Skip to main content
SpringerLink
Log in

Stationary Determinantal Processes on \({\mathbb {Z}}^d\) with N Labeled Objects per Site, Part I: Basic Properties and Full Domination

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

We study a class of stationary determinantal processes on configurations of N labeled objects that may be present or absent at each site of \({\mathbb {Z}}^d\). Our processes, which include the uniform spanning forest as a principal example, arise from the block Toeplitz matrices of matrix-valued functions on the d-torus. We find the maximum level of uniform insertion tolerance for these processes, extending a result of Lyons and Steif from the \(N = 1\) case to \(N > 1\). We develop a method for conditioning determinantal processes in the general discrete setting to be as large as possible in a fixed set as an approach to determining uniform insertion tolerance. The method of conditioning on maximality developed here is used in a subsequent paper to study stochastic domination, strong domination and phase uniqueness for the same class of processes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Benjamini, I., Lyons, R., Peres, Y., Schramm, O.: Uniform spanning forests. Ann. Probab. 29(1), 1–65 (2001). https://doi.org/10.1214/aop/1008956321

    Article  MathSciNet  MATH  Google Scholar 

  2. Borodin, A., Okounkov, A., Olshanski, G.: Asymptotics of Plancherel measures for symmetric groups. J. Am. Math. Soc. 13, 481–515 (2000)

    Article  MathSciNet  Google Scholar 

  3. Borwein, J.M., Zucker, I.J.: Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind. IMA J. Numer. Anal. 12(4), 519–526 (1992). https://doi.org/10.1093/imanum/12.4.519

    Article  MathSciNet  MATH  Google Scholar 

  4. Bufetov, A.I., Qiu, Y.: Equivalence of Palm measures for determinantal point processes associated with Hilbert spaces of holomorphic functions. C. R. Math. Acad. Sci. Paris 353(6), 551–555 (2015). https://doi.org/10.1016/j.crma.2015.03.018

    Article  MathSciNet  MATH  Google Scholar 

  5. Burton, R., Pemantle, R.: Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab. 21(3), 1329–1371 (1993)

    Article  MathSciNet  Google Scholar 

  6. Ghosh, S.: Determinantal processes and completeness of random exponentials: the critical case. Probab. Theory Relat. Fields 163(3–4), 643–665 (2015). https://doi.org/10.1007/s00440-014-0601-9

    Article  MathSciNet  MATH  Google Scholar 

  7. Glasser, M.L., Zucker, I.J.: Extended Watson integrals for the cubic lattices. Proc. Nat. Acad. Sci. USA 74(5), 1800–1801 (1977)

    Article  MathSciNet  Google Scholar 

  8. Heicklen, D., Lyons, R.: Change intolerance in spanning forests. J. Theor. Probab. 16(1), 47–58 (2003). https://doi.org/10.1023/A:1022222319655

    Article  MathSciNet  MATH  Google Scholar 

  9. Helson, H., Lowdenslager, D.: Prediction theory and Fourier series in several variables. Acta Math. 99, 165–202 (1958). https://doi.org/10.1007/BF02392425

    Article  MathSciNet  MATH  Google Scholar 

  10. Helson, H., Lowdenslager, D.: Prediction theory and Fourier series in several variables. II. Acta Math. 106, 175–213 (1961). https://doi.org/10.1007/BF02545786

    Article  MathSciNet  MATH  Google Scholar 

  11. Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  12. Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Zeros of Gaussian Analytic Functions and Determinantal Point Processes. University Lecture Series, vol. 51. American Mathematical Society, Providence (2009). https://doi.org/10.1090/ulect/051

  13. Johansson, K.: Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. Math. (2) 153(1), 259–296 (2001). https://doi.org/10.2307/2661375

    Article  MathSciNet  MATH  Google Scholar 

  14. Kolmogoroff, A.: Interpolation und Extrapolation von stationären zufälligen Folgen. Bull. Acad. Sci. URSS Sér. Math. [Izvestia Akad. Nauk. SSSR] 5, 3–14 (1941)

