A Human-AI Theorem Connecting Spontaneous and Field-Induced Mechanisms of Collective Behavior in One Dimension

Can an artificial intelligence (AI) generate a scientific hypothesis outside a human collaborator's active hypothesis space (AHS), and can human-AI research be organized to make such breakthroughs more likely? We document such a case while proving a theorem that connects two basic organizing mechanisms of statistical physics: collective behavior arising in zero field from competing interactions and that induced or controlled by an external field. A zero-field $O(n)$-vector open chain with arbitrary inhomogeneous nearest- and next-nearest-neighbor interaction functions $U_i(S_i\cdot{S}_{i+1})$ and $V_i(S_i\cdot{S}_{i+2})$ is microscopically, via a temperature-independent mapping at the Hamiltonian level, equivalent to a simpler $O(n)$ open chain with nearest-neighbor interaction $V_i( σ_i\cdot σ_{i+1})$ and axial single-spin potential $U_i(σ_i^z)$ for every integer $n\ge1$ and every system size $L\ge1$. The homogeneous linear specialization maps the foundational frustrated $J_1$-$J_2$ model onto the canonical $J$-$h$ field model---with $n=1,2,3$ being the Ising, XY, and Heisenberg classical spin models, respectively. An analogous theorem holds when the continuous $O(n)$ spins are replaced by the $q$-state Potts spins with the standard Potts interaction, implying a closed-form exact solution of the $J_1$-$J_2$ Potts open chain for every $q\ge2$ and every $L\ge1$. The emergence of the theorems from sustained human-AI collaboration suggests that involving AI throughout a systematic research program may incubate autonomous scientific breakthroughs.

Landau theory of quenched criticality in linear in-context learning

In-context learning (ICL) allows a pretrained model to infer a new task from examples supplied in its prompt without updating its parameters. In linear models of ICL, the prediction error develops a double-descent singularity when the number of pretraining samples becomes comparable to the number of learnable parameters. We formulate this interpolation singularity as a critical phenomenon of a quenched disordered system. By comparing annealed and quenched descriptions of the same linear ICL model, we identify the connected sample-to-sample fluctuations of the learned parameters as the microscopic origin of the singular error. A Landau potential is constructed by integrating the cavity self-consistency equation for the renormalized ridge parameter $ξ$. The role of (magnetization) order parameter is played by $ξ$, while the bare ridge parameter $λ$ becomes its conjugate magnetic field. The normalized sample complexity $τ$ acts as a temperature and the double-descent singularity occurs at the critical temperature $τ_c =1$. The Landau susceptibility is precisely the quantity that diverges in the fluctuation contribution to the prediction error. The order parameter is closely related to the fraction of zero eigenvalues of the empirical relaxation matrix in the ridgeless limit, which define flat directions in the learning dynamics. The Landau theory is generically cubic in the order parameter with critical exponents $(β_{\rm cr},δ_{\rm cr},γ_{\rm cr})=(1,2,1)$. In the large-context regime, there appears a pseudogap-like regime characterized by suppressed order parameter. Predictions of the Landau theory are independently confirmed from numerical solutions of the original learning problem with good quantitative agreement. Our results pave the way for solid statistical-physics understanding of the interpolation criticality in linear in-context learning.

Characterizing Full Nonequilibrium Dynamics of Simple Exclusion Processes

The simple exclusion process (SEP) is a paradigmatic model for nonequilibrium transport, yet the rich dynamics of its time-dependent joint distribution over an exponentially large configuration space remain notoriously intractable. Here, we leverage variational autoregressive networks to systematically characterize the nonequilibrium dynamics of symmetric (SSEP), asymmetric (ASEP), and totally asymmetric (TASEP) cases from one to three dimensions. We first validate the approach by reproducing the previous finite-time results for the 1D SSEP and long-time tensor-network results for the 2D SSEP, and then provide richer finite-time dynamics of the SSEP, ASEP, and TASEP in 1D and 2D, and a new finite-time analysis in 3D. Specifically, in 1D, we reveal that finite-time dynamical-activity maps directly correspond to the classical three-phase TASEP steady-state organization, and, in the long-time limit, boundary and bulk effects separately govern the dynamical susceptibility during the crossover from diffusive to ballistic transport. In 2D, we establish a mean-field directional-density criterion, supported by our neural-network calculations, and show that long-time boundary and bulk effects mirror their 1D counterparts. In 3D, we uncover new finite-time scaling relations for the active-inactive phase transition of the SSEP, and reveal a broadly consistent scaling exponent of the phase-transition point versus system size, implying that the phase-transition point is asymptotically controlled by the characteristic length scale ($s_c\sim L^{-2}$) regardless of dimension. This work thus establishes a unified framework for characterizing the nonequilibrium dynamics of representative transport systems.

