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Torsion-weighted spanning acycle entropy in cubical lattices and Mahler measures

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Abstract

We compute the eigenvalues of up-Laplacians on cubical lattices and derive the torsion-weighted count of spanning acycles in cubical lattices by using the matrix-tree theorem for simplicial/cell complexes. As a corollary, we show that the torsion-weighted spanning acycle entropy of a cubical lattice defined on \(\mathbb {Z}^b \times \prod _{j=b+1}^q \{0,1,\dots , n_j\}\) is expressed as a linear combination of logarithmic Mahler measures.

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Notes

  1. \(\Lambda ^{(\ell )}_{A_0}\) and \(\Lambda ^{(\ell ')}_{A_0'}\) might be the same for different indices \((A_0, \ell )\) and \((A_0', \ell ')\), so the multiplicity here means the dimension of the eigenspace corresponding to the index \((A_0, \ell )\).

References

  • Adhikari, K., Kumar, K., Saha, K.: On the spectrum of random simplicial complexes in thermodynamic regime. https://arxiv.org/abs/2301.09062

  • Aizenman, M., Chayes, J.T., Chayes, L., Fröhlich, J., Russo, L.: On a sharp transition from area law to perimeter law in a system of random surfaces. Commun. Math. Phys. 92, 19–69 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  • Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: An explanation of the \(1/f\) noise. Phys. Rev. Lett. 59, 381–384 (1987)

    Article  ADS  CAS  PubMed  Google Scholar 

  • Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality. Phys. Rev. A 38(3), 364–374 (1988)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  • Benjamini, I., Lyons, R., Peres, Y., Schramm, O.: Uniform spanning forests. Ann. Probab. 29, 1–65 (2001)

    Article  MathSciNet  Google Scholar 

  • Bertin, M.J., Feaver, A., Fuselier, J., Lalìn, M., Manes, M.: Mahler measure of some singular K3-surfaces. Contemporary Math. 606, CRM Proceedings, Women in Numbers 2, Research Directions in Number Theory, 149–169 (2013)

  • Bobrowski, O., Skraba, P.: Homological percolation and the Euler characteristic. Phys. Rev. E 101(3), 032304 (2020a)

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  • Bobrowski, O., Skraba, P.: Homological percolation: The formation of giant \(k\)-cycles. https://arxiv.org/abs/2005.14011. (2020b)

  • Bolker, E.D.: Simplicial geometry and transportation polytopes. Trans. A. M. S. 217, 121–142 (1976)

    MathSciNet  Google Scholar 

  • Boyd, D.: Mahler’s measure and special values of \(L\)-functions. Exp. Math. 7, 37–82 (1998)

    Article  MathSciNet  Google Scholar 

  • Burton, R., Pemantle, R.: Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab. 21, 1329–1371 (1993)

    Article  MathSciNet  Google Scholar 

  • Catanzaro, M.J., Chernyak, V.Y., Klein, J.R.: Kirchhoff’s theorems in higher dimensions and Reidemeister torsion. Homol. Homotopy Appl. 17, 165–189 (2015)

    Article  MathSciNet  Google Scholar 

  • Dhar, D.: Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64, 1613–1616 (1990)

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  • Dhar, D.: The abelian sandpiles and related models. Phys. A 263, 4–25 (1999)

    Article  Google Scholar 

  • Duncan, P., Kahle, M., Schweinhart, B.: Homological percolation on a torus: plaquettes and permutohedra. https://arxiv.org/abs/2011.11903. (2020)

  • Duval, A., Klivans, C.J., Martin, J.L.: Simplicial matrix-tree theorems. Trans. Am. Math. Soc. 361, 6073–6114 (2009)

    Article  MathSciNet  Google Scholar 

  • Duval, A., Klivans, C.J., Martin, J.L.: Simplicial and cellular trees. Recent Trends Combinat. 713–752, IMA Vol. Math. Appl., 159, Springer, (2016)

  • Fisher, M.E.: On the dimer solution of planar Ising models. J. Math. Phys. 7, 1776–1781 (1966)

    Article  ADS  CAS  Google Scholar 

  • Grimmett, G.R., Holroyd, A.E.: Plaquettes, spheres, and entanglement. Electron. J. Proba. 15, 1415–1428 (2010)

