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26203 results in Theoretical Physics and Mathematical Physics

Page 2 of 1311

  • Wayfinding behaviours in natural environments

  • Ole Edward Wattne, Frode Volden
  • Journal:
    The Journal of Navigation , First View
    Published online by Cambridge University Press:
    25 November 2025, pp. 1-15
    • Article
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      • Open access
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  • Adapting Barker’s ((2019). The Journal of Navigation, 72(3), 539–554) taxonomy of wayfinding behaviours – originally developed for man-made environments, paper and screen – we examined which behaviours are also found in the outdoors. In the analysis of the collected data from a questionnaire (n=401), we find that participants employ every category in Barker’s framework of social, semantic and spatial behaviours. Our respondents report the use of digital maps on a mobile phone as the most common behaviour, with following directional signs as the second most used. Furthermore, social wayfinding behaviours figure prominently and the participants express preferences for various information sources. We demonstrate similarities of behaviours across the different types of environments and we confirm the applicability of Barker’s taxonomy of wayfinding behaviours also in nature. Our study generates knowledge that potentially can make navigation simpler and more efficient through wayfinding design, and lead to heightened feeling of safety in the outdoors. Wayfinding behaviour studies, like this one, can serve as a bridge between human psychology and practical design.

Causal Fermion Systems
  • An Introduction to Fundamental Structures, Methods and Applications
  • Felix Finster, Sebastian Kindermann, Jan‐Hendrik Treude
  • Published online:
    15 November 2025
    Print publication:
    23 October 2025
  • The theory of causal fermion systems represents a novel approach to fundamental physics and is a promising candidate for a unified physical theory. This book offers a comprehensive overview of the theory, structured in four parts: the first lays the necessary mathematical and physical foundations; the second offers an introduction to the theory and the causal action principle; the third describes the mathematical tools for analyzing causal fermion systems; and the fourth gives an outlook on the key physical applications. With relevance across mathematical and theoretical physics, the book is aimed at graduate students and researchers interested in novel approaches to the structure of spacetime and alternative perspectives to the more established quantum field theories. It can be used for advanced courses in the subject or as a reference for research and self-guided study. Exercises are included at the end of each chapter to build and develop key concepts.

  • Local central limit theorem for triangle counts in sparse random graphs

  • Part of
  • PEDRO ARAÚJO, LETÍCIA MATTOS
  • Journal:
    Mathematical Proceedings of the Cambridge Philosophical Society , First View
    Published online by Cambridge University Press:
    14 November 2025, pp. 1-17

  • Let $X_H$ be the number of copies of a fixed graph H in G(n,p). In 2016, Gilmer and Kopparty conjectured that a local central limit theorem should hold for $X_H$ as long as H is connected, $p\gg n^{-1/m(H)}$ and $n^2(1-p)\gg 1$, where m(H) denotes the m-density of H. Recently, Sah and Sawhney showed that the Gilmer–Kopparty conjecture holds for constant p. In this paper, we show that the Gilmer–Kopparty conjecture holds for triangle counts in the sparse range. More precisely, if $p \in (4n^{-1/2}, 1/2)$, then

    \[ {} {} {}\sup_{x\in \mathcal{L}}\left| \dfrac{1}{\sqrt{2\pi}}e^{-x^2/2}-\sigma\cdot \mathbb{P}(X^* = x)\right|=n^{-1/2+o(1)}p^{1/2}, {} {}\]
    where $\sigma^2 = \mathbb{V}\text{ar}(X_{K_3})$, $X^{*}=(X_{K_3}-\mathbb{E}(X_{K_3}))/\sigma$ and $\mathcal{L}$ is the support of $X^*$. By combining our result with the results of Röllin–Ross and Gilmer–Kopparty, this establishes the Gilmer–Kopparty conjecture for triangle counts for $n^{-1}\ll p \lt c$, for any constant $c\in (0,1)$. Our quantitative result is enough to prove that the triangle counts converge to an associated normal distribution also in the $\ell_1$-distance. This is the first local central limit theorem for subgraph counts above the so-called $m_2$-density threshold.

