Infinite 𝑝-adic random matrices and ergodic decomposition of 𝑝-adic Hua measures

Assiotis, Theodoros

doi:10.1090/tran/8526

Infinite $p$-adic random matrices and ergodic decomposition of $p$-adic Hua measures
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Trans. Amer. Math. Soc. 375 (2022), 1745-1766 Request permission

Abstract:

Neretin in [Izv. Ross. Akad. Nauk Ser. Mat. 77 (2013), pp. 95–108] constructed an analogue of the Hua measures on the infinite $p$-adic matrices $\mathrm {Mat}\left (\mathbb {N},\mathbb {Q}_p\right )$. Bufetov and Qiu in [Compos. Math. 153 (2017), pp. 2482–2533] classified the ergodic measures on $\mathrm {Mat}\left (\mathbb {N},\mathbb {Q}_p\right )$ that are invariant under the natural action of $\mathrm {GL}(\infty ,\mathbb {Z}_p)\times \mathrm {GL}(\infty ,\mathbb {Z}_p)$. In this paper we solve the problem of ergodic decomposition for the $p$-adic Hua measures introduced by Neretin. We prove that the probability measure governing the ergodic decomposition has an explicit expression which identifies it with a Hall-Littlewood measure on partitions. Our arguments involve certain Markov chains.

