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Boolean cumulants and subordination in free probability

Franz Lehner

Institute of Discrete Mathematics, Graz University of Technology, Steyrergasse 30, A-8010 Graz, Austria

E-mail Address: lehner@math.tu-graz.ac.at

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and

Kamil Szpojankowski

https://orcid.org/0000-0003-0926-1285

Wydział Matematyki i Nauk Informacyjnych, Politechnika Warszawska ul. Koszykowa 75, 00-662 Warszawa, Poland

E-mail Address: k.szpojankowski@mini.pw.edu.pl

Corresponding author.

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https://doi.org/10.1142/S2010326321500362Cited by:1 (Source: Crossref)

Abstract

Subordination is the basis of the analytic approach to free additive and multiplicative convolution. We extend this approach to a more general setting and prove that the conditional expectation $𝔼_{φ} [{(z - X - f (X) Y f^{*} (X))}^{- 1} | X]$ for free random variables $X, Y$ and a Borel function $f$ is a resolvent again. This result allows the explicit calculation of the distribution of noncommutative polynomials of the form $X + f (X) Y f^{*} (X)$ . The main tool is a new combinatorial formula for conditional expectations in terms of Boolean cumulants and a corresponding analytic formula for conditional expectations of resolvents, generalizing subordination formulas for both additive and multiplicative free convolutions. In the final section, we illustrate the results with step by step explicit computations and an exposition of all necessary ingredients.

Supported by the Austrian Federal Ministry of Education, Science and Research and the Polish Ministry of Science and Higher Education, grants $N^{os}$ PL 08/2016 and PL 06/2018. KSz: research partially supported by NCN grant 2016/23/D/ST1/01077.

Keywords:

