
SIGMA 19 (2023), 004, 11 pages arXiv:2208.11297
https://doi.org/10.3842/SIGMA.2023.004
Law of Large Numbers for Roots of Finite Free Multiplicative Convolution of Polynomials
Katsunori Fujie ^{a} and Yuki Ueda ^{b}
^{a)} Department of Mathematics, Hokkaido University, North 10 West 8, KitaKu, Sapporo, Hokkaido, 0600810, Japan
^{b)} Department of Mathematics, Hokkaido University of Education, Hokumoncho 9, Asahikawa, Hokkaido, 0708621, Japan
Received August 25, 2022, in final form January 09, 2023; Published online January 14, 2023
Abstract
We provide the law of large numbers for roots of finite free multiplicative convolution of polynomials which have only nonnegative real roots. Moreover, we study the empirical root distributions of limit polynomials obtained through the law of large numbers of finite free multiplicative convolution when their degree tends to infinity.
Key words: finite free probability; finite free multiplicative convolution; law of large numbers.
pdf (326 kb)
tex (13 kb)
References
 Arizmendi O., GarzaVargas J., Perales D., Finite free cumulants: multiplicative convolutions, genus expansion and infinitesimal distributions, arXiv:2108.08489.
 Bercovici H., Voiculescu D., Free convolution of measures with unbounded support, Indiana Univ. Math. J. 42 (1993), 733773.
 Haagerup U., Möller S., The law of large numbers for the free multiplicative convolution, in Operator Algebra and Dynamics,Springer Proc. Math. Stat., Vol. 58, Springer, Heidelberg, 2013, 157186, arXiv:1211.4457.
 Hardy G.H., Littlewood J.E., Pólya G., Inequalities, 2nd ed., Cambridge University Press, Cambridge, 1952.
 Lindsay J.M., Pata V., Some weak laws of large numbers in noncommutative probability, Math. Z. 226 (1997), 533543.
 Marcus A.W., Polynomial convolutions and (finite) free probability, arXiv:2108.07054.
 Marcus A.W., Spielman D.A., Srivastava N., Finite free convolutions of polynomials, Probab. Theory Related Fields 182 (2022), 807848, arXiv:1504.00350.
 Tucci G.H., Limits laws for geometric means of free random variables, Indiana Univ. Math. J. 59 (2010), 113, arXiv:0802.4226.
 Ueda Y., Maxconvolution semigroups and extreme values in limit theorems for the free multiplicative convolution, Bernoulli 27 (2021), 502531, arXiv:2003.05382.
 Voiculescu D., Multiplication of certain noncommuting random variables, J. Operator Theory 18 (1987), 223235.

