Skip to main content

Limits and fluctuations of p-adic random matrix products


We show that singular numbers (also known as elementary divisors, invariant factors or Smith normal forms) of products and corners of random matrices over \({\mathbb {Q}}_p\) are governed by the Hall–Littlewood polynomials, in a structurally identical manner to the known relations between singular values of complex random matrices and Heckman–Opdam hypergeometric functions. This implies that the singular numbers of a product of corners of Haar-distributed elements of \(\mathrm {GL}_N({\mathbb {Z}}_p)\) form a discrete-time Markov chain distributed as a Hall–Littlewood process, with the number of matrices in the product playing the role of time. We give an exact sampling algorithm for the Hall–Littlewood processes which arise by relating them to an interacting particle system similar to PushTASEP. By analyzing the asymptotic behavior of this particle system, we show that the singular numbers of such products obey a law of large numbers and their fluctuations converge dynamically to independent Brownian motions. In the limit of large matrix size, we also show that the analogues of the Lyapunov exponents for matrix products have universal limits within this class of \(\mathrm {GL}_N({\mathbb {Z}}_p)\) corners.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4


  1. 1.

    For background on the p-adic numbers and matrix groups over them, see Sect. 3.

  2. 2.

    Here we view A as a map \({\mathbb {Z}}_p^n \rightarrow {\mathbb {Z}}_p^n\) and identify \(\lambda = {{\,\mathrm{SN}\,}}(A)\) with the abelian p-group \({{\,\mathrm{coker}\,}}(A) \cong \bigoplus _i {\mathbb {Z}}/p^{\lambda _i}{\mathbb {Z}}\).

  3. 3.

    To obtain \(0,-1,-2,\ldots \), take \(N \rightarrow \infty \) in (1.7) of [46] with scaling and use that the digamma function \(\psi (z)\) is asymptotic to \(\log (z)\).

  4. 4.

    These operations also make sense for signatures with some negative parts, but we refer to Sect. 2 for conventions on these.

  5. 5.

    These have one Plancherel specialization, while ours have both specializations geometric progressions in t.

  6. 6.

    Conjecturally, this is true if \(q,t \in [0,1)\) or if \(q,t \in (-1,0]\), see Matveev [50]. For our application we will only need the case \(q=0, t = 1/p \in (0,1)\), for which the nonnegativity follows from the interpretation of the structure coefficients in terms of the Hall algebra [47, Ch. III], or alternatively from Theorem 1.3 Part 3.

  7. 7.

    These are related by duality to the structure coefficients \(c_{\lambda ,\mu }^\nu \), see [47, Ch. VI], but we will not elaborate on this because we do not need it and due to our conventions with signatures it would be somewhat cumbersome to state.

  8. 8.

    Defined on \(\Omega _{\hat{x}_1} \times \Omega _{\hat{x}_2} \times \cdots \) via the Kolmogorov extension theorem.

  9. 9.

    Note that (5.23), (5.24) are the same as (5.21) and (5.22).

  10. 10.

    Many versions in print require that the increments \({\bar{Y}}_i(j)\) be identically distributed as well as independent, but a version for random walks with distinct independent increments may be obtained by specializing Donsker’s theorem for martingales [18, Thm. 3] to this case.


  1. 1.

    Achter, J.D.: The distribution of class groups of function fields. J. Pure Appl. Algebra 204(2), 316–333 (2006)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Ahn, A.: Fluctuations of $\beta $-Jacobi product processes (2019). arXiv preprint arXiv:1910.00743

  3. 3.

    Ahn, A., Strahov, E.: Product matrix processes with symplectic and orthogonal invariance via symmetric functions (2020). arXiv preprint arXiv:2007.11979

  4. 4.

    Ahn, A., Van Peski, R.: Lyapunov exponents for truncated unitary and Ginibre matrices (in preparation)

  5. 5.

    Akemann, G., Burda, Z., Kieburg, M.: Universal distribution of Lyapunov exponents for products of Ginibre matrices. J. Phys. A Math. Theor. 47(39), 395202 (2014)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Akemann, G., Burda, Z., Kieburg, M.: From integrable to chaotic systems: universal local statistics of Lyapunov exponents. EPL (Europhys. Lett.) 126(4), 40001 (2019)

    Article  Google Scholar 

  7. 7.

    Akemann, G., Burda, Z., Kieburg, M.: Universality of local spectral statistics of products of random matrices (2020). arXiv preprint arXiv:2008.11470

  8. 8.

    Akemann, G., Ipsen, J.R.: Recent exact and asymptotic results for products of independent random matrices (2015). arXiv preprint arXiv:1502.01667

  9. 9.

    Akemann, G., Ipsen, J.R., Kieburg, M.: Products of rectangular random matrices: singular values and progressive scattering. Phys. Rev. E 88(5), 052118 (2013)

    Article  Google Scholar 

  10. 10.

    Akemann, G., Kieburg, M., Wei, L.: Singular value correlation functions for products of Wishart random matrices. J. Phys. A Math. Theor. 46(27), 275205 (2013)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Assiotis, T.: Infinite p-adic random matrices and ergodic decomposition of p-adic Hua measures (2020). arXiv preprint arXiv:2009.04762

  12. 12.

