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Mean-Field Bound on the 1-Arm Exponent for Ising Ferromagnets in High Dimensions

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Sojourns in Probability Theory and Statistical Physics - I

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 298))

Abstract

The 1-arm exponent \(\rho \) for the ferromagnetic Ising model on \(\mathbb {Z}^d\) is the critical exponent that describes how fast the critical 1-spin expectation at the center of the ball of radius r surrounded by plus spins decays in powers of r. Suppose that the spin-spin coupling J is translation-invariant, \(\mathbb {Z}^d\)-symmetric and finite-range. Using the random-current representation and assuming the anomalous dimension \(\eta =0\), we show that the optimal mean-field bound \(\rho \le 1\) holds for all dimensions \(d>4\). This significantly improves a bound previously obtained by a hyperscaling inequality.

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Acknowledgements

The work of AS is supported by the JSPS Grant-in-Aid for Challenging Exploratory Research 15K13440. The work of SH is supported by the Ministry of Education, Culture, Sports, Science and Technology through Program for Leading Graduate Schools (Hokkaido University “Ambitious Leader’s Program”). We thank Aernout van Enter for providing references about the history of the problem.

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Authors and Affiliations

  1. Graduate School of Mathematics, Hokkaido University, Sapporo, Japan

    Satoshi Handa

  2. Mathematisches Institut, Ludwig-Maximilians-Universität München, Munich, Germany

    Markus Heydenreich

  3. Faculty of Science, Hokkaido University, Sapporo, Japan

    Akira Sakai

Authors
  1. Satoshi Handa
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  2. Markus Heydenreich
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  3. Akira Sakai
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Corresponding author

Correspondence to Akira Sakai .

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Editors and Affiliations

  1. NYU Shanghai, Shanghai, China

    Vladas Sidoravicius

Additional information

We dedicate this work to Chuck Newman on the occasion of his 70th birthday.

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© 2019 Springer Nature Singapore Pte Ltd.

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Handa, S., Heydenreich, M., Sakai, A. (2019). Mean-Field Bound on the 1-Arm Exponent for Ising Ferromagnets in High Dimensions. In: Sidoravicius, V. (eds) Sojourns in Probability Theory and Statistical Physics - I. Springer Proceedings in Mathematics & Statistics, vol 298. Springer, Singapore. https://doi.org/10.1007/978-981-15-0294-1_8

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