbf{SKILD}$, a $\textbf{S}$cale-invariant $\textbf{K}$-Space $\textbf{I}$mage $\textbf{L}$earning $\textbf{D}$iffusion model that unifies generation and continuous super-resolution within a single unconditional framework. Both natural images and critical physical systems exhibit scale invariance, and we leverage it to design a forward process that attenuates image content from fine to coarse scales while injecting spectrum-matched Gaussian noise, making scale an explicit coordinate of the diffusion dynamics. The same trained reverse process performs generation and continuous super-resolution by varying only the starting timestep: $\textit{no task-specific architecture, no conditioning branch, no classifier-free guidance, no retraining per scale factor}$. Empirically, SKILD reaches FID $2.61$ and Inception Score $9.65$ on unconditional CIFAR-10, performs $2\times$--$8\times$ super-resolution on ImageNet from a single unconditional checkpoint while outperforming conditional models across perceptual metrics, and reconstructs critical Ising models whose connected four-point correlations closely track the ground truth.

版本更新 · v2arXiv:2609.23942

费米子的李-杨定理

Lee-Yang theorem for fermions

Chaithanya Rayudu, Takahiro Misawa, Andrew Zhao, Jun Takahashi

AI 速读 基于摘要

作者证明了一类广泛相互作用的费米子模型配分函数的李-杨零点不存在定理，并据此给出估计基态能量的可证明高效量子算法。

quant-ph · cond-mat.stat-mech · cond-mat.str-el · cs.CC · math-ph

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Lee-Yang theorems are a powerful tool for studying many-body systems, with applications ranging from analyzing phase transitions to proving the efficiency of certain classical and quantum algorithms. In this work, we prove a Lee-Yang zero-freeness theorem for the partition function of a broad class of interacting fermion models, implying the existence of a provably efficient quantum algorithm for estimating their ground-state energies. This class includes several well-known models such as the attractive Hubbard model, repulsive Hubbard model on bipartite graphs, and the interacting Hofstadter model. Our results also rigorously establish the nonexistence of phase transitions in these models in the presence of a nonzero local external field.

版本更新 · v2arXiv:2609.08219

Speed limits for information acquisition in stochastic learning dynamics

Shuta Kobayashi, Andreas Dechant

原文摘要

Neural networks acquire internal representations through learning. In this work, we formulate stochastic gradient descent (SGD) as a Markovian stochastic process and derive a Fisher-information flow speed limit that bounds the rate at which trainable parameters can acquire information about data-generating parameters. The resulting inequality decomposes the upper bound on the information flow into drift and noise contributions, thereby quantifying the roles of deterministic learning forces and SGD-induced fluctuations from an information-theoretic perspective. We verify the bound in analytically tractable basis-function linear regression, where the information budget predicted by the bound reproduces the ordering and characteristic time scales with which different data-generating parameters are encoded in the learned parameters. These results establish Fisher-information speed limits as a quantitative framework for understanding which properties of the data-generating process are acquired, in what order, and how rapidly they can be encoded in the trainable parameters.

cond-mat.stat-mech · cs.LG · stat.ML

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新论文 · v1arXiv:2611.07845

Masked diffusion as random sequential adsorption: an exactly solvable model for parallel decoding

Adrian Baule

原文摘要

Masked diffusion models generate text by reversing a stochastic masking process. Unlike autoregressive LLMs, which generate text one token at a time, they can reveal many tokens in parallel, at the price of an error, since tokens revealed together are drawn independently. We show that this trade-off maps onto the statistical physics of driven many-particle systems. (i) For any data, the error of parallel generation is the entropy production of the generation process relative to the reversed masking process. (ii) For Markov data, generation is a random sequential adsorption process with labelled gaps, which we solve exactly for any number of steps. (iii) For many steps, the error becomes the dissipation of a slowly driven system: the unmasking schedule plays the role of the driving protocol, the number of steps that of its duration, and the error is set by a thermodynamic metric that we obtain in closed form from the mutual information between tokens. Optimal schedules are geodesics of this metric. The exact solution shows how correlations in the text shape both the error and the quality of the output. The longer the correlations, the more the error concentrates at the start of generation. Schedules that start with small steps then gain more, both in path error and in perplexity, and need fewer steps to reach their asymptotic behaviour.

