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Mirror Symmetry of Height-Periodic Gradient Gibbs Measures of an SOS Model on Cayley Trees

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Abstract

For the solid-on-solid model with spin values from the set of all integers on a Cayley tree we give gradient Gibbs measures (GGMs). Such a measure corresponds to a boundary law (which is an infinite-dimensional vector-valued function defined on vertices of the Cayley tree) satisfying an infinite system of functional equations. We give several concrete GGMs of boundary laws which are independent from vertices of the Cayley tree and (as an infinite-dimensional vector) have periodic, (non-)mirror-symmetric coordinates. Namely, the particular class of height-periodic boundary laws of period \(q\le 5\) is studied, where solutions are classified by their period and (two-)mirror-symmetry.

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Data Availability

The datasets generated during and/or analysed during the current study are available from the author (U.A.Rozikov) on reasonable request.

Notes

  1. All GGMs mentioned in this theorem are different from such measures of [8, 12].

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Acknowledgements

The author thanks both referees for their useful and helpful comments. The work supported by the fundamental project (Number: F-FA-2021-425) of The Ministry of Innovative Development of the Republic of Uzbekistan.

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Authors and Affiliations

  1. V. I. Romanovskiy Institute of Mathematics, 9, Universitet Str., 100174, Tashkent, Uzbekistan

    U. A. Rozikov

  2. AKFA University, National Park Street, Barkamol MFY, Mirzo-Ulugbek District, Tashkent, Uzbekistan

    U. A. Rozikov

  3. National University of Uzbekistan, 4, Universitet Str., 100174, Tashkent, Uzbekistan

    U. A. Rozikov

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  1. U. A. Rozikov
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Correspondence to U. A. Rozikov.

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Communicated by Aernout van Enter.

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Rozikov, U.A. Mirror Symmetry of Height-Periodic Gradient Gibbs Measures of an SOS Model on Cayley Trees. J Stat Phys 188, 26 (2022). https://doi.org/10.1007/s10955-022-02953-z

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