    Bhargava, M., Kane, D.M., Lenstra, H.W., Poonen, B., Rains, E.: Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves. Camb. J. Math. 3(3), 275–321 (2015)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Borodin, A.M.: The law of large numbers and the central limit theorem for the Jordan normal form of large triangular matrices over a finite field. J. Math. Sci. 96(5), 3455–3471 (1999)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Borodin, A., Corwin, I.: Macdonald processes. Probab. Theory Relat. Fields 158(1–2), 225–400 (2014)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Borodin, A., Gorin, V.: General $\beta $-Jacobi corners process and the Gaussian free field. Commun. Pure Appl. Math. 68(10), 1774–1844 (2015)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Borodin, A.: On a family of symmetric rational functions. Adv. Math. 306, 973–1018 (2017)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Brofferio, S., Schapira, B.: Poisson boundary of $GL_d({\mathbb{Q}}_p)$. Israel J. Math. 185(1), 125 (2011)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Brown, B.M.: Martingale central limit theorems. Ann. Math. Stat. 42(1), 59–66 (1971)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Bufetov, A., Petrov, L.: Law of large numbers for infinite random matrices over a finite field. Sel. Math. New Ser. 21(4), 1271–1338 (2015)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Bufetov, A., Matveev, K.: Hall-Littlewood RSK field. Sel. Math. New Ser. 24(5), 4839–4884 (2018)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Cartwright, D.I., Woess, W.: Isotropic random walks in a building of type. Math. Z. 247(1), 101–135 (2004)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Chhaibi, R.: Non-Archimedean Whittaker functions as characters: a probabilistic approach to the Shintani–Casselman–Shalika formula. Int. Math. Res. Not. 2017(7), 2100–2138 (2017)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Cohen, H., Lenstra, H.W.: Heuristics on class groups of number fields. In: Number Theory Noordwijkerhout 1983, pp. 33–62. Springer (1984)

  24. 24.

    Collins, B.: Product of random projections, Jacobi ensembles and universality problems arising from free probability. Probab. Theory Relat. Fields 133(3), 315–344 (2005)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Corwin, I., Dimitrov, E.: Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall-Littlewood Gibbsian line ensembles. Commun. Math. Phys. 363(2), 435–501 (2018)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Crisanti, A., Paladin, G., Vulpiani, A.: Products of Random Matrices: In Statistical Physics, vol. 104. Springer, Berlin (2012)

    MATH  Google Scholar 

  27. 27.

    Dimitrov, Evgeni: KPZ and Airy limits of Hall-Littlewood random plane partitions. In: Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, vol. 54, pp. 640–693. Institut Henri Poincaré (2018)

  28. 28.

    Ellenberg, J.S., Jain, S., Venkatesh, A.: Modeling $\lambda $-invariants by p-adic random matrices. Commun. Pure Appl. Math. 64(9), 1243–1262 (2011)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Evans, S.N.: Local fields, Gaussian measures, and Brownian motions. Top. Probab. Lie Groups Bound. Theory 28, 11–50 (2001)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Evans, S.N.: Elementary divisors and determinants of random matrices over a local field. Stoch. Process. Appl. 102(1), 89–102 (2002)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Forrester, P.J.: Asymptotics of finite system Lyapunov exponents for some random matrix ensembles. J. Phys. A Math. Theor. 48(21), 215205 (2015)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Forrester, P.J., Liu, D.-Z.: Singular values for products of complex Ginibre matrices with a source: hard edge limit and phase transition. Commun. Math. Phys. 344(1), 333–368 (2016)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Forrester, P.J., Rains, E.M.: Interpretations of some parameter dependent generalizations of classical matrix ensembles. Probab. Theory Relat. Fields 131(1), 1–61 (2005)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Friedman, E., Washington, L.C.: On the distribution of divisor class groups of curves over a finite field. In: Théorie des Nombres/Number Theory Laval (1987)

  35. 35.

    Fulman, J.: A probabilistic approach toward conjugacy classes in the finite general linear and unitary groups. J. Algebra 212(2), 557–590 (1999)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Fulman, J.: Random matrix theory over finite fields. Bull. Am. Math. Soc. 39(1), 51–85 (2002)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Fulman, J.: Cohen–Lenstra heuristics and random matrix theory over finite fields. J. Group Theory 17(4), 619–648 (2014)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Furstenberg, H., Kesten, H.: Products of random matrices. Ann. Math. Stat. 31(2), 457–469 (1960)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Gol’dsheid, I.Ya., Margulis, G.A.: Lyapunov indices of a product of random matrices. RuMaS 44(5), 11–71 (1989)

  40. 40.

    Gorin, V., Kleptsyn, V.: Universal objects of the infinite beta random matrix theory (2020). arXiv preprint arXiv:2009.02006

  41. 41.

    Gorin, V., Marcus, A.W.: Crystallization of random matrix orbits. Int. Math. Res. Not. 2020(3), 883–913 (2020)

    MathSciNet  Article  Google Scholar 

  42. 42.

    Gorin, V., Sun, Y.: Gaussian fluctuations for products of random matrices (2018). arXiv preprint arXiv:1812.06532

  43. 43.

    Jones, L., O’Connell, N.: Weyl chambers, symmetric spaces and number variance saturation. ALEA Lat. Am. J. Probab. Math. Stat 2, 91–118 (2006)

  44. 44.

    Kieburg, M., Kösters, H., et al.: Products of random matrices from polynomial ensembles. In: Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, vol. 55, pp. 98–126. Institut Henri Poincaré (2019)

  45. 45.