    MathSciNet  MATH  Google Scholar 

  15. Kolmogoroff, A.N.: Stationary sequences in Hilbert’s space. Bolletin Moskovskogo Gosudarstvenogo Universiteta. Matematika 2, 40 (1941)

    Google Scholar 

  16. Liggett, T.M.: Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 276. Springer, New York (1985). https://doi.org/10.1007/978-1-4613-8542-4

  17. Lyons, R.: Determinantal probability measures. Publ. Math. Inst. Hautes Études Sci. 98, 167–212 (2003). https://doi.org/10.1007/s10240-003-0016-0

    Article  MathSciNet  MATH  Google Scholar 

  18. Lyons, R.: Determinantal probability: basic properties and conjectures. In: Proceedings of International Congress of Mathematicians 2014, Seoul, Korea vol. IV, pp. 137–161 (2014)

  19. Lyons, R., Peres, Y.: Probability on Trees and Networks. Cambridge University Press, New York (2016)

    Book  Google Scholar 

  20. Lyons, R., Steif, J.E.: Stationary determinantal processes: phase multiplicity, Bernoullicity, entropy, and domination. Duke Math. J. 120(3), 515–575 (2003). https://doi.org/10.1215/S0012-7094-03-12032-3

    Article  MathSciNet  MATH  Google Scholar 

  21. Lyons, R., Thom, A.: Invariant coupling of determinantal measures on sofic groups. Ergodic Theory Dyn. Syst. 36(2), 574–607 (2016). https://doi.org/10.1017/etds.2014.70

    Article  MathSciNet  MATH  Google Scholar 

  22. Pemantle, R.: Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19(4), 1559–1574 (1991)

    Article  MathSciNet  Google Scholar 

  23. Pemantle, R.: Towards a theory of negative dependence. J. Math. Phys. 41(3), 1371–1390 (2000). https://doi.org/10.1063/1.533200. (Probabilistic techniques in equilibrium and nonequilibrium statistical physics)

    Article  MathSciNet  MATH  Google Scholar 

  24. 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). https://doi.org/10.1016/S0022-1236(03)00171-X

    Article  MathSciNet  MATH  Google Scholar 

  25. Soshnikov, A.: Determinantal random point fields. Russ. Math. Surv. 55(5), 923–975 (2000)

    Article  MathSciNet  Google Scholar 

  26. Szegő, G.: Beiträge zur Theorie der Toeplitzschen Formen. Math. Z. 6(3–4), 167–202 (1920)

    Article  MathSciNet  Google Scholar 

  27. Watson, G.N.: Three triple integrals. Q. J. Math. Oxf. Ser. 10, 266–276 (1939)

    Article  MathSciNet  Google Scholar 

  28. Weiss, G.H.: Aspects and Applications of the Random Walk. Random Materials and Processes. North-Holland Publishing Co., Amsterdam (1994)

    MATH  Google Scholar 

  29. Zucker, I.J.: 70+ years of the Watson integrals. J. Stat. Phys. 145(3), 591–612 (2011). https://doi.org/10.1007/s10955-011-0273-0

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I thank Russell Lyons for recommending this project and for his guidance. I thank Jeff Steif for his feedback on a preliminary version of this paper. This work was partially supported by the National Science Foundation (NSF) under the Grants DMS-1007244 and DMS-1612363.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indiana University, Rawles Hall, Bloomington, IN, 47405, USA

    Justin Cyr

Authors
  1. Justin Cyr
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Justin Cyr.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cyr, J. Stationary Determinantal Processes on \({\mathbb {Z}}^d\) with N Labeled Objects per Site, Part I: Basic Properties and Full Domination. J Theor Probab 34, 1321–1365 (2021). https://doi.org/10.1007/s10959-020-01062-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-020-01062-5

Keywords

Mathematics Subject Classification (2020)

Search

Navigation