  • 2 - Correlation Inequalities

  • Barry Simon, California Institute of Technology
  • Book:
    Phase Transitions in the Theory of Lattice Gases
    Published online:
    17 December 2025
    Print publication:
    13 November 2025, pp 39-205

  • Summary

    The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

    Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

    This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

  • 4 - Peierls Argument

  • Barry Simon, California Institute of Technology
  • Book:
    Phase Transitions in the Theory of Lattice Gases
    Published online:
    17 December 2025
    Print publication:
    13 November 2025, pp 283-345

  • Summary

    The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

    Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

    This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

  • 7 - Stochastic Geometric Representations

  • Barry Simon, California Institute of Technology
  • Book:
    Phase Transitions in the Theory of Lattice Gases
    Published online:
    17 December 2025
    Print publication:
    13 November 2025, pp 427-507

  • Summary

    The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

    Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

    This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

  • 9 - Final Thoughts

  • Barry Simon, California Institute of Technology
  • Book:
    Phase Transitions in the Theory of Lattice Gases
    Published online:
    17 December 2025
    Print publication:
    13 November 2025, pp 557-570

  • Summary

    The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

    Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

    This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

  • 1 - A Framework for Lattice Gases

  • Barry Simon, California Institute of Technology
  • Book:
    Phase Transitions in the Theory of Lattice Gases
    Published online:
    17 December 2025
    Print publication:
    13 November 2025, pp 1-38

  • Summary

    The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

    Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

    This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

  • 3 - The Lee–Yang Theorem

  • Barry Simon, California Institute of Technology
  • Book:
    Phase Transitions in the Theory of Lattice Gases
    Published online:
    17 December 2025
    Print publication:
    13 November 2025, pp 206-282

  • Summary

    The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

    Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

    This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

  • 8 - Berezinskii–Kosterlitz–Thouless Transition

  • Barry Simon, California Institute of Technology
  • Book:
    Phase Transitions in the Theory of Lattice Gases
    Published online:
    17 December 2025
    Print publication:
    13 November 2025, pp 508-556

  • Summary

    The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

    Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

    This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

  • 5 - No Continuous Symmetry Breaking in 2D (and Other Situations)

  • Barry Simon, California Institute of Technology
  • Book:
    Phase Transitions in the Theory of Lattice Gases
    Published online:
    17 December 2025
    Print publication:
    13 November 2025, pp 346-367

  • Summary

    The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

    Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

    This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

  • 6 - Infrared Bounds

  • Barry Simon, California Institute of Technology
  • Book:
    Phase Transitions in the Theory of Lattice Gases
    Published online:
    17 December 2025
    Print publication:
    13 November 2025, pp 368-426

  • Summary

    The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

    Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

    This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

  • Quantization dimensions of negative order

  • Part of
  • MARC KESSEBÖHMER, ALJOSCHA NIEMANN
  • Journal:
    Mathematical Proceedings of the Cambridge Philosophical Society , First View
    Published online by Cambridge University Press:
    11 November 2025, pp. 1-25
    • Article
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      • Open access
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  • We investigate the possibility of defining meaningful upper and lower quantization dimensions for a compactly supported Borel probability measure of order r, including negative values of r. To this end, we employ the concept of partition functions, which generalises the notion of the $L^q$-spectrum, thus extending the authors’ earlier work with Sanguo Zhu in a natural way. In particular, we derive inherent fractal-geometric bounds and easily verifiable necessary conditions for the existence of quantization dimensions. We state the exact asymptotics of the quantization error of negative order for absolutely continuous measures, thereby providing an affirmative answer to an open question regarding the geometric mean error posed by Graf and Luschgy in this journal in 2004.

Page 2 of 1311