cond-mat.stat-mech · cs.AI · cs.CL · cs.LG

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新论文 · v1arXiv:2611.07337

Frustration-freeness is decidable but unprovable

Naoto Shiraishi

原文摘要

The frustration-free property is valuable in quantum many-body systems, which states that its ground states simultaneously minimize all local interaction terms, even when these terms act on overlapping degrees of freedom. Prominent examples include the AKLT model and Kitaev's toric code model, which serve as paradigmatic models of symmetry-protected topological phases and topological order, respectively. A fundamental question is whether a given Hamiltonian is frustration-free for all system sizes. Here we establish two seemingly contradictory results concerning this decision problem. First, we prove that frustration-freeness is decidable for any fixed local Hilbert space dimension. Specifically, there exists a finite, Hamiltonian-independent system-size cutoff such that frustration-freeness at this cutoff size guarantees frustration-freeness at all system sizes. Second, on the other hand, we also show that there exists a frustration-free Hamiltonian whose frustration-freeness cannot be proved within the standard axioms of mathematics (ZFC). We reconcile these seemingly contradictory results by showing that the cutoff, although finite for each local Hilbert space dimension, is uncomputable as a function of that dimension. Our findings reveal an unexpected fundamental distinction between decidability and provability arising in quantum many-body physics.

quant-ph · cond-mat.stat-mech · cond-mat.str-el · cs.LO

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版本更新 · v3arXiv:2011.04672

Principles of Quantum Communication Theory: A Modern Approach

Sumeet Khatri, Ludovico Lami, Mark M. Wilde

原文摘要

This is a preliminary version of a book in progress on the theory of quantum communication. We adopt an information-theoretic perspective throughout and give a comprehensive account of fundamental results in quantum communication theory from the past decade (and earlier), with an emphasis on the modern one-shot-to-asymptotic approach that underlies much of today's state-of-the-art research in this field. In Part I, we cover mathematical preliminaries and provide a detailed study of quantum mechanics from an information-theoretic perspective. We also provide an extensive and thorough review of quantum entropies, and we devote an entire chapter to the study of entanglement measures. Equipped with these essential tools, in Part II we study classical communication (with and without entanglement assistance), entanglement distillation, quantum communication, secret key distillation, and private communication. In Part III, we cover the latest developments in feedback-assisted communication tasks, such as quantum and classical feedback-assisted communication, LOCC-assisted quantum communication, and secret key agreement.

quant-ph · cond-mat.stat-mech · cs.IT · hep-th · math-ph

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2605.10795

Factual recall in linear associative memories: sharp asymptotics and mechanistic insights

Alessio Giorlandino, Sebastian Goldt, Antoine Maillard

原文摘要

Large language models excel at factual recall, yet the fundamental limits of storing input--output associations with neural networks remain unclear. We study these limits in a minimal setting: a linear associative memory that maps $p$ input embeddings in $\mathbb{R}^d$ to their $d$-dimensional targets via a linear layer, requiring each mapped input to score its own target above all other outputs. Since all inputs compete for the same outputs, these constraints are strongly correlated, which makes the storage capacity $p_c$ --- the largest number of associations that can be stored --- hard to characterise. We address this in three steps. First, we introduce a decoupled model, in which each input competes with its own independent set of outputs, and provide numerical and analytical evidence that it is equivalent to the original model, both in capacity and in the structure of the learnt solution. Second, using tools from statistical physics, we show that the decoupled model has capacity $p_c \log p_c / d^2 = 1/2$, extend this computation to linear two-layer networks, and go beyond the capacity by predicting the spectrum of the learnt weights and the storage mechanism. The latter explains how the optimal solution outperforms a naïve Hebbian rule: rather than boosting input--output alignments with broad fluctuations, it raises the correct scores just above the extreme-value threshold set by the competing outputs. Third, we prove the decoupled model's capacity thresholds rigorously, including in the non-convex two-layer case; combined with a recent proof for the original model, this shows that both models have the same capacity for full-rank weights. These findings give a sharp characterisation of factual storage in linear networks, and suggest decoupling as a general tool for memorisation and learning problems in which each input must be matched to one of an extensive number of outputs.