References

Sergio Albeverio and Witold Karwowski, Diffusion on $p$-adic numbers, Gaussian random fields (Nagoya, 1990) Ser. Probab. Statist., vol. 1, World Sci. Publ., River Edge, NJ, 1991, pp. 86–99. MR 1163602
Sergio Albeverio and Witold Karwowski, A random walk on $p$-adics—the generator and its spectrum, Stochastic Process. Appl. 53 (1994), no. 1, 1–22. MR 1290704, DOI 10.1016/0304-4149(94)90054-X
Theodoros Assiotis, Hua-Pickrell diffusions and Feller processes on the boundary of the graph of spectra, Ann. Inst. Henri Poincaré Probab. Stat. 56 (2020), no. 2, 1251–1283 (English, with English and French summaries). MR 4076783, DOI 10.1214/19-AIHP1001
Theodoros Assiotis, A matrix Bougerol identity and the Hua-Pickrell measures, Electron. Commun. Probab. 23 (2018), Paper No. 7, 11. MR 3771765, DOI 10.1214/18-ECP107
Theodoros Assiotis, Ergodic decomposition for inverse Wishart measures on infinite positive-definite matrices, SIGMA Symmetry Integrability Geom. Methods Appl. 15 (2019), Paper No. 067, 24. MR 4002268, DOI 10.3842/SIGMA.2019.067
T. Assiotis and J. Najnudel, The boundary of the orbital beta process, Moscow Mathematical Journal 21 (2021), 659–694., DOI 10.17323/1609-4514-2021-21-4-659-694
Alexei Borodin and Ivan Corwin, Macdonald processes, Probab. Theory Related Fields 158 (2014), no. 1-2, 225–400. MR 3152785, DOI 10.1007/s00440-013-0482-3
Alexei Borodin and Grigori Olshanski, Infinite random matrices and ergodic measures, Comm. Math. Phys. 223 (2001), no. 1, 87–123. MR 1860761, DOI 10.1007/s002200100529
Alexei Borodin and Grigori Olshanski, Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, Ann. of Math. (2) 161 (2005), no. 3, 1319–1422. MR 2180403, DOI 10.4007/annals.2005.161.1319
Alexei Borodin and Leonid Petrov, Nearest neighbor Markov dynamics on Macdonald processes, Adv. Math. 300 (2016), 71–155. MR 3534830, DOI 10.1016/j.aim.2016.03.016
Alexey Bufetov and Leonid Petrov, Law of large numbers for infinite random matrices over a finite field, Selecta Math. (N.S.) 21 (2015), no. 4, 1271–1338. MR 3397450, DOI 10.1007/s00029-015-0179-9
Alexey Bufetov and Leonid Petrov, Yang-Baxter field for spin Hall-Littlewood symmetric functions, Forum Math. Sigma 7 (2019), Paper No. e39, 70. MR 4031106, DOI 10.1017/fms.2019.36
A. I. Bufetov, Ergodic decomposition for measures quasi-invariant under Borel actions of inductively compact groups, Mat. Sb. 205 (2014), no. 2, 39–70 (Russian, with Russian summary); English transl., Sb. Math. 205 (2014), no. 1-2, 192–219. MR 3204667, DOI 10.1070/sm2014v205n02abeh004371
A. I. Bufetov, Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures. I. Construction of infinite determinantal measures, Izv. Ross. Akad. Nauk Ser. Mat. 79 (2015), no. 6, 18–64 (Russian, with Russian summary); English transl., Izv. Math. 79 (2015), no. 6, 1111–1156. MR 3438464, DOI 10.4213/im8383
A. I. Bufetov, Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures. II. Convergence of determinantal measures, Izv. Ross. Akad. Nauk Ser. Mat. 80 (2016), no. 2, 16–32 (Russian, with Russian summary); English transl., Izv. Math. 80 (2016), no. 2, 299–315. MR 3507376, DOI 10.4213/im8384
A. I. Bufetov, Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures III. The infinite Bessel process as the limit of radial parts of finite-dimensional projections of infinite Pickrell measures, Izv. Ross. Akad. Nauk Ser. Mat. 80 (2016), no. 6, 43–64 (Russian, with Russian summary); English transl., Izv. Math. 80 (2016), no. 6, 1035–1056. MR 3588812, DOI 10.4213/im8385
Alexander I. Bufetov and Yanqi Qiu, The explicit formulae for scaling limits in the ergodic decomposition of infinite Pickrell measures, Ark. Mat. 54 (2016), no. 2, 403–435. MR 3546359, DOI 10.1007/s11512-016-0230-x
Alexander I. Bufetov and Yanqi Qiu, Ergodic measures on spaces of infinite matrices over non-Archimedean locally compact fields, Compos. Math. 153 (2017), no. 12, 2482–2533. MR 3705296, DOI 10.1112/S0010437X17007412
Cesar Cuenca, BC type $z$-measures and determinantal point processes, Adv. Math. 334 (2018), 1–80. MR 3828733, DOI 10.1016/j.aim.2018.06.003
Manon Defosseux, Orbit measures, random matrix theory and interlaced determinantal processes, Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 1, 209–249 (English, with English and French summaries). MR 2641777, DOI 10.1214/09-AIHP314
Mauro Del Muto and Alessandro Figà-Talamanca, Diffusion on locally compact ultrametric spaces, Expo. Math. 22 (2004), no. 3, 197–211. MR 2069670, DOI 10.1016/S0723-0869(04)80005-7
Steven N. Evans, Local properties of Lévy processes on a totally disconnected group, J. Theoret. Probab. 2 (1989), no. 2, 209–259. MR 987578, DOI 10.1007/BF01053411
Steven N. Evans, Local field Brownian motion, J. Theoret. Probab. 6 (1993), no. 4, 817–850. MR 1245397, DOI 10.1007/BF01049177
