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


SIGMA 17 (2021), 008, 35 pages      arXiv:2005.02837      https://doi.org/10.3842/SIGMA.2021.008

Pfaffian Point Processes from Free Fermion Algebras: Perfectness and Conditional Measures

Shinji Koshida
Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo, Tokyo 112-8551, Japan

Received July 23, 2020, in final form January 16, 2021; Published online January 26, 2021

Abstract
The analogy between determinantal point processes (DPPs) and free fermionic calculi is well-known. We point out that, from the perspective of free fermionic algebras, Pfaffian point processes (PfPPs) naturally emerge, and show that a positive contraction acting on a ''doubled'' one-particle space with an additional structure defines a unique PfPP. Recently, Olshanski inverted the direction from free fermions to DPPs, proposed a scheme to construct a fermionic state from a quasi-invariant probability measure, and introduced the notion of perfectness of a probability measure. We propose a method to check the perfectness and show that Schur measures are perfect as long as they are quasi-invariant under the action of the symmetric group. We also study conditional measures for PfPPs associated with projection operators. Consequently, we show that the conditional measures are again PfPPs associated with projection operators onto subspaces explicitly described.

Key words: Pfaffian point process; determinantal point process; CAR algebra; quasi-free state.

pdf (597 kb)   tex (43 kb)  