stat.ML · cond-mat.dis-nn · cond-mat.stat-mech · cs.LG

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2607.15215

Stochastic Ising networks with asymmetric and time-delayed interactions

Hantao Zhang, Sidra Gibeault, Matthew W. Daniels, Philippe Talatchian, Ursula Ebels, Advait Madhavan, Mark D. Stiles

原文摘要

Stochastic Ising networks are widely used to describe collective dynamics in complex systems and to compute, yet physical implementations of these networks often contain both asymmetric interactions and nonzero signal propagation times that fall outside conventional theories of their behavior. Here, we study stochastic Ising networks with asymmetric and time-delayed interactions motivated by experimental observations in coupled superparamagnetic tunnel junctions. Time delay fundamentally reshapes the dynamics induced by antisymmetric couplings, producing strong oscillatory temporal correlations consistent with experiment. Sufficiently long delays strongly suppress the non-uniformity of steady-state probability distributions, a behavior explained by an exponential suppression of the first-order equal-time cross-correlation between pairs of spins. Residual non-uniformity is due to the equal-time cross-correlation mediated by a third spin that contributes only at higher order in the coupling strength. These relatively featureless probability distributions coexist with pronounced delayed temporal correlations, distinguishing them from equilibrium high-temperature behavior. We further show analytically that these delay-induced near-uniform distributions emerge in a broad class of stochastic networks that lack symmetry-breaking bias fields. Introducing symmetry-breaking bias fields restores a strong dependence of the steady state on interactions with qualitatively modified behavior. Simulations of networks with five coupled spins demonstrate that these effects persist beyond minimal systems with only two spins.

physics.app-ph · cond-mat.mes-hall · cond-mat.stat-mech · cs.ET · physics.data-an

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.05446

A Fully Polynomial Approximation Algorithm for the Sherrington-Kirkpatrick Model

Dylan Herman, Shouvanik Chakrabarti, Junhyung Lyle Kim, Guneykan Ozgul

原文摘要

Montanari's incremental approximate message passing (IAMP) algorithm [FOCS 2019; SIAM J. Comput., 2025] and the zero-temperature formulation of El Alaoui, Montanari and Sellke (AMS) [Ann. Probab., 2021] find a $(1-\varepsilon)$ approximation to the maximum Sherrington-Kirkpatrick (SK) energy in $C(\varepsilon)N^2$ operations for fixed accuracy, with probability tending to one as $N\to\infty$, under their respective full-support and access-model assumptions. However, it was unknown whether this approximation could be achieved with work polynomial in both $N$ and $1/\varepsilon$, together with quantitative guarantees at finite dimension. Under the same assumptions as AMS, we introduce grouped IAMP with exact Gaussian query outputs (GEG-IAMP), which is the first SK approximation algorithm with total work polynomial in $N$ and $1/\varepsilon$ and a finite-size guarantee above a dimension threshold polynomial in $1/\varepsilon$ and $\log(1/α)$. More precisely, for every $α\in(0,1/2)$, it returns a $(1-\varepsilon)$ approximation with probability at least $1-α$ whenever $N\ge \mathsf{K}\varepsilon^{-2}\log^2(e/\varepsilon)\log(\mathsf{K}/α)$, using $O(N^2/\varepsilon)$ work and $O(N^2)$ storage. GEG-IAMP replaces the Onsager correction by Reeves's coupling construction [arXiv:2508.10782] and groups Euler-Maruyama steps into $O(1/\varepsilon)$ matrix-vector products. We also prove a relative guarantee for AMS IAMP with $O(N^2\varepsilon^{-3}\log^2(e/\varepsilon))$ work but a sufficient dimension exponential in inverse accuracy. By returning a fixed configuration below its dimension threshold, GEG-IAMP also provides an asymptotic fully polynomial randomized approximation scheme (AFPRAS), with an additive error independent of dimension.

cs.DS · cond-mat.stat-mech · math.OC · math.PR · quant-ph

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