    Koepf, W.: Hypergeometric summation. Vieweg, Braunschweig/Wiesbaden, 5(6) (1998)

  46. 46.

    Liu, D.-Z., Wang, D., Wang, Y.: Lyapunov exponent, universality and phase transition for products of random matrices. arXiv preprint arXiv:1810.00433 (2018)

  47. 47.

    Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  48. 48.

    Macdonald, I.G.: Symmetric Functions and Orthogonal Polynomials, vol. 12. American Mathematical Society, Providence (1998)

    MATH  Google Scholar 

  49. 49.

    Marchenko, V.A., Pastur, L.A.: Distribution of eigenvalues for some sets of random matrices. Matematicheskii Sbornik 114(4), 507–536 (1967)

    MATH  Google Scholar 

  50. 50.

    Matveev, K.: Macdonald-positive specializations of the algebra of symmetric functions: proof of the Kerov conjecture. Ann. Math. 189(1), 277–316 (2019)

    MathSciNet  Article  Google Scholar 

  51. 51.

    Neretin, Y.A.: Hua measures on the space of p-adic matrices and inverse limits of Grassmannians. Izv. Math. 77(5), 941–953 (2013)

    MathSciNet  Article  Google Scholar 

  52. 52.

    Okounkov, A., Olshanski, G.: Asymptotics of Jack polynomials as the number of variables goes to infinity. Int. Math. Res. Not. 13, X–682 (1998)

    MATH  Google Scholar 

  53. 53.

    Oseledets, I.: A multiplicative ergodic theorem. Characteristic Ljapunov exponents of dynamical systems. Trudy Moskovskogo Matematicheskogo Obshchestva 19, 179–210 (1968)

    MathSciNet  Google Scholar 

  54. 54.

    Parkinson, J.: Buildings, groups of lie type, and random walks. Groups Graphs Random Walks 436, 391 (2017)

    MathSciNet  Article  Google Scholar 

  55. 55.

    Raghunathan, M.S.: A proof of Oseledec’s multiplicative ergodic theorem. Israel J. Math. 32(4), 356–362 (1979)

  56. 56.

    Schapira, B.: Random walk on a building of type $ {{\tilde{A}}}_r $ and Brownian motion of the Weyl chamber. Annales de l’IHP Probabilités et Statistiques 45, 289–301 (2009)

  57. 57.

    Shiryaev, A.N.: Probability, 2nd edn, Volume 95 of Graduate Texts in Mathematics. Springer, New York (1996). Translated from the Russian by R. P. Boas

  58. 58.

    Sun, Y.: Matrix models for multilevel Heckman-Opdam and multivariate Bessel measures (2016). arXiv preprint arXiv:1609.09096

  59. 59.

    Tao, T.: Tate’s proof of the functional equation. (2008)

  60. 60.

    Van Peski, R.: Random matrices over integers of local fields. Undergraduate Thesis (2018)

  61. 61.

    Wood, M.M.: Random integral matrices and the Cohen-Lenstra heuristics. arXiv preprint arXiv:1504.04391 (2015)

  62. 62.

    Wood, M.M.: Asymptotics for number fields and class groups. In: Directions in Number Theory, pp. 291–339. Springer, Berlin (2016)

    Chapter  Google Scholar 

  63. 63.

    Wood, M.M.: Cohen–Lenstra heuristics and local conditions. Res. Number Theory 4(4), 41 (2018)

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Roger Van Peski.

Additional information

Publisher's Note

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

I thank my advisor Alexei Borodin for many helpful conversations throughout this project and feedback on several drafts; Vadim Gorin, for early encouragement, pointers to the matrix product literature, and other helpful discussions and comments; and Andrew Ahn, for many illuminating discussions on the complex case and its relation to this work, and detailed feedback on the exposition. Some preliminary material appeared in my unpublished undergraduate thesis [60], and it is my pleasure to thank Ju-Lee Kim for her guidance on that project. I also wish to thank Adam Block, Jason Fulman, Nathan Kaplan, Mario Kieburg, Sergei Korotkikh, and Kevin Lin for additional helpful conversations. This material is based on work supported by the National Science Foundation Graduate Research Fellowship under Grant No. #1745302.

Appendix A: Relations to the Archimedean case and alternate proof of Proposition 3.2

Appendix A: Relations to the Archimedean case and alternate proof of Proposition 3.2

As mentioned in the Introduction, Theorem 1.3 is exactly analogous to results on singular values of corners and products in the real, complex and quaternion cases. We will first informally state these results in more detail than in the Introduction in order to highlight the parallel, and give references to more complete treatments. We will then give an alternate proof of Proposition 3.2 which is simpler, but valid only under additional assumptions which do not cover the Hall–Littlewood case \(q=0\). This was the first proof we found, but we were unable to justify the \(q \rightarrow 0\) limit and hence resorted to the stronger results proven in Sect. 3. However, the proof below has the advantage that it survives the limit to the real/complex/quaternion cases which we are about to describe, and hence could be used to adapt the convolution-of-projectors method of Theorem 1.3 to prove the analogous result in this setting. At the end of the appendix we will outline how this could be carried out.

Fix a parameter \(\beta > 0\) and let \(q=e^{-\epsilon }\), \(t = q^{\beta /2}\). In all cases below, we assume that the integers nmNk satisfy the same constraints as in Theorem 1.3. Below we give an informal statement of the analogue of Theorem 1.3 in the real, complex and quaternion setting.