Leveraging generative models to assist Monte Carlo sampling

Sampling high-dimensional probability distributions is a central task in scientific computing, with applications ranging from Bayesian inference to statistical physics and molecular simulation. Despite decades of methodological developments, two major challenges remain: scaling to high dimensions and efficiently exploring multimodal distributions characterized by metastable states. Classical approaches such as Markov chain Monte Carlo, tempering methods, or enhanced sampling based on collective variables have achieved major successes, but they also face intrinsic limitations. This tutorial review explores a new paradigm that has recently emerged at the interface of machine learning and computational statistical physics: the use of generative models as tools for sampling. In this context, models such as normalizing flows and diffusion models are not used in their traditional data-driven setting, but rather as flexible probabilistic models that can assist the sampling of distributions known only up to a normalization constant. This manuscript reviews the early development of this rapidly evolving field and discusses several methodological directions, including exact samplers based on generative models and strategies to train such models in the absence of data. While an exhaustive survey of the literature is not attempted, we present a selection of key ideas and methods, along with a discussion of their strengths and limitations. The review is intended to be an accessible tutorial for both physics and machine learning audiences, and it aims to provide a starting point for researchers interested in exploring this exciting area of research.

Variational Bounds for Perceptron Learning from Structured Data

We introduce a variational approach to a finite-temperature continuous-spin perceptron trained on a Gaussian mixture. The model allows for a broad class of concave utilities and log-concave separable prior measures on the spins. By combining the interpolation method with log-concavity and concentration estimates, we derive lower and upper minimax variational bounds for the limiting quenched pressure. Remarkably, the two bounds differ only in the order of optimization of two variational parameters, while all remaining extrema are controlled by the concave--convex structure of the variational potential. Whenever the two optimizations commute, the two bounds match and identify the solution of the model. The same potential yields the fixed-point equations as stationarity conditions and provides a unified route to the computation of the ground-state energy, training loss, and generalization error.

Poor Man's Agentic Modeling: Simulating Large LLM-Agent Societies on a Laptop

Simulating societies of many large language model (LLM) agents is expensive, yet the questions asked of such simulations are usually macroscopic: phase behaviour, stylised facts, and scaling with the number of agents $N$, not the cognition of any single agent. We turn a statistical-physics observation into a method: replace each LLM agent by a low-parameter model fitted from a few hundred to a few thousand cheap queries, then run the society at any $N$ on a laptop. Whether this works is decided before the simulation runs, chiefly by what each agent perceives. We introduce an [interaction order x memory] taxonomy that maps perception and memory to an effective theory and a predicted $N$-trend of the surrogate error. We validate it on a faithful reimplementation of the LLM macroeconomy EconAgent and seven further named LLM simulations, with agent decisions cloned from genuine LLM elicitations (primarily DeepSeek) for a few dollars; the predicted error trends hold cell by cell, and the two refuted predictions, both on a strongly saturating response and traced to its curvature, are themselves matched quantitatively by the theory with no free parameters.

Scaling limit of the Random Language Model

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 condensation transition at a critical value $x_c=1/8$, below which rule usage concentrates and language statistics acquire a nontrivial dependence on corpus length. A second characteristic scale at $x=1/2$ marks the onset of entropy reduction from its maximal value. Across these regimes, we derive explicit scaling laws for the number of distinct rules, entropy, and related observables, identifying distinct scaling, saturation, 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.

Generative Criticality in Large Language Model Temperature Scaling

We propose a statistical-field framework for text generated by large language models (LLMs), treating token embeddings as continuous spin variables on a one-dimensional chain. Defining a susceptibility from the connected two-point correlator and an order parameter from the ensemble-averaged embedding field, we vary the \texttt{softmax} temperature $T$ and observe a sharp susceptibility peak near a characteristic $T_c$ with power-law-like scaling, a concurrent rapid change in the order parameter, and a collapse onto a single semantic direction below $T_c$. The intrinsic dimension estimated by the two nearest neighbor (TwoNN) method independently corroborates these findings, reaching a minimum near $T_c$. Results are robust across model scales (Qwen3: 0.6B--32B) and prompt categories. While the phenomenology closely resembles a continuous phase transition, the non-equilibrium nature of autoregressive generation warrants further investigation. Our framework provides quantitative tools for probing the collective statistical structure of LLM outputs and suggests connections between decoding strategies and critical phenomena.