    MathSciNet  Google Scholar 

  • Guttmann, A.J., Rogers, M.D.: Spanning tree generating functions and Mahler measures. J. Phys. A: Math. Theor. 45, 494001 (2012)

    Article  MathSciNet  Google Scholar 

  • Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  • Hino, M., Kanazawa, S.: Asymptotic behavior of lifetime sums for random simplicial complex process. J. Math. Soc. Jpn 71, 765–804 (2019)

    Article  Google Scholar 

  • Hiraoka, Y., Ochiai, H., Shirai, T.: Zeta functions of periodic cubical lattices and cyclomatic-like polynomials. Adv. Stud. Pure Math. 84, 93–121 (2020)

    Article  Google Scholar 

  • Hiraoka, Y., Shirai, T.: Tutte polynomials and random-cluster models in Bernoulli cell complexes. RIMS Kokyuroku Bessatsu B 59, 289–304 (2016)

    MathSciNet  Google Scholar 

  • Hiraoka, Y., Shirai, T.: Minimum spanning acycle and lifetime of Persistent homology in the Linial-Meshulam process. Random Struct. Algorithm. 51, 315–340 (2017)

    Article  MathSciNet  Google Scholar 

  • Hiraoka, Y., Tsunoda, K.: Limit theorems for random cubical homology. Discrete Comput. Geom. 60, 665–687 (2018)

    Article  MathSciNet  Google Scholar 

  • Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Zeros of Gaussian analytic functions and determinantal point processes. University Lecture Series, 51. American Mathematical Society, Providence, RI (2009)

  • Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational homology. Applied Mathematical Sciences 157. Springer, Berlin (2004)

  • Kalai, G.: Enumeration of \(Q\)-acyclic simplicial complexes. Israel J. Math. 45, 337–351 (1983)

    Article  MathSciNet  Google Scholar 

  • Kasteleyn, P.W.: The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice. Physica 27, 1209–1225 (1961)

    Article  ADS  Google Scholar 

  • Kurokawa, N., Ochiai, H.: Mahler measures via the crystalization. Comment. Math. Univ. St. Pauli 54, 121–137 (2005)

    MathSciNet  Google Scholar 

  • Linial, N., Meshulam, R.: Homological connectivity of random \(2\)-complexes. Combinatorica 26, 475–487 (2006)

    Article  MathSciNet  Google Scholar 

  • Linial, N., Peled, Y.: On the phase transition in random simplicial complexes. Ann. Math. 184, 745–773 (2016)

    Article  MathSciNet  Google Scholar 

  • Lind, D., Schmidt, K., Ward, T.: Mahler measure and entropy for commuting automorphisms of compact groups. Invent. Math. 101, 593–629 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  • Lyons, R.: Determinantal probability measures. Publ. Math. Inst. Hautes Études Sci. 98, 167–212 (2003)

    Article  MathSciNet  Google Scholar 

  • Lyons, R.: Asymptotic enumeration of spanning trees. Combin. Probab. Comput. 14, 491–522 (2005)

    Article  MathSciNet  Google Scholar 

  • Lyons, R.: Random complexes and \(\ell ^2\)-Betti numbers. J. Topol. Anal. 1(2), 153–175 (2009)

    Article  MathSciNet  Google Scholar 

  • Mahler, K.: On some inequalities for polynomials in several variables. J. Lond. Math. Soc. 37, 341–344 (1962)

    Article  MathSciNet  Google Scholar 

  • Majumdar, S.N., Dhar, D.: Equivalence between the Abelian sandpile model and the \(q \rightarrow 0\) limit of the Potts model. Phys. A 185, 129–145 (1992)

    Article  Google Scholar 

  • Meshulam, R., Wallach, N.: Homological connectivity of random \(k\)-dimensional complexes. Random Struct. Algorithm. 34, 408–417 (2009)

    Article  MathSciNet  Google Scholar 

  • Onsager, L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117–149 (1944)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  • Pemantle, R.: Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19, 1559–1574 (1991)