AMSC: 46L54, 60B20

References

1. O. Arizmendi, T. Hasebe, F. Lehner and C. Vargas . Relations between cumulants in noncommutative probability, Adv. Math. 282 (2015) 56–92. Crossref, Google Scholar
2. S. T. Belinschi and H. Bercovici , Partially defined semigroups relative to multiplicative free convolution, Int. Math. Res. Not. (2) (2005) 65–101. Crossref, Google Scholar
3. S. T. Belinschi and H. Bercovici , A new approach to subordination results in free probability, J. Anal. Math. 101 (2007) 357–365. Crossref, Google Scholar
4. S. T. Belinschi, T. Mai and R. Speicher , Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem, J. Reine Angew. Math. 732 (2017) 21–53. Crossref, Google Scholar
5. S. T. Belinschi and A. Nica , $η$ -series and a Boolean Bercovici--Pata bijection for bounded $k$ -tuples, Adv. Math. 217(1) (2008) 1–41. Crossref, Google Scholar
6. S. T. Belinschi, R. Speicher, J. Treilhard and C. Vargas , Operator-valued free multiplicative convolution: analytic subordination theory and applications to random matrix theory, Int. Math. Res. Not. (14) (2015) 5933–5958. Crossref, Google Scholar
7. H. Bercovici and D. Voiculescu , Free convolution of measures with unbounded support, Indiana Univ. Math. J. 42(3) (1993) 733–773. Crossref, Google Scholar
8. P. Biane , Processes with free increments, Math. Z. 227(1) (1998) 143–174. Crossref, Google Scholar
9. A. Bostan, F. Chyzak, B. Salvy, G. Lecerf and E. Schost , Differential equations for algebraic functions, in ISSAC 2007 (ACM, New York, 2007), pp. 25–32. Crossref, Google Scholar
10. E. Brieskorn and H. Knörrer , Plane Algebraic Curves (Birkhäuser Verlag, Basel, 1986). Crossref, Google Scholar
11. G. Cébron , Free convolution operators and free Hall transform, J. Funct. Anal. 265(11) (2013) 2645–2708. Crossref, Google Scholar
12. J. Cockle , On transcendental and algebraic solution, Philos. Mag. XXI(CXLI) (1861) 379–383. Crossref, Google Scholar
13. J. B. Conway , A Course in Operator Theory, Graduate Studies in Mathematics, Vol. 21 (American Mathematical Society, Providence, RI, 2000). Google Scholar
14. D. A. Cox, J. Little and D. O’Shea , Using Algebraic Geometry, 2nd edn. Graduate Texts in Mathematics (Springer, New York, 2005). Google Scholar
15. D. A. Cox, J. Little and D. O’Shea , Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Undergraduate Texts in Mathematics, 4th edn. (Springer, Cham, 2015). Crossref, Google Scholar
16. N. Dunford and J. T. Schwartz , Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7 (Interscience Publishers, Inc., New York, 1958), with the assistance of W. G. Bade and R. G. Bartle . Google Scholar
17. K. Ebrahimi-Fard and F. Patras , Cumulants, free cumulants and half-shuffles, Proc. R. Soc. A. 471(2176) (2015). Crossref, Google Scholar
18. D. Eisenbud , Commutative Algebra, Graduate Texts in Mathematics, Vol. 150 (Springer-Verlag, New York, 1995). Crossref, Google Scholar
19. M. Fevrier, M. Mastnak, A. Nica and K. Szpojankowski , Using Boolean cumulants to study multiplication and anticommutators of free random variables, Trans. Amer. Math. Soc. 373 (2020) 7167–7205. Crossref, Google Scholar
20. U. Haagerup , On Voiculescu’s $R$ - and $S$ -transforms for free non-commuting random variables, in Free Probability Theory, Fields Institute Communication, Vol. 12 (American Mathematical Society, Providence, RI, 1997), pp. 127–148. Google Scholar
21. P. R. Halmos , A Hilbert Space Problem Book, 2nd edn., Graduate Texts in Mathematics, Vol. 19. (Springer-Verlag, New York, 1982). Crossref, Google Scholar
22. D. Jekel and W. Liu, An operad of non-commutative independences defined by trees, preprint (2019), arXiv:1901.09158 Google Scholar
23. M. Kauers and P. Paule , The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates, Texts and Monographs in Symbolic Computation (Springer, New York, 2011). Crossref, Google Scholar
24. F. Lehner , Free cumulants and enumeration of connected partitions, European J. Combin. 23(8) (2002) 1025–1031. Crossref, Google Scholar
25. J. A. Mingo and R. Speicher , Free Probability and Random Matrices, Fields Institute Monographs, Vol. 35 (Springer, New York, 2017). Crossref, Google Scholar
26. A. Nica and R. Speicher , Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series, Vol. 335 (Cambridge University Press, Cambridge, 2006). Crossref, Google Scholar
27. B. Sturmfels , Solving Systems of Polynomial Equations, CBMS Regional Conference Series in Mathematics, Vol. 97 (American Mathematical Society, Providence, RI, 2002). Crossref, Google Scholar
28. K. Szpojankowski and J. Wesołowski , Conditional expectations through boolean cumulants and subordination towards a better understanding of the lukacs property in free probability, ALEA, Lat. Am. J. Probab. Math. Stat. 17 (2020) 253–272. Crossref, Google Scholar
29. M. Takesaki , Theory of Operator Algebras. I, Encyclopaedia of Mathematical Sciences, Vol. 124 (Springer-Verlag, Berlin, 2002). Google Scholar
30. A. E. Taylor and D. C. Lay , Introduction to Functional Analysis, 2nd edn. (John Wiley & Sons, New York, 1980). Google Scholar
31. V. Vasilchuk , Asymptotic distribution of the spectrum of symmetrically deformed unitary invariant random matrix ensemble, in Proc. Int. Conf. Days on Diffraction (IEEE, 2018), pp. 288–293. Crossref, Google Scholar
32. D. Voiculescu , Multiplication of certain noncommuting random variables, J. Operator Theory 18(2) (1987) 223–235. Google Scholar
33. D. Voiculescu , The analogues of entropy and of Fisher’s information measure in free probability theory. I, Comm. Math. Phys. 155(1) (1993) 71–92. Crossref, Google Scholar
34. D. Voiculescu , The coalgebra of the free difference quotient and free probability, Internat. Math. Res. Not. (2) (2000) 79–106. Crossref, Google Scholar
35. D. V. Voiculescu , Addition of certain noncommuting random variables, J. Funct. Anal. 66(3) (1986) 323–346. Crossref, Google Scholar
36. D. V. Voiculescu, K. J. Dykema and A. Nica , Free Random Variables, CRM Monograph Series, Vol. 1 (American Mathematical Society, Providence, RI, 1992). Crossref, Google Scholar
37. R. J. Walker , Algebraic Curves. Princeton Mathematical Series, Vol. 13 (Princeton University Press, Princeton, N. J., 1950). Google Scholar
38. W. Woess , Nearest neighbour random walks on free products of discrete groups, Boll. Un. Mat. Ital. B (6), 5(3) (1986) 961–982. Google Scholar