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Non-Archimedean GUE corners and Hecke modules

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Pith's one-line read Random p-adic alternating and Hermitian matrices have corner singular-number spectra given explicitly by Hall-Littlewood processes.

desk verdict A genuinely new unification of p-adic GUE corners with Hall-Littlewood processes, but the external Hecke-module extraction and a likely typo in (5.29) need referee attention. read the letter →

arxiv 2412.05999 v2 pith:6SUY5CQ6 submitted 2024-12-08 math.PR math.COmath.NTmath.RT

Jiahe Shen , Roger Van Peski This is my paper · ORCID
classification math.PRmath.COmath.NTmath.RT MSC 60B2005E0522E5011S80
keywords non-archimedeanGUEHall-LittlewoodprocesssphericalHeckealgebrap-adicrandommatricescornersalternatingHermitianMacdonaldprocesses
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper shows that the non-archimedean analogues of the GUE and anti-symmetric GUE corners processes are governed by Hall-Littlewood polynomials, not just in single-matrix marginals but as full joint laws. For a random alternating $2n\times 2n$ matrix with additive Haar entries over the ring of integers of a p-adic field, the singular numbers of all principal corners form a Hall-Littlewood process with parameter $t=1/q$, given explicitly by (1.5). For a random Hermitian $n\times n$ matrix, the joint law is a marginal of a formal Hall-Littlewood process whose intermediate weights may be negative, given by (1.6). This places the models in the Macdonald-process framework and explains why Hall-Littlewood polynomials appear in previously known formulas.

What carries the argument

The load-bearing object is the spherical Hecke algebra of $\mathrm{GL}_n(F)$ with respect to $\mathrm{GL}_n(\mathfrak o)$---the convolution algebra of bi-invariant compactly supported functions---together with its modules on alternating and Hermitian matrices. The structure theorems say that the Hecke action is, under the usual spherical-function isomorphism, multiplication by Hall-Littlewood polynomials, with parameters $t^2$ in the alternating case and $-t$ in the Hermitian case. This turns the random-matrix operation of multiplying by a conjugation factor into a product of Hall-Littlewood polynomials, and the explicit probabilities follow from principal specializations, skew Cauchy identities, and the modified structure coefficients $c^{\mathrm{alt},\lambda}_{\mu,\nu}(t)$ and $c^{\mathrm{her},\lambda}_{\mu,\nu}(t)$ defined in (1.13) and (1.14).

What would settle it

Enumerate, modulo $\mathfrak p^N$, the joint distribution of singular numbers of all principal corners of a random alternating $4\times4$ matrix over $\mathbb{Z}_p$ for $p=2$ or $3$, and check that the empirical ratios converge to the right-hand side of (1.5). A mismatch would falsify the Hecke-module bridge; the same test can be run for the Hermitian $2\times2$ case against (1.6).

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.2. In the alternating case, for $A_{2n}$ with i.i.d. above-diagonal Haar entries in $\mathfrak o$, the joint distribution of $\mathrm{SN}_{\mathrm{alt}}(A_2),\dots,\mathrm{SN}_{\mathrm{alt}}(A_{2n})$ equals the explicit Hall-Littlewood process (1.5); in the Hermitian case, for $A_n$ with i.i.d. Haar entries above the diagonal and on the diagonal, the joint distribution of $\mathrm{SN}_{\mathrm{her}}(A_1),\dots,\mathrm{SN}_{\mathrm{her}}(A_n)$ equals the explicit formal Hall-Littlewood marginal (1.6). The route is algebraic: the action $A\mapsto B^TAB$ or $B^*AB$ is converted into multiplication by Hall-Littlewood polynomials through the module structure of the spherical Hecke algebra, and symmetric-function identities then produce the probabilities. Single-matrix marginals (1.7) and (1.8) recover earlier results, and the method also yields Hall-Littlewood product processes for iterated conjugation by random matrices.

Load-bearing premise

Everything depends on the structural theorem that the symmetry-algebra action on alternating and Hermitian matrices is exactly multiplication by Hall-Littlewood polynomials; if that theorem fails, or cannot be extended to positive characteristic and ramified extensions, the probability formulas do not follow.