  1. (1)

    Define the random signature \(\lambda (\epsilon )\) by

    $$\begin{aligned} \Pr (\lambda (\epsilon )=\lambda ) = \frac{P_\lambda (1,t,\ldots ,t^{n-1};q,t)Q_\lambda (t^{m-n+1},\ldots ,t^{N-n};q,t)}{\Pi _{(q,t)}(1,t,\ldots ,t^{n-1};t^{m-n+1},\ldots ,t^{N-n})} \end{aligned}$$

    for any \(\lambda \in {{\,\mathrm{Sig}\,}}_n^+\), with qt depending on \(\epsilon \) as above. Then as \(\epsilon \rightarrow 0\), the random real signature \(\epsilon \lambda (\epsilon ) = (\epsilon \lambda _1(\epsilon ),\ldots ,\epsilon \lambda _n(\epsilon ))\) converges in distribution to some limiting random real signature \(\lambda (0)\). When \(\beta = 1,2,4\), \(\lambda (0)\) has the same distribution as \((-\log (r_n),\ldots ,-\log (r_1))\), where \(r_1 \ge \cdots \ge r_n\) are the squared singular values of an \(n \times m\) corner of a Haar-distributed element of \({\mathcal {O}}(n), U(n)\) or \({{\,\mathrm{Sp}\,}}^*(n)\) respectively. This is due to Forrester-Rains [33], see also Borodin-Gorin [15, Thm. 2.8].

  2. (2)

    Fix a real signature \(\ell \) of length n and define the nonrandom signature

    $$\begin{aligned} \lambda (\epsilon ) := (\lfloor \ell _1/\epsilon \rfloor , \ldots , \lfloor \ell _n/\epsilon \rfloor ) \in {{\,\mathrm{Sig}\,}}_n. \end{aligned}$$

    Define the random signature \(\nu (\epsilon )\) by

    $$\begin{aligned} \Pr (\nu (\epsilon )=\nu ) = \frac{Q_{\nu /\lambda (\epsilon )}(1,\ldots ,t^{-(k-1)};q,t)P_\nu (t^{N-n},\ldots ,t^{N-1};q,t)}{P_\lambda (t^{N-n},\ldots ,t^{N-1};q,t) \Pi _{(q,t)}(1,\ldots ,t^{-(k-1)};t^{N-n},\ldots ,t^{N-1})} \end{aligned}$$

    for any \(\nu \in {{\,\mathrm{Sig}\,}}_n\). Then as \(\epsilon \rightarrow 0\), \(\nu (\epsilon )\) converges to a random real signature \(\nu (0)\). Suppose \(\beta = 1,2,4\) and \({\mathbb {F}}= {\mathbb {R}},{\mathbb {C}},{\mathbb {H}}\) respectively, and \(A_{col} \in M_{n \times (N-k)}({\mathbb {F}})\) is the first \(N-k\) columns of \(A \in M_{n \times N}({\mathbb {F}})\) with fixed singular values \(e^{-\ell }:=(e^{-\ell _1},\ldots ,e^{-\ell _n})\) and distribution invariant under the orthogonal, unitary or symplectic groups acting on the right and left. Then the distribution of the negative logarithms of the squared singular values of \(A_{col}\) is given by \(\nu (0)\). The statement for (1.4) is exactly analogous. We could not locate these exact statements in the literature but essentially equivalent ones appear in Borodin-Gorin [15] and Sun [58] when considering the Jacobi corners process.

  3. (3)

    Fix real signatures \(r,\ell \) of length n and define nonrandom integer signatures \(\lambda (\epsilon )\) as above and \(\rho (\epsilon )\) similarly with r in place of \(\ell \). Then as \(\epsilon \rightarrow 0\), \(\epsilon \cdot (\rho (\epsilon ) \boxtimes _{(1,\ldots ,t^{n-1})} \lambda (\epsilon ))\) (where we abuse notation and use \(\boxtimes \) to refer to the convolution operation with Macdonald polynomials instead of Hall–Littlewood) converges to a random real signature s. When \(\beta = 1,2,4\), \(e^{-s}\) gives the distribution of singular values of AB where AB are bi-invariant under the orthogonal, unitary or symplectic group and have fixed singular values \(e^{-r}\) and \(e^{-\ell }\). See Gorin-Marcus [41, Prop. 2.2].

More general background on these limits may be found in Ahn [2], Borodin-Gorin [15], Gorin-Marcus [41], and Sun [58].

Remark 13

The explicit formulas for the above distributions are uniform expressions in terms of \(\beta \), and the distributions for general \(\beta \in [0,\infty )\) are referred to as \(\beta \)-ensembles. \(\beta \) is then seen as an inverse temperature parameter, and the zero-temperature limit \(\beta \rightarrow \infty \) has in particular been studied, both because it provides tractable though accurate approximations to \(\beta =1,2,4\), and because it exhibits asymptotic behaviors interesting in their own right. In particular, the product convolution and corners operation–the analogues of Theorem 1.3 Parts 3 and 2 respectively–become deterministic in this limit and are controlled by certain orthogonal polynomials. See Gorin-Marcus [41] and Gorin-Kleptsyn [40] for a discussion of the eigenvalue (as opposed to singular value) case, and Borodin-Gorin [15, Cor. 5.4] for the deterministic \(\beta \rightarrow \infty \) limit of Jacobi corners; we are not aware of anywhere the \(\beta \rightarrow \infty \) limits of general corners and products (the analogues of Parts 2, 3 of Theorem 1.3) are worked out explicitly in the literature. In our setting, viewing the measures and operations of Theorem 1.3 for arbitrary \(t \in (0,1)\) not necessarily a prime power is exactly analogous to this extrapolation to general \(\beta \).