    Article  MathSciNet  Google Scholar 

  • Rodriguez Villegas, F.: Modular Mahler measures I, Topics in number theory (University Park, PA 1997), 17-48, Math. Appl., 467, Kluwer Acad. Publ. Dordrecht (1999)

  • Shrock, R., Wu, F.Y.: Spanning trees on graphs and lattices in \(d\) dimensions. J. Phys. A 33, 3881–3902 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  • Smyth, C.J.: On measures of polynomials in several variables. Bull. Austral. Math. Soc. 23, 49–63 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  • Schmidt, K., Verbitskiy, E.: Abelian sandpiles and the harmonic model. Commun. Math. Phys. 292(3), 721–759 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  • Shirai, T., Verbitskiy, E.: Solvable and algebraic systems on infinite ladder. Indag. Math. 27, 1162–1183 (2016)

    Article  MathSciNet  Google Scholar 

  • Shirai, T., Takahashi, Y.: Fermion process and Fredholm determinant. In: Proceedings of the Second ISAAC Congress, vol. I, 15–23, Kluwer (2000)

  • Shirai, T., Takahashi, Y.: Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes. J. Funct. Anal. 205(2), 414–463 (2003)

    Article  MathSciNet  Google Scholar 

  • Shirai, T., Takahashi, Y.: Random point fields associated with certain Fredholm determinants II Fermion shifts and their ergodic and Gibbs properties. Ann. Probab. 31(3), 1533–1564 (2003)

    Article  MathSciNet  Google Scholar 

  • Skraba, P., Thoppe, G., Yogeshwaran, D.: Randomly weighted \(d\)-complexes: minimal spanning acycles and persistence diagrams. Electron. J. Combin. 27, 47 (2020)

    Article  MathSciNet  Google Scholar 

  • Skraba, P., Yogeshwaran, D.: Central limit theorem for Euclidean minimal spanning acycles. arXiv:2205.12348v2

  • Soshnikov, A.: Determinantal random point fields. Russian Math. Surv. 55, 923–975 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  • Temperley, H.N.V.: The enumeration of graphs on large periodic lattices. In: Combinatorics: Proceedings of the British Combinatorial Conference, pp. 285–294. Inst. Math. Appl., Southend (1972)

  • Temperley, H.N.V., Enumeration of graphs on a large periodic lattice. In: Combinatorics (Proc. British Combinatorial Conf., Univ. Coll. Wales, Aberystwyth,: London Math. Soc. Lecture Note Ser., No. 13, London: Cambridge Univ. Press 1974, 155–159 (1973)

  • Temperley, H.N.V., Fisher, M.E.: Dimer problem in statistical mechanics – an exact result. Philos. Mag. 6, 1061–1063 (1961)

    Article  ADS  MathSciNet  Google Scholar 

  • Tzeng, W.-J., Wu, F.Y.: Spanning trees on hypercubic lattices and non-orientable surfaces. Appl. Math. Lett. 13(7), 19–25 (2000)

    Article  MathSciNet  Google Scholar 

  • Walter, M., Truemper, K.: Implementation of a unimodularity test. Math. Program. Ser. C 5, 57–73 (2013)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to express their gratitute to the anonymous referee for the careful reading and valuable suggestions that contributed to the improvement of this paper. This work was partially supported by JST CREST Mathematics 15656429, Y.H. was supported in part by JSPS KAKENHI Grant Numbers JP20H00119 and JP22H05107, T.S. was supported in part by JPSP KAKENHI Grant Numbers JP17K18740, JP18H01124, JP20K20884, JP22H05105, and JP23H01077.

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  1. Kyoto University Institute for Advanced Study, Kyoto University, Japan

    Yasuaki Hiraoka

  2. Institute of Mathematics for Industry, Kyushu University, Japan

    Tomoyuki Shirai

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  1. Yasuaki Hiraoka
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  2. Tomoyuki Shirai
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Y.H. and T.S. started this work together and wrote the main manuscript text. All authors reviewed the manuscript.

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Correspondence to Tomoyuki Shirai.

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Hiraoka, Y., Shirai, T. Torsion-weighted spanning acycle entropy in cubical lattices and Mahler measures. J Appl. and Comput. Topology (2024). https://doi.org/10.1007/s41468-024-00163-y

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