Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 17 (2021), 104, 22 pages      arXiv:2106.14497      https://doi.org/10.3842/SIGMA.2021.104

Scaling Limits for the Gibbs States on Distance-Regular Graphs with Classical Parameters

Masoumeh Koohestani a, Nobuaki Obata b and Hajime Tanaka b
a)  Department of Mathematics, K.N. Toosi University of Technology, Tehran 16765-3381, Iran
b)  Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan

Received July 19, 2021, in final form November 22, 2021; Published online November 26, 2021

Abstract
We determine the possible scaling limits in the quantum central limit theorem with respect to the Gibbs state, for a growing distance-regular graph that has so-called classical parameters with base unequal to one. We also describe explicitly the corresponding weak limits of the normalized spectral distribution of the adjacency matrix. We demonstrate our results with the known infinite families of distance-regular graphs having classical parameters and with unbounded diameter.

Key words: quantum probability; quantum central limit theorem; distance-regular graph; Gibbs state; classical parameters.

pdf (565 kb)   tex (33 kb)  

References

  1. Accardi L., Lu Y.G., Volovich I., Quantum theory and its stochastic limit, Springer-Verlag, Berlin, 2002.
  2. Askey R., Ismail M., Recurrence relations, continued fractions, and orthogonal polynomials, Mem. Amer. Math. Soc. 49 (1984), iv+108 pages.
  3. Atakishiyev N.M., Klimyk U., Duality of $q$-polynomials, orthogonal on countable sets of points, Electron. Trans. Numer. Anal. 24 (2006), 108-180, arXiv:math.CA/0411249.
  4. Bang S., Dubickas A., Koolen J.H., Moulton V., There are only finitely many distance-regular graphs of fixed valency greater than two, Adv. Math. 269 (2015), 1-55, arXiv:0909.5253.
  5. Bannai E., Ito T., Algebraic combinatorics. I. Association schemes, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984.
  6. Biggs N., Algebraic graph theory, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993.
  7. Bogachev V.I., Measure theory, Vols. I, II, Springer-Verlag, Berlin, 2007.
  8. Brouwer A.E., Cohen A.M., Neumaier A., Distance-regular graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 18, Springer-Verlag, Berlin, 1989.
  9. Van Dam E.R., Koolen J.H., A new family of distance-regular graphs with unbounded diameter, Invent. Math. 162 (2005), 189-193.
  10. Van Dam E.R., Koolen J.H., Tanaka H., Distance-regular graphs, Electron. J. Combin. (2016), #DS22, 156 pages, arXiv:1410.6294.
  11. Fu A.M., A combinatorial proof of the Lebesgue identity, Discrete Math. 308 (2008), 2611-2613.
  12. Gudder S.P., Quantum probability, Probability and Mathematical Statistics, Academic Press, Inc., Boston, MA, 1988.
  13. Hashimoto Y., Quantum decomposition in discrete groups and interacting Fock spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2001), 277-287.
  14. Hashimoto Y., Hora A., Obata N., Central limit theorems for large graphs: method of quantum decomposition, J. Math. Phys. 44 (2003), 71-88.
  15. Hashimoto Y., Obata N., Tabei N., A quantum aspect of asymptotic spectral analysis of large Hamming graphs, in Quantum Information, III (Nagoya, 2000), Editors T. Hida, K. Saitô, World Sci. Publ., River Edge, NJ, 2001, 45-57.
  16. Hora A., Central limit theorems and asymptotic spectral analysis on large graphs, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998), 221-246.
  17. Hora A., Gibbs state on a distance-regular graph and its application to a scaling limit of the spectral distributions of discrete Laplacians, Probab. Theory Related Fields 118 (2000), 115-130.