References

  1. Aissen M., Schoenberg I.J., Whitney A.M., On the generating functions of totally positive sequences. I, J. Anal. Math. 2 (1952), 93-103.
  2. Araki H., On quasifree states of ${\rm CAR}$ and Bogoliubov automorphisms, Publ. Res. Inst. Math. Sci. 6 (1970), 385-442.
  3. Binnenhei C., Implementation of endomorphisms of the CAR algebra, Rev. Math. Phys. 7 (1995), 833-869, arXiv:physics/9701009.
  4. Borodin A., Multiplicative central measures on the Schur graph, J. Math. Sci. 96 (1999), 3472-3477.
  5. Borodin A., Determinantal point processes, in The Oxford Handbook of Random Matrix Theory, Oxford University Press, Oxford, 2011, 231-249, arXiv:0911.1153.
  6. Borodin A., Olshanski G., Point processes and the infinite symmetric group. Part II: Higher correlation functions, arXiv:math.RT/9804087.
  7. Borodin A., Olshanski G., Point processes and the infinite symmetric group. Part III: Fermion point processes, arXiv:math.RT/9804088.
  8. Borodin A., Olshanski G., Point processes and the infinite symmetric group. Part IV: Matrix Whittaker kernel, arXiv:math.RT/9810013.
  9. Borodin A., Olshanski G., Point processes and the infinite symmetric group, Math. Res. Lett. 5 (1998), 799-816, arXiv:math.RT/9810015.
  10. Borodin A., Rains E.M., Eynard-Mehta theorem, Schur process, and their Pfaffian analogs, J. Stat. Phys. 121 (2005), 291-317, arXiv:math-ph/0409059.
  11. Bratteli O., Robinson D.W., Operator algebras and quantum statistical mechanics. 1. $C^*$- and $W^\ast$-algebras, symmetry groups, decomposition of states, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, New York, 1987.
  12. Bratteli O., Robinson D.W., Operator algebras and quantum statistical mechanics. 2. Equilibrium states. Models in quantum statistical mechanics, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997.
  13. Bufetov A.I., Cunden F.D., Qiu Y., $J$-Hermitian determinantal point processes: balanced rigidity and balanced Palm equivalence arXiv:1912.10743.
  14. Bufetov A.I., Olshanski G., A hierarchy of Palm measures for determinantal point processes with gamma kernels, arXiv:1904.13371.
  15. Edrei A., On the generating functions of totally positive sequences. II, J. Anal. Math. 2 (1952), 104-109.
  16. Ferrari P.L., Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues, Comm. Math. Phys. 252 (2004), 77-109, arXiv:math-ph/0402053.
  17. Hough J.B., Krishnapur M., Peres Y., Virág B., Determinantal processes and independence, Probab. Surv. 3 (2006), 206-229, arXiv:math.PR/0503110.
  18. Ivanov V.N., Dimensions of skew-shifted Young diagrams and projective characters of the infinite symmetric group, J. Math. Sci. 96 (1999), 3517-3530.
  19. Kargin V., On Pfaffian random point fields, J. Stat. Phys. 154 (2014), 681-704, arXiv:1210.6603.
  20. Kassel A., Lévy T., Determinantal probability measures on Grassmannians, arXiv:1910.06312.
  21. Katori M., Tanemura H., Infinite systems of noncolliding generalized meanders and Riemann-Liouville differintegrals, Probab. Theory Related Fields 138 (2007), 113-156, arXiv:math.PR/0506187.
  22. König W., Orthogonal polynomial ensembles in probability theory, Probab. Surv. 2 (2005), 385-447, arXiv:math.PR/0403090.
  23. Koshmanenko V., Singular quadratic forms in perturbation theory, Mathematics and its Applications, Vol. 474, Kluwer Academic Publishers, Dordrecht, 1999.
  24. Lenard A., States of classical statistical mechanical systems of infinitely many particles. I, Arch. Rational Mech. Anal. 59 (1975), 219-239.
  25. Lenard A., States of classical statistical mechanical systems of infinitely many particles. II. Characterization of correlation measures, Arch. Rational Mech. Anal. 59 (1975), 240-256.
  26. Lyons R., Determinantal probability measures, Publ. Math. Inst. Hautes Études Sci. (2003), 167-212, arXiv:math.PR/0204325.
  27. Lytvynov E., Fermion and boson random point processes as particle distributions of infinite free Fermi and Bose gases of finite density, Rev. Math. Phys. 14 (2002), 1073-1098, arXiv:math-ph/0112006.
  28. Lytvynov E., Mei L., On the correlation measure of a family of commuting Hermitian operators with applications to particle densities of the quasi-free representations of the CAR and CCR, J. Funct. Anal. 245 (2007), 62-88, arXiv:math.PR/0608334.
  29. Macdonald I.G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979.
  30. Matsumoto S., $\alpha$-Pfaffian, Pfaffian point process and shifted Schur measure, Linear Algebra Appl. 403 (2005), 369-398, arXiv:math.CO/0411277.
  31. Matsumoto S., Correlation functions of the shifted Schur measure, J. Math. Soc. Japan 57 (2005), 619-637, arXiv:math.CO/0312373.
  32. Matsumoto S., Shirai T., Correlation functions for zeros of a Gaussian power series and Pfaffians, Electron. J. Probab. 18 (2013), no. 49, 18 pages, arXiv:1212.6108.
  33. Nagao T., Pfaffian expressions for random matrix correlation functions, J. Stat. Phys. 129 (2007), 1137-1158, arXiv:0708.2036.
  34. Nazarov M.L., Factor representations of the infinite spin-symmetric group, Russian Math. Surveys 43 (1988), 229-230.
  35. Okounkov A., Infinite wedge and random partitions, Selecta Math. (N.S.) 7 (2001), 57-81, arXiv:math.RT/9907127.
  36. Olshanski G., Point processes and the infinite symmetric group. Part V: Analysis of the matrix Whittaker kernel, arXiv:math.RT/9810014.
  37. Olshanski G., Point processes related to the infinite symmetric group, in The Orbit Method in Geometry and Physics (Marseille, 2000), Progr. Math., Vol. 213, Birkhäuser Boston, Boston, MA, 2003, 349-393.
  38. Olshanski G., The quasi-invariance property for the Gamma kernel determinantal measure, Adv. Math. 226 (2011), 2305-2350, arXiv:0910.0130.
  39. Olshanski G., Determinantal point processes and fermion quasifree states, Comm. Math. Phys. 378 (2020), 507-555, arXiv:2002.10723.
  40. Petrov L., Random strict partitions and determinantal point processes, Electron. Commun. Probab. 15 (2010), 162-175, arXiv:1002.2714.
  41. Petrov L., Pfaffian stochastic dynamics of strict partitions, Electron. J. Probab. 16 (2011), no. 82, 2246-2295, arXiv:1011.3329.
  42. Powers R.T., Størmer E., Free states of the canonical anticommutation relations, Comm. Math. Phys. 16 (1970), 1-33.
  43. Rains E.M., Correlation functions for symmetrized increasing subsequences, arXiv:math.CO/0006097.
  44. Rédei M., Summers S.J., Quantum probability theory, Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Modern Phys. 38 (2007), 390-417, arXiv:quant-ph/0601158.
  45. Ruijsenaars S.N.M., On Bogoliubov transformations. II. The general case, Ann. Physics 116 (1978), 105-134.
  46. Shale D., Stinespring W.F., Spinor representations of infinite orthogonal groups, J. Math. Mech. 14 (1965), 315-322.
  47. Shirai T., Takahashi Y., Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes, J. Funct. Anal. 205 (2003), 414-463.
  48. Shirai T., Takahashi Y., Random point fields associated with certain Fredholm determinants. II. Fermion shifts and their ergodic and Gibbs properties, Ann. Probab. 31 (2003), 1533-1564.
  49. Soshnikov A., Determinantal random point fields, Russian Math. Surveys 55 (2000), 923-975.
  50. Spohn H., Interacting Brownian particles: a study of Dyson's model, in Hydrodynamic Behavior and Interacting Particle Systems (Minneapolis, Minn., 1986), IMA Vol. Math. Appl., Vol. 9, Springer, New York, 1987, 151-179.
  51. Takesaki M., Theory of operator algebras. I, Springer-Verlag, New York - Heidelberg, 1979.
  52. Thoma E., Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe, Math. Z. 85 (1964), 40-61.
  53. Tracy C.A., Widom H., A limit theorem for shifted Schur measures, Duke Math. J. 123 (2004), 171-208, arXiv:math.PR/0210255.
  54. Vuletić M., The shifted Schur process and asymptotics of large random strict plane partitions, Int. Math. Res. Not. 2007 (2007), rnm043, 53 pages, arXiv:math-ph/0702068.
  55. Wang Z.-L., Li S.-H., BKP hierarchy and Pfaffian point process, Nuclear Phys. B 939 (2019), 447-464, arXiv:1807.02259.

Previous article  Next article  Contents of Volume 17 (2021)