Theoretical and Computational Physics

Latest papers

• Spectral origin of the topological gap exponent d + {\eta}: mechanism, kernel, decomposition, and scope

  Matthew Loftus · 10 September 2026

  The topological gap $\Delta$ -- the excess $H_1$ total persistence of a critical point cloud over a density-matched null -- scales as $\Delta \sim L^{d+\eta}$. We derive this analytically: the spectral integral $I(\alpha) = \sum_{k\neq 0} S_{\mathrm{conn}}(k)\,|k|^{\alpha}$ scales as $L^{2-\alpha-\e…

• A Theory of Speciation in Generative Diffusion Models on Compact Riemannian Manifolds

  Alessio Marta, Paola Causin · 26 August 2026

  Speciation in generative diffusion models denotes the emergence of distinct stable branches during denoising, through which initially undifferentiated trajectories progressively commit to different data classes. In this work we develop an intrinsic theory of speciation for diffusion models supported…

• Relevant and Irrelevant: A Renormalization Group Analysis of Transformer Attention

  Parviz Haggi-Mani, Irina Rish · 20 July 2026

  Using the language of Wilsonian renormalization group theory (RG), we treat the Transformer's attention mechanism as a perturbation of the trained MLP residual-stack fixed point and ask whether it constitutes a relevant, marginal, or irrelevant operator. We derive a fixed-point shift formula and obt…

• Why DDIM Hallucinates More Than DDPM: A Theoretical Analysis of Reverse Dynamics

  Muhammad H. Ashiq, Samanyu Arora, Abhinav N. Harish, Ishaan Kharbanda, Hung Yun Tseng, Grigorios G. Chrysos · 1 June 2026

  We theoretically study the hallucination phenomena in two canonical diffusion samplers: the stochastic Denoising Diffusion Probabilistic Model (DDPM) and the deterministic Denoising Diffusion Implicit Model (DDIM). We analyze the reverse ODE (DDIM) and SDE (DDPM) for a Gaussian mixture target, provi…

• Saturating Scaling Laws for Equational Discovery: A Phenomenology of Growth Dynamics in Three Toy Substrates with Two Real-World Replications

  Fabio Rovai · 26 May 2026

  We investigate growth dynamics in deterministic equational discovery substrates. Across three toy domains (arithmetic, boolean, higher-order list; n=592 trajectories), short-range substrate sizes fit a power-law N(t) proportional to t^b. Within each substrate b is architecture-sensitive (cross-valid…

• Why DDIM Hallucinates More than DDPM: A Theoretical Analysis of Reverse Dynamics

  Muhammad H. Ashiq, Samanyu Arora, Abhinav N. Harish, Ishaan Kharbanda, Hung Yun Tseng, Grigorios G. Chrysos · 11 May 2026

  We theoretically study the hallucination phenomena in two canonical diffusion samplers: the stochastic Denoising Diffusion Probabilistic Model (DDPM) and the deterministic Denoising Diffusion Implicit Model (DDIM). We analyze the reverse ODE (DDIM) and SDE (DDPM) for a Gaussian mixture target, provi…

• How Long Does Infinite Width Last? Signal Propagation in Long-Range Linear Recurrences

  Mariia Seleznova · 7 May 2026

  We study signal propagation in linear recurrent models at finite width. While existing signal propagation theory relies predominantly on the infinite-width limit, it remains unclear for how long that approximation remains accurate when recurrent depth $t$ grows jointly with width $n$. This question …

• Concurrence of Symmetry Breaking and Nonlocality Phase Transitions in Diffusion Models

  Yifan F. Zhang, Fangjun Hu, Guangkuo Liu, Mert Okyay, Xun Gao · 7 May 2026

  Diffusion models undergo a phase transition in a critical time window during generation dynamics, with two complementary diagnoses of criticality. The symmetry breaking picture views the critical window as when trajectories bifurcate into different semantic minima of the energy landscape, whereas th…

• Barren Plateaus as Destructive Interference: A Diagnostic Framework and Implications for Structured Ansatzes

  Pilsung Kang · 5 May 2026

  Barren plateaus (BPs) are usually described by the exponential suppression of gradient variance, but the mechanism by which gradient signal disappears remains unclear. We show that this phenomenon can be understood as destructive interference among termwise gradient contributions. To make this persp…

• Phase Transitions as the Breakdown of Statistical Indistinguishability

  Taiyo Narita, Hideyuki Miyahara · 20 April 2026

  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, …

• The topological gap at criticality: scaling exponent d + {\eta}, universality, and scope

  Matthew Loftus · 3 April 2026

  The topological gap $\Delta = TP_{H_1}^{real} - TP_{H_1}^{shuf}$ -- the excess $H_1$ total persistence of the majority-spin alpha complex over a density-matched null -- encodes critical correlations in spin models. We establish finite-size scaling: $\Delta(L,T) = A L^{d+\eta} G_-(L|t/T_c|)$, with $G…

• A Monte Carlo estimator of flow fields for sampling and noise problems

  Michael S. Albergo, Gurtej Kanwar · 3 March 2026

  Learned field transformations may help address ubiquitous critical slowing down and signal-to-noise problems in lattice field theory. In the context of an annealed sequence of distributions, field transformations are defined by integrating flow fields that exactly solve a local transport problem. Th…

• Generalization of Gibbs and Langevin Monte Carlo Algorithms in the Interpolation Regime

  Andreas Maurer, Erfan Mirzaei, Massimiliano Pontil · 13 February 2026

  This paper provides data-dependent bounds on the expected error of the Gibbs algorithm in the overparameterized interpolation regime, where low training errors are also obtained for impossible data, such as random labels in classification. The results show that generalization in the low-temperature …

• Theory of Speciation Transitions in Diffusion Models with General Class Structure

  Beatrice Achilli, Marco Benedetti, Giulio Biroli, Marc M\'ezard · 5 February 2026

  Diffusion Models generate data by reversing a stochastic diffusion process, progressively transforming noise into structured samples drawn from a target distribution. Recent theoretical work has shown that this backward dynamics can undergo sharp qualitative transitions, known as speciation transiti…

• Optimized Machine Learning Methods for Studying the Thermodynamic Behavior of Complex Spin Systems

  Dmitrii Kapitan, Pavel Ovchinnikov, Konstantin Soldatov, Petr Andriushchenko, Vitalii Kapitan · 9 December 2025

  This paper presents a systematic study of the application of convolutional neural networks (CNNs) as an efficient and versatile tool for the analysis of critical and low-temperature phase states in spin system models. The problem of calculating the dependence of the average energy on the spatial dis…

• Complete asymptotic type-token relationship for growing complex systems with inverse power-law count rankings

  Pablo Rosillo-Rodes, Laurent Hébert-Dufresne, Peter Sheridan Dodds · 5 November 2025

  The growth dynamics of complex systems often exhibit statistical regularities involving power-law relationships. For real finite complex systems formed by countable tokens (animals, words) as instances of distinct types (species, dictionary entries), an inverse power-law scaling $S \sim r^{-α}$ betw…