Steven N. Evans, Elementary divisors and determinants of random matrices over a local field, Stochastic Process. Appl. 102 (2002), no. 1, 89–102. MR 1934156, DOI 10.1016/S0304-4149(02)00187-4
Jason Fulman, A probabilistic approach toward conjugacy classes in the finite general linear and unitary groups, J. Algebra 212 (1999), no. 2, 557–590. MR 1676854, DOI 10.1006/jabr.1998.7659
Jason Fulman, The Rogers-Ramanujan identities, the finite general linear groups, and the Hall-Littlewood polynomials, Proc. Amer. Math. Soc. 128 (2000), no. 1, 17–25. MR 1657747, DOI 10.1090/S0002-9939-99-05292-2
Jason Fulman, A probabilistic proof of the Rogers-Ramanujan identities, Bull. London Math. Soc. 33 (2001), no. 4, 397–407. MR 1832551, DOI 10.1017/S0024609301008207
Jason Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51–85. MR 1864086, DOI 10.1090/S0273-0979-01-00920-X
Jason Fulman, Cohen-Lenstra heuristics and random matrix theory over finite fields, J. Group Theory 17 (2014), no. 4, 619–648. MR 3228936, DOI 10.1515/jgt-2014-0005
Vadim Gorin, Sergei Kerov, and Anatoly Vershik, Finite traces and representations of the group of infinite matrices over a finite field, Adv. Math. 254 (2014), 331–395. MR 3161102, DOI 10.1016/j.aim.2013.12.028
Vadim Gorin and Grigori Olshanski, A quantization of the harmonic analysis on the infinite-dimensional unitary group, J. Funct. Anal. 270 (2016), no. 1, 375–418. MR 3419766, DOI 10.1016/j.jfa.2015.06.006
L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, Translations of Mathematical Monographs, vol. 6, American Mathematical Society, Providence, R.I., 1979. Translated from the Russian, which was a translation of the Chinese original, by Leo Ebner and Adam Korányi; With a foreword by M. I. Graev; Reprint of the 1963 edition. MR 598469
Sergei Kerov, Grigori Olshanski, and Anatoly Vershik, Harmonic analysis on the infinite symmetric group, Invent. Math. 158 (2004), no. 3, 551–642. MR 2104794, DOI 10.1007/s00222-004-0381-4
Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw, Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder. MR 2656096, DOI 10.1007/978-3-642-05014-5
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
Yurii A. Neretin, Hua-type integrals over unitary groups and over projective limits of unitary groups, Duke Math. J. 114 (2002), no. 2, 239–266. MR 1920189, DOI 10.1215/S0012-7094-02-11423-9
Yu. A. Neretin, The beta function of the Bruhat-Tits building and the deformation of the space $l^2$ on the set of $p$-adic lattices, Mat. Sb. 194 (2003), no. 12, 31–62 (Russian, with Russian summary); English transl., Sb. Math. 194 (2003), no. 11-12, 1775–1805. MR 2052695, DOI 10.1070/SM2003v194n12ABEH000786
Yu. A. Neretin, Hua measures on the space of $p$-adic matrices and inverse limits of Grassmannians, Izv. Ross. Akad. Nauk Ser. Mat. 77 (2013), no. 5, 95–108 (Russian, with Russian summary); English transl., Izv. Math. 77 (2013), no. 5, 941–953. MR 3137196, DOI 10.1070/im2013v077n05abeh002665
Grigori Olshanski, The problem of harmonic analysis on the infinite-dimensional unitary group, J. Funct. Anal. 205 (2003), no. 2, 464–524. MR 2018416, DOI 10.1016/S0022-1236(02)00022-8
Grigori Olshanski and Anatoli Vershik, Ergodic unitarily invariant measures on the space of infinite Hermitian matrices, Contemporary mathematical physics, Amer. Math. Soc. Transl. Ser. 2, vol. 175, Amer. Math. Soc., Providence, RI, 1996, pp. 137–175. MR 1402920, DOI 10.1090/trans2/175/09
Doug Pickrell, Mackey analysis of infinite classical motion groups, Pacific J. Math. 150 (1991), no. 1, 139–166. MR 1120717, DOI 10.2140/pjm.1991.150.139
Doug Pickrell, Measures on infinite-dimensional Grassmann manifolds, J. Funct. Anal. 70 (1987), no. 2, 323–356. MR 874060, DOI 10.1016/0022-1236(87)90116-9
A. Soshnikov, Determinantal random point fields, Uspekhi Mat. Nauk 55 (2000), no. 5(335), 107–160 (Russian, with Russian summary); English transl., Russian Math. Surveys 55 (2000), no. 5, 923–975. MR 1799012, DOI 10.1070/rm2000v055n05ABEH000321
Yanqi Qiu, Infinite random matrices & ergodic decomposition of finite and infinite Hua-Pickrell measures, Adv. Math. 308 (2017), 1209–1268. MR 3600086, DOI 10.1016/j.aim.2017.01.003
A. M. Veršik, A description of invariant measures for actions of certain infinite-dimensional groups, Dokl. Akad. Nauk SSSR 218 (1974), 749–752 (Russian). MR 0372161
A. M. Vershik and S. V. Kerov, Asymptotic theory of the characters of a symmetric group, Funktsional. Anal. i Prilozhen. 15 (1981), no. 4, 15–27, 96 (Russian). MR 639197
A. M. Vershik and S. V. Kerov, Characters and factor-representations of the infinite unitary group, Dokl. Akad. Nauk SSSR 267 (1982), no. 2, 272–276 (Russian). MR 681202