Editorial extensions

If this is right

  • The joint distribution of all principal corner singular numbers of a Haar-random p-adic alternating matrix is exactly the Hall-Littlewood process (1.5), not merely a limiting or approximate law.
  • The single-matrix alternating formula (1.7) recovers the known Fulman-Kaplan result, now as a corollary of the process-level statement.
  • The Hermitian joint law (1.6) is a genuine probability measure even though it is written with signed intermediate weights; its marginal (1.8) agrees with the previously known formula.
  • The product processes of Corollary 5.6 put iterated conjugation by random p-adic matrices into the Hall-Littlewood framework.
  • The connection to Macdonald processes gives a structured route to asymptotic questions about boundaries and local limits for these symmetry classes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves implicit that the module-theoretic bridge is likely to extend to other symmetry classes whose Hecke modules have been computed; if so, analogous Hall-Littlewood or Macdonald-process descriptions would hold for those random matrix ensembles.
  • The formal duality in which the Hermitian case uses $-1/q$ in place of $1/q$ suggests a deeper relation to unitary-group phenomena over finite fields; the authors note the similarity but do not establish the mechanism.
  • The alternating dynamics admit an explicit Markov sampling algorithm, whereas the Hermitian transitions involve cancellations; one testable extension would be a direct sampling scheme for the Hermitian corners process.
  • If the suspected validity in positive characteristic holds, the same formulas govern random matrices over function fields, where cokernel statistics are studied.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper computes the joint distribution of singular numbers of all principal corners of Haar-random matrices over the ring of integers of a non-archimedean local field, in two symmetry classes. Theorem 1.2(1) shows that for a random alternating 2n × 2n matrix the corner singular numbers form a Hall-Littlewood process (display (1.5)); Theorem 1.2(2) shows that for a random Hermitian n × n matrix the corners form a marginal of a formal Hall-Littlewood process with signed transition weights (display (1.6)); the parameter is t = 1/q throughout. The proofs pass through the module structure of the spherical Hecke algebra acting on alternating and Hermitian matrices (Theorems 3.3 and 4.3, extracted from Hironaka-Sato [HS88] and Hironaka [Hir99, Hir88a]), producing product-type distribution formulas (Theorem 1.4), corner transitions (Theorem 1.3), single-matrix marginals (Corollary 5.5, recovering Fulman-Kaplan [FK19, Theorem 3.2] and matching Lee [Lee23]), and a product-convolution process (Theorem 2.13). An appendix discusses the status of the characteristic-zero assumption.

Significance. If the stated results hold, this is a substantial contribution to non-archimedean random matrix theory: it places the p-adic GUE and aGUE corners processes in the Macdonald-process framework and provides a structural (Hecke-module) explanation for formulas previously obtained by direct computation. The strengths are real: t = 1/q is an input with no free parameters; the derivations are detailed and mostly self-contained conditional on the cited module-structure theorems; the paper is candid about its characteristic-zero assumption and about the external status of Theorems 3.3 and 4.3; and the worked examples (3.5, 4.5, 5.7, 5.8) give concrete anchors. The negative-t 'formal Hall-Littlewood' phenomenon in the Hermitian case is a novel and interesting feature. The main risk identified below is that the Hermitian normalization chain and the finite-field rank product in (5.29) contain errors as printed, exactly where the probabilistic statements are derived from the algebra; both concerns are local and checkable, but they need to be fixed.

major comments (3)
  1. [Section 4, display (4.14).] The displayed chain in (4.14) is internally inconsistent. The q-exponent in the second line, ⟨λ−ν−2µ, ρ_n⟩, simplifies to n(λ)−n(ν)−2n(µ) using ⟨λ, ρ_n⟩ = n(λ) − (n−1)|λ|/2 and |λ| = 2|µ| + |ν| (Proposition 4.1), but the third line displays q^{n(ν)−n(λ)+2n(µ)}, which is the opposite exponent and is unequal in general (for n = 2, µ = (1,1), ν = (0,0), λ = (4,0), the two exponents are −2 and +2). The final displayed formula of (4.14) agrees with the third line and is the formula used in (5.6)–(5.7) to derive Theorem 1.3(2) and hence Theorem 1.2(2). The checks in Example 4.5 all satisfy n(λ)−n(ν)−2n(µ) = 0, so they cannot detect this mismatch; moreover, the simplification in Example 4.5(2) implicitly uses V_ν(−q^{−1})/V_λ(−q^{−1}) = 1, which holds only under additional multiplicity assumptions. Because this chain is the load-bearing translation from the Hermitian Hecke module to probabilities, the authors should re-derive the normalization from Theorem 4.3 through Corollary 4.4 and (4.14), fix the sign, and add a check with a generic signature.
  2. [Section 5, display (5.29).] The product for the full-rank probability of C′ ∈ M_{l,(m−n)}(F_{q^2}) has the exponents in the wrong order. The probability that an l × (m−n) matrix over F_{q^2} has rank l is ∏_{i=0}^{l−1}(1 − q^{2n−2m+2i}), with increasing exponents, whereas the displayed product (1 − q^{2n−2m})···(1 − q^{2n−2m−2l+2}) has decreasing exponents and is already incorrect for l = 2. This formula is used in the proof of Corollary 5.5, which supplies the base marginal distribution for Theorem 1.2, so the proof as written does not go through. In addition, the equality chain in (5.30) passes to m → ∞ without stating it: the final display is independent of m and equals the preceding expression only in the limit, consistent with Remark 5.2. The final formula (5.25) is consistent with the literature and recoverable through that limit, so the issue is local and fixable, but the displayed derivation must be corrected in the published version.
  3. [Theorem 4.3 and footnote 6.] The extraction of Theorem 4.3 from [Hir99] and [Hir88a] is load-bearing but not fully checkable from the manuscript. Footnote 6 changes Hironaka's normalization, asserting that the z-variables in [Hir88a, (1.6)] and [Hir99, (2.2)] 'are twice as different' and that the displayed formula of [Hir88a] must be revised accordingly; the revision is then used in (4.5)–(4.7) and feeds Corollary 4.4, the volume factors (4.13), and (4.14). Given that the sign errors in (4.14) sit precisely in the chain fed by this theorem, the paper should provide the complete dictionary between the present variables and Hironaka's conventions, including the sign (−1)^{n(ν)+|ν|} in (4.6), rather than asserting the result is 'essentially contained' in the cited works. A concrete test would be to verify the module map (4.7) on generic signatures for n = 2 and n = 3, beyond the Pieri cases of Example 4.5.
minor comments (6)
  1. [Theorem 1.4(2).] Theorem 1.4(2) states that B ∈ GL_{2n}(F), but the setting (A ∈ Hern(F), μ ∈ Sig_n, and the proof) requires B ∈ GL_n(F); the '2n' appears to be a typo.
  2. [Abstract.] The abstract attributes the recovered result to 'Fulman-Kaplan [Ful16]'; the result recovered in (1.7) is [FK19, Theorem 3.2], while [Ful16] is a single-author paper by Fulman.
  3. [Corollary 4.4, display (4.11).] In (4.11), the second equality displays q^{n(λ)−n(ν)−2(µ)}; the exponent should be n(ν)−n(λ)+2n(µ) (the value obtained by simplifying ⟨ν−λ+2µ, ρ_n⟩ using the degree condition in Proposition 4.1), and '2(µ)' should read '2n(µ)'.
  4. [Definition 4.1.] In Definition 4.1, the phrase 'the convolution of the Hecke algebra over the symplectic Hecke module' should read 'Hermitian Hecke module'.
  5. [Example 4.5(2) and proof of Theorem 1.4.] In the display at the end of Example 4.5(2) and in the final display of the proof of Theorem 1.4, the Hall-Littlewood parameter of Pλ and Pν is printed as 'q^{−1}' in two places and should be '−q^{−1}'.
  6. [Footnote 6.] The phrase 'twice as different' is unclear; the intended meaning is presumably that the z-variables in [Hir88a, (1.6)] and [Hir99, (2.2)] differ by a factor of two, and this should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Hall-Littlewood process formulas are derived from external Hecke-module structure results and independent symmetric-function identities.