  18. Hora A., Asymptotic spectral analysis on the Johnson graphs in infinite degree and zero temperature limit, Interdiscip. Inform. Sci. 10 (2004), 1-10.
  19. Hora A., Obata N., Quantum probability and spectral analysis of graphs, Theoretical and Mathematical Physics, Springer, Berlin, 2007.
  20. Hora A., Obata N., Asymptotic spectral analysis of growing regular graphs, Trans. Amer. Math. Soc. 360 (2008), 899-923.
  21. Ismail M.E.H., Masson D.R., $q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals, Trans. Amer. Math. Soc. 346 (1994), 63-116.
  22. Ivanov A.A., Shpectorov S.V., The association schemes of dual polar spaces of type ${}^2A_{2d-1}(p^f)$ are characterized by their parameters if $d\geq 3$, Linear Algebra Appl. 114/115 (1989), 133-139.
  23. Kelley J.L., General topology, Graduate Texts in Mathematics, Vol. 27, Springer-Verlag, New York - Berlin, 1975.
  24. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
  25. Koekoek R., Swarttouw R.F., The Askey scheme of hypergeometric orthogonal polynomials and its $q$-analog, Report 98-17, Delft University of Technology, 1998, available at http://aw.twi.tudelft.nl/ koekoek/askey.html.
  26. Leonard D.A., Orthogonal polynomials, duality and association schemes, SIAM J. Math. Anal. 13 (1982), 656-663.
  27. Meyer P.-A., Quantum probability for probabilists, Lecture Notes in Math., Vol. 1538, Springer-Verlag, Berlin, 1993.
  28. Nica A., Speicher R., Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, Vol. 335, Cambridge University Press, Cambridge, 2006.
  29. Obata N., Spectral analysis of growing graphs. A quantum probability point of view, SpringerBriefs in Mathematical Physics, Vol. 20, Springer, Singapore, 2017.
  30. Pan Y.-J., Weng C.-W., A note on triangle-free distance-regular graphs with $a_2\not=0$, J. Combin. Theory Ser. B 99 (2009), 266-270.
  31. Parthasarathy K.R., An introduction to quantum stochastic calculus, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1992.
  32. Tanaka H., Vertex subsets with minimal width and dual width in $Q$-polynomial distance-regular graphs, Electron. J. Combin. 18 (2011), #P167, 32 pages, arXiv:1011.2000.
  33. Terwilliger P., $Q$-polynomial distance-regular graphs containing a singular line with cardinality at least $3$, unpublished manuscript.
  34. Terwilliger P., The subconstituent algebra of an association scheme. I, J. Algebraic Combin. 1 (1992), 363-388.
  35. Terwilliger P., The subconstituent algebra of an association scheme. II, J. Algebraic Combin. 2 (1993), 73-103.
  36. Terwilliger P., The subconstituent algebra of an association scheme. III, J. Algebraic Combin. 2 (1993), 177-210.
  37. Terwilliger P., Kite-free distance-regular graphs, European J. Combin. 16 (1995), 405-414.
  38. Terwilliger P., Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear Algebra Appl. 330 (2001), 149-203, arXiv:math.RA/0406555.
  39. Terwilliger P., Leonard pairs and the $q$-Racah polynomials, Linear Algebra Appl. 387 (2004), 235-276, arXiv:math.QA/0306301.
  40. Terwilliger P., Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array, Des. Codes Cryptogr. 34 (2005), 307-332, arXiv:math.RA/0306291.
  41. Terwilliger P., Zitnik A., The quantum adjacency algebra and subconstituent algebra of a graph, J. Combin. Theory Ser. A 166 (2019), 297-314, arXiv:1710.06011.
  42. Voiculescu D.V., Dykema K.J., Nica A., Free random variables, CRM Monograph Series, Vol. 1, Amer. Math. Soc., Providence, RI, 1992.
  43. Weng C.-W., $D$-bounded distance-regular graphs, European J. Combin. 18 (1997), 211-229.
  44. Weng C.-W., Classical distance-regular graphs of negative type, J. Combin. Theory Ser. B 76 (1999), 93-116.

Previous article  Next article  Contents of Volume 17 (2021)