Phase Transitions as the Breakdown of Statistical Indistinguishability

Taiyo Narita; Hideyuki Miyahara

cond-mat.stat-mechApr 17, 2026

Phase Transitions as the Breakdown of Statistical Indistinguishability

Authors: Taiyo Narita, Hideyuki Miyahara

Organizations: Graduate School of Information Science and Technology, Hokkaido University, Sapporo, Hokkaido 060-0814, Japan

Abstract

We introduce a novel characterization of phase transitions based on hypothesis testing. In our formulation, a phase transition is defined as the breakdown of statistical indistinguishability under vanishing parameter perturbations in the thermodynamic limit. This perspective provides a general, order-parameter-free framework that does not rely on model-specific insights or learning procedures. We show that conventional approaches, such as those based on the Binder parameter, can be reinterpreted as special cases within this framework. As a concrete realization, we employ a distribution-free two-sample run test and demonstrate that the critical point of the two-dimensional Ising model is accurately identified without prior knowledge of the order parameter.

Explore similar work

A computational phase transition for learning-to-sample from Ising models
May 23, 2026Andrej Risteski, Thuy-Duong VuongTwo-Dimensional Ising Model Phase Transitions
Siamese Neural Network for Label-Efficient Critical Phenomena Prediction in 3D Percolation Models
Jul 5, 2025Shanshan Wang, Dian Xu, Jianmin Shen +3Phase Transitions
Phase Transitions in Driven Informational Systems: A Two-Field Perspective on Learning Theory and Non-Equilibrium Chemistry
May 5, 2026Truong Xuan KhanhPhase Transitions Statistical Mechanics

May 23, 2026 · cs.LGJ/K move · Enter open · S save

A computational phase transition for learning-to-sample from Ising models

Andrej Risteski, Thuy-Duong Vuong

We study \emph{learning-to-sample} -- a basic algorithmic task underlying generative modeling -- for Ising models, a standard testbed for algorithmic ideas in both theoretical computer science and machine learning. Given i.i.d. samples of an unknown target distribution, the goal of learning-to-sample is to learn a computationally efficient generation procedure that produces new samples following approximately the same distribution. We construct a family of Ising models of constantly bounded-width which lie just beyond the spectral threshold

λ_{\max}(J)-λ_{\min}(J)=1

, and show that learning-to-sample for this family is computationally hard under standard cryptographic assumptions, even when the learner is given both polynomially many i.i.d. samples from the model and explicit access to its parameters. Combined with results of [AJKPV24,KLV25] showing tractability of learning-to-sample below the spectral threshold, this establishes a sharp computational phase transition at the spectral threshold. Moreover, combined with prior results on parameter learning for bounded-width Ising models [KM17,WSD19,VML20], this shows that learning-to-sample can be more difficult than parameter learning. Finally, we show that any efficient learner for these hard instances exhibits a natural memorization-hallucination dichotomy: the learner must either output configurations that, after a simple transformation, match the (transformed) training data or place substantial mass on configurations of negligible probability under the target distribution.

Two-Dimensional Ising Model Phase Transitions Adaptive Sampling Generative Models

Open paper Ask about it PDF

Abstract

Explore similar work

A computational phase transition for learning-to-sample from Ising models

Siamese Neural Network for Label-Efficient Critical Phenomena Prediction in 3D Percolation Models

Phase Transitions in Driven Informational Systems: A Two-Field Perspective on Learning Theory and Non-Equilibrium Chemistry