We observe the exact same freezing to a deterministic operation in the p-adic case of products and corners in the limit \(p \rightarrow \infty \), i.e. \(t \rightarrow 0\). It is interesting to note that while the \(\beta \rightarrow \infty \) limit requires extrapolation away from the usual matrix models, the \(t \rightarrow 0\) limit does not because one can find arbitrarily large primes. In the corners case, the partition \(\nu \) in the notation of Theorem 1.3 concentrates around \(\lambda \), and the partition \(\mu \) concentrates around \((\lambda _{d+1},\ldots ,\lambda _n)\). In the product case, \(\nu \) concentrates around \((\lambda _1+\mu _1,\ldots ,\lambda _n+\mu _n)\). These facts may be easily verified using the explicit formulas for Hall–Littlewood polynomials in Sect. 2.3, and may also be seen heuristically directly from the matrix models without any formulas.

Below we prove Proposition 3.2 under the additional assumptions that \({\mathbf {a}}= (1,t,\ldots ,t^{N-1})\) and \(q,t \in (0,1)\). We remark that Proposition 3.3 may be proven by similar label-variable duality manipulations under the restricted hypotheses as above; the modifications to the proof below are not difficult.


For the remainder of the proof, we will denote \(\boxtimes _{(1,\ldots ,t^{N-1})}\) by \(\boxtimes _t\) and use \({{\,\mathrm{Supp}\,}}\) for the support of a measure. Let \(\lambda (D) = (D[N-n],\lambda )\) and \(\mu (D) = (D[N-m],0[m])\). Recall that

$$\begin{aligned} M_D^{Cauchy}(\nu ):= & {} \sum _{\begin{array}{c} \kappa \in {{\,\mathrm{Sig}\,}}_N \\ \kappa _{N-n+i}=\nu _i \text { for all }i=1,\ldots ,n \end{array}} c_{\lambda (D),\mu (D)}^\kappa (q,t) \frac{P_\kappa (t^{N-1},\ldots ,1)}{P_{\lambda (D)}(t^{N-1},\ldots ,1)P_{\mu (D)}(t^{N-1},\ldots ,1)} \end{aligned}$$

and we wish to show

$$\begin{aligned} M_D^{Cauchy}(\nu ) \rightarrow \frac{P_\nu (t^{n-1},\ldots ,1)Q_{\nu /\lambda }(t^{N-n},\ldots ,t^{m-n+1})}{P_\lambda (t^{n-1},\ldots ,1) \Pi (t^{n-1},\ldots ,1;t^{N-n},\ldots ,t^{m-n+1})} \end{aligned}$$

(note that we have written the measure in a different form from Proposition 3.2 by using homogeneity to rearrange powers of t).

Denote the limiting measure of (A.1) by \(\mathcal {M}\). The proof is by a kind of moments method which consists of showing the convergence of expectations of observables

$$\begin{aligned} {\mathbb {E}}_{\nu \sim M_D^{Cauchy}}[P_\alpha (q^{\nu _1}t^{n-1},\ldots ,q^{\nu _n})] \rightarrow {\mathbb {E}}_{\nu \sim \mathcal {M}}[P_\alpha (q^{\nu _1}t^{n-1},\ldots ,q^{\nu _n})] \end{aligned}$$

as \(D \rightarrow \infty \) for each \(\alpha \in {{\,\mathrm{Sig}\,}}_n^+\), followed by an argument that these ‘moments’ are sufficient to give convergence of measures. We rely on the nontrivial label-variable duality satisfied by these observables, see [47, Section 6]:

$$\begin{aligned} \frac{P_\nu (q^{\alpha _1}t^{n-1},\ldots ,q^{\alpha _n})}{P_\nu (t^{n-1},\ldots ,1)} = \frac{P_\alpha (q^{\nu _1}t^{n-1},\ldots ,q^{\nu _n})}{P_\alpha (t^{n-1},\ldots ,1)}. \end{aligned}$$

Such a strategy is used to prove similar statements in [41, Section 4].

We first show

$$\begin{aligned}&\left| {\mathbb {E}}_{\nu \sim M_D^{Cauchy}}[P_\alpha (q^{\nu _1}t^{n-1},\ldots ,q^{\nu _n})] - {\mathbb {E}}_{\kappa \sim \lambda (D) \boxtimes _t \mu (D)}[P_{(\alpha ,0[N-n])}(q^{\kappa _1}t^{N-1},\ldots ,q^{\kappa _N})]\right| \rightarrow 0\nonumber \\ \end{aligned}$$

as \(D \rightarrow \infty \). To show (A.4) it suffices to show that there exist constants \(C(\alpha ,D)\) independent of \(\kappa \in {{\,\mathrm{Supp}\,}}(\lambda (D) \boxtimes _t \mu (D))\) such that \(C(\alpha ,D) \rightarrow 0\) as \(D \rightarrow \infty \) and

$$\begin{aligned} |P_{(\alpha ,0[N-n])}(q^{\kappa _1}t^{N-1},\ldots ,q^{\kappa _N}) -P_\alpha (q^{\nu _1}t^{n-1},\ldots ,q^{\nu _n})| < C(\alpha ,D). \end{aligned}$$

where \(\nu \) is defined by \(\nu _i = \kappa _{N-n+i}\). This suffices because the support \({{\,\mathrm{Supp}\,}}(\lambda (D) \boxtimes _t \mu (D))\) of this measure contains only \(\kappa \) for which \(\kappa \supset \lambda (D)\) by basic properties of the structure coefficients, hence

$$\begin{aligned}&\left| {\mathbb {E}}_{\nu \sim M_D^{Cauchy}}[P_\alpha (q^{\nu _1}t^{n-1},\ldots ,q^{\nu _n})] - {\mathbb {E}}_{\kappa \sim \lambda (D) \boxtimes _t \mu (D)}[P_{(\alpha ,0[N-n])}(q^{\kappa _1}t^{N-1},\ldots ,q^{\kappa _N})]\right| \\\&\quad < C(\alpha ,D) \end{aligned}$$

by (A.5) and linearity of expectation. So let us prove (A.5).

\(P_{(\alpha ,0[N-n])}\) is a polynomial in N variables \(q^{\kappa _1}t^{N-1},\ldots ,q^{\kappa _N}\), which we split into two collections of variables, the first \(N-n\) and the last n. As \(D \rightarrow \infty \), the first \(N-n\) variables go to 0 because \(\kappa _i \ge \lambda (D)_i = D\) for \(i=1,\ldots ,N-n\), for any \(\kappa \in {{\,\mathrm{Supp}\,}}(\lambda (D) \boxtimes _t \mu (D))\), hence

$$\begin{aligned}&P_{(\alpha ,0[N-n])}(q^{\kappa (D)_1}t^{N-1},\ldots ,q^{\kappa (D)_N}) \\&\qquad \rightarrow P_{(\alpha ,0[N-n])}(0[N-n],t^{n-1}q^{\kappa (D)_{N-n+1}},\ldots ,q^{\kappa (D)_N}) \\&\quad = P_\alpha (q^{\nu _1}t^{n-1},\ldots ,q^{\nu _n}) \end{aligned}$$

for any sequence \(\kappa (D) \in {{\,\mathrm{Supp}\,}}(\lambda (D) \boxtimes _t \mu (D))\) with last n parts given by \(\nu \). The last n variables always lie in a compact interval \([0,q^{\lambda _n}]\) because \(\kappa _i \ge \lambda _n\) for \(i=N-n+1,\ldots ,N\) by interlacing, for any \(\kappa \in {{\,\mathrm{Supp}\,}}(\lambda (D) \boxtimes _t \mu (D))\). Hence the above convergence is uniform over \(\nu \) and \(\kappa \), i.e. (A.5) holds.

Thus to show (A.2), it suffices to show

$$\begin{aligned} {\mathbb {E}}_{\kappa \sim \lambda (D) \boxtimes _t \mu (D)}[P_{(\alpha ,0[N-n])}(q^{\kappa _1}t^{N-1},\ldots ,q^{\kappa _N})] \rightarrow {\mathbb {E}}_{\nu \sim \mathcal {M}}[P_\alpha (q^{\nu _1}t^{n-1},\ldots ,q^{\nu _n})].\nonumber \\ \end{aligned}$$

Now, using label-variable duality (A.3),

$$\begin{aligned}&{\mathbb {E}}_{\kappa \sim \lambda (D) \boxtimes _t \mu (D)}[P_{(\alpha ,0[N-n])}(q^{\kappa _1}t^{N-1},\ldots ,q^{\kappa _N})] \\&\quad ={\mathbb {E}}_{\kappa \sim \lambda (D) \boxtimes _t \mu (D)}\left[ \frac{P_\kappa (q^{\alpha _1}t^{N-1},\ldots ,1)}{P_\kappa (t^{N-1},\ldots ,1)}P_{(\alpha ,0[N-n])}(t^{N-1},\ldots ,1)\right] \\&\quad = \sum _{\kappa \in {{\,\mathrm{Sig}\,}}_N} c_{\lambda (D),\mu (D)}^\kappa (q,t) \frac{P_\kappa (t^{N-1},\ldots ,1)P_{(\alpha ,0[N-n])}(t^{N-1},\ldots ,1)}{P_{\lambda (D)}(t^{N-1},\ldots ,1)P_{\mu (D)}(t^{N-1},\ldots ,1)}\frac{P_\kappa (q^{\alpha _1}t^{N-1},\ldots ,1)}{P_\kappa (t^{N-1},\ldots ,1)} \\&\quad = \frac{P_{(\alpha ,0[N-n])}(t^{N-1},\ldots ,1)}{P_{\lambda (D)}(t^{N-1},\ldots ,1)P_{\mu (D)}(t^{N-1},\ldots ,1)} \sum _{\kappa \in {{\,\mathrm{Sig}\,}}_N} c_{\lambda (D),\mu (D)}^\kappa (q,t)P_\kappa (q^{\alpha _1}t^{N-1},\ldots ,1) \\&\quad = \frac{P_{(\alpha ,0[N-n])}(t^{N-1},\ldots ,1)}{P_{\lambda (D)}(t^{N-1},\ldots ,1)P_{\mu (D)}(t^{N-1},\ldots ,1)} P_{\lambda (D)}(q^{\alpha _1}t^{N-1},\ldots ,1)P_{\mu (D)}(q^{\alpha _1}t^{N-1},\ldots ,1) \\&\quad = \frac{P_{(\alpha ,D[N-n])}(q^Dt^{N-1},\ldots ,q^D t^n, q^{\lambda _1}t^{n-1},\ldots ,q^{\lambda _n})P_{(\alpha ,D[N-n])}(q^Dt^{N-1},\ldots ,q^D t^m, t^{m-1},\ldots ,1)}{P_{(\alpha ,D[N-n])}(t^{N-1},\ldots ,1)}. \end{aligned}$$

As \(D \rightarrow \infty \), the above clearly converges to

$$\begin{aligned} \frac{P_{\alpha }(q^{\lambda _1}t^{n-1},\ldots ,q^{\lambda _n}) P_{\alpha }(t^{m-1},\ldots ,1)}{P_\alpha (t^{N-1},\ldots ,1)}, \end{aligned}$$

so we must show

$$\begin{aligned} {\mathbb {E}}_{\nu \sim \mathcal {M}}[P_\alpha (q^{\nu _1}t^{n-1},\ldots ,q^{\nu _n})] = \frac{P_{\alpha }(q^{\lambda _1}t^{n-1},\ldots ,q^{\lambda _n}) P_{\alpha }(t^{m-1},\ldots ,1)}{P_\alpha (t^{N-1},\ldots ,1)}.\qquad \end{aligned}$$

Again using label-variable duality, and the Cauchy identity Lemma 2.3, we have

$$\begin{aligned}&{\mathbb {E}}_{\nu \sim \mathcal {M}}[P_\alpha (q^{\nu _1}t^{n-1},\ldots ,q^{\nu _n})] \\&\quad = {\mathbb {E}}_{\nu \sim \mathcal {M}}\left[ \frac{P_\alpha (t^{n-1},\ldots ,1) P_\nu (q^{\alpha _1}t^{n-1},\ldots ,q^{\alpha _n})}{P_\nu (t^{n-1},\ldots ,1)}\right] \\&\quad = \frac{P_\alpha (t^{n-1},\ldots ,1)\sum _{\nu \in {{\,\mathrm{Sig}\,}}_n} P_\nu (q^{\alpha _1}t^{n-1},\ldots ,q^{\alpha _n})Q_{\nu /\lambda }(t^{N-n},\ldots ,t^{m-n+1})}{\Pi (t^{n-1},\ldots ,1;t^{N-n},\ldots ,t^{m-n+1})P_\lambda (t^{n-1},\ldots ,1)} \\&\quad = \frac{\Pi (q^{\alpha _1}t^{n-1},\ldots ,q^{\alpha _n}; t^{N-n},\ldots ,t^{m-n+1})}{\Pi (t^{n-1},\ldots ,1;t^{N-n},\ldots ,t^{m-n+1})} \frac{P_\alpha (t^{n-1},\ldots ,1)P_\lambda (q^{\alpha _1}t^{n-1},\ldots ,q^{\alpha _n})}{P_\lambda (t^{n-1},\ldots ,1)} \\&\quad = \frac{\Pi (q^{\alpha _1}t^{n-1},\ldots ,q^{\alpha _n}; t^{N-n},\ldots ,t^{m-n+1})}{\Pi (t^{n-1},\ldots ,1;t^{N-n},\ldots ,t^{m-n+1})} P_\alpha (q^{\lambda _1}t^{n-1},\ldots ,q^{\lambda _n}). \end{aligned}$$

Hence (A.7) is equivalent to

$$\begin{aligned} \frac{ \Pi (q^{\alpha _1}t^{n-1},\ldots ,q^{\alpha _n}; t^{N-n},\ldots ,t^{m-n+1})}{\Pi (t^{n-1},\ldots ,1;t^{N-n},\ldots ,t^{m-n+1})} = \frac{P_{\alpha }(t^{m-1},\ldots ,1)}{P_\alpha (t^{N-1},\ldots ,1)}. \end{aligned}$$

(A.8) follows by applying the explicit formula for principally specialized Macdonald polynomials, [47, (6.11’)], to the numerator and denominator of the RHS, expanding the LHS into infinite products and noting that all but finitely many terms cancel, and comparing the resulting expressions.

We have proven convergence of ‘moments’, so let us upgrade this to convergence of measures. Consider the compact set

$$\begin{aligned} U^n := \{(u_1,\ldots ,u_n) \in {\mathbb {R}}^n: 0 \le u_1 \le \cdots \le u_n \le q^{\lambda _n}\}. \end{aligned}$$

Then we have a map \(\phi : {{\,\mathrm{Sig}\,}}_n \rightarrow U^n\) given by \(\phi (\nu _1,\ldots ,\nu _n) = (q^{\nu _1},\ldots ,q^{\nu _n})\). Also,

$$\begin{aligned} f_\alpha (u_1,\ldots ,u_n) := \frac{P_\alpha (u_1t^{n-1},\ldots ,u_n)}{P_\alpha (t^{n-1},\ldots ,1)} \end{aligned}$$

defines a function on \(U^n\). The subalgebra of \(\mathcal C(U^n)\) generated by the functions \(f_\alpha \) is just the set of finite linear combinations of \(f_\alpha \) because products of Macdonald polynomials may be expanded as linear combinations of Macdonald polynomials. This algebra contains the constant functions (\(f_{(0[n])}\) is constant) and separates points, so by the Stone-Weierstrass theorem it is dense in \(\mathcal C(U^n)\) with sup norm.