full rationale

The central derivation chain is: Theorem 3.3 / Theorem 4.3, extracted from Hironaka-Sato [HS88] and Hironaka [Hir99, Hir88a], identify the Hecke-module action with multiplication by Hall-Littlewood polynomials; Proposition 3.2 / Proposition 4.2 convert the relevant convolution integrals into structure coefficients and orbit volumes; Theorem 1.4 follows by principal specialization; Theorem 2.13 and Section 5 turn these structure-coefficient identities into Hall-Littlewood process formulas; and Theorem 1.2 iterates Theorem 1.3 with Corollary 5.5 supplying the base marginal. None of these steps defines the random variable or the predicted distribution in terms of the answer. The coefficients calt and cher are defined in (1.13) and (1.14) by purely symmetric-function expansions, independent of random matrices; the paper then proves that the same coefficients govern the matrix-product probabilities, which is a substantive identification rather than a tautology. The only author-overlap citations are to [VP21], used as a tool for the unrestricted (no-symmetry) product/corner process; those results are external published statements about a different ensemble and do not assume the alternating or Hermitian corner formulas obtained here. The recoveries of Fulman-Kaplan and Lee are corollaries, not inputs. Appendix A explicitly lists the characteristic-zero hypotheses inherited from [HS88], [Hir99], and [Hir88a]; these are external-validity caveats, not circular steps. Equation (5.29) contains what appears to be a local sign/direction issue in the exponents of the full-rank probability product, but that is a checkable computational matter, not a circular reduction. Overall, the paper's probabilistic conclusions are not equivalent by construction to its inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central derivations are conditional on published module-structure theorems of Hironaka and Hironaka-Sato; the paper translates these into probabilistic statements. No free parameters are fitted; t=1/q is determined by the residue field. No new postulated entities are introduced.

assumptions (5)
  • domain assumption Module structure of spherical Hecke algebra on alternating matrices (Theorem 3.3, extracted from Hironaka-Sato [HS88]).
    This is the key structural input for the alternating case; the paper does not reprove it, only translates it.
  • domain assumption Module structure of spherical Hecke algebra on Hermitian matrices (Theorem 4.3, extracted from Hironaka [Hir99, Hir88a]).
    Key structural input for the Hermitian case.
  • domain assumption Local field F has characteristic zero; in the Hermitian case, F/F0 is unramified.
    Stated in Section 1.2; positive characteristic not fully checked (Appendix A).
  • standard math Standard Hall-Littlewood combinatorics: basis property, skew Cauchy identity, and principal specialization formulas.
    Used throughout Sections 2 and 5, citing Macdonald [Mac98] and [VP21].
  • standard math Satake isomorphism