By hypothesis, the structure coefficients are nonnegative and hence \(M_D^{Cauchy}\) is indeed a probability measure for each D. To show weak convergence \(M_D^{Cauchy}\rightarrow \mathcal {M}\), we must show for any \(f \in \mathcal C(U^n)\) that \(\int _{U^n} f d\phi _*(M_D^{Cauchy}) \rightarrow \int _{U^n} f d\phi _*(\mathcal {M})\). By the above, there exists a linear combination g of \(f_\alpha \)s such that \(\sup _{u \in U^n}|f(u)-g(u)| < \epsilon /3\). Since \(\mathcal {M}\) and \(M_D^{Cauchy}\) are probability measures it follows that \( \int _{U^n} |f-g| d\phi _*(\mathcal {M}) < \epsilon /3\) and similarly with \(\mathcal {M}\) replaced by any \(M_D^{Cauchy}\). By (A.2), we may choose D such that

$$\begin{aligned} \left| \int _{U^n} g d\phi _*(M_D^{Cauchy}) - \int _{U^n} g d\phi _*(\mathcal {M})\right| < \epsilon /3. \end{aligned}$$

Putting together the three inequalities yields

$$\begin{aligned} \left| \int _{U^n} f d\phi _*(M_D^{Cauchy}) -\int _{U^n} f d\phi _*(\mathcal {M})\right| < \epsilon , \end{aligned}$$

hence \(\phi _*(M_D^{Cauchy})\) converges weakly to \(\phi _*(\mathcal {M})\). Because both measures are supported on a discrete subset \(\phi (\{\nu \in {{\,\mathrm{Sig}\,}}_n: \nu _n \ge \lambda _n\})\) of \(U^n\), this implies \(M_D^{Cauchy}(\nu ) = \phi _*(M_D^{Cauchy})(\phi (\nu )) \rightarrow \phi _*(\mathcal {M})(\phi (\nu )) = \mathcal {M}(\nu )\) for each \(\nu \in {{\,\mathrm{Sig}\,}}_n\), completing the proof. \(\square \)

The proofs of Propositions 3.2 and 3.3 in Sect. 4 heavily used the discrete structure of the set of integer signatures, and we have no idea how they would be modified to the continuum limit to real signatures described earlier. However, we claim that the above proof could be modified with no substantial changes. Let us briefly outline why this is so.

Definition 17

Let \(r = (r_1,\ldots ,r_n) \in {{\,\mathrm{Sig}\,}}_n^{\mathbb {R}}\) have distinct parts, \(\theta > 0\) a parameter, and \(y_1,\ldots ,y_n\) complex variables. Setting \(\lambda (\epsilon ) = \lfloor \epsilon ^{-1}(r_1,\ldots ,r_n) \rfloor \), we define the (type A) Heckman–Opdam hypergeometric function

$$\begin{aligned} \mathcal F_r(y_1,\ldots ,y_n;\theta ) := \lim _{\epsilon \rightarrow 0} \epsilon ^{\theta \left( {\begin{array}{c}n\\ 2\end{array}}\right) } P_\lambda (e^{\epsilon y_1},\ldots ,e^{\epsilon y_n}; q=e^{-\epsilon }, t = e^{-\theta \epsilon }) \end{aligned}$$

The dual Heckman–Opdam function may be obtained similarly by degenerating Q. Instead of defining the measures appearing in the singular value setting as limits of Macdonald measures, as we did earlier in this “Appendix”, one may instead first take the limit to Heckman–Opdam functions and then define measures in terms of these. When one takes the limit of (A.3) in the above regime, one obtains

$$\begin{aligned} \frac{\mathcal F_r(-\lambda _1-(n-1)\theta , -\lambda _2-(n-2)\theta ,\ldots ,-\lambda _n;\theta )}{\mathcal F_r(-(n-1)\theta ,\ldots ,0;\theta )} = \frac{J_\lambda (e^{-r_1},\ldots ,e^{-r_n};\theta )}{J_\lambda (1,\ldots ,1;\theta )} \end{aligned}$$

where \(J_\lambda \) is the classical Jack polynomial. The same argument used to prove Proposition 3.2 above may be used after this limit, with the Macdonald polynomials replaced by Heckman–Opdam functions or Jack polynomials as appropriate given the above, and the sums replaced by integrals. This post-limit version of Proposition 3.2 may then be used to implement the convolution-of-projectors strategy we used in Sect. 3 to prove the analogue of Theorem 1.3 in the real/complex/quaternion setting. We refer to [41] for similar random matrix arguments utilizing label-variable duality and Jack/Heckman–Opdam functions.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Van Peski, R. Limits and fluctuations of p-adic random matrix products. Sel. Math. New Ser. 27, 98 (2021).

Download citation


  • p-adic random matrices
  • Hall–Littlewood polynomials
  • particle systems

Mathematics Subject Classification

  • 15B52 (primary); 15B33
  • 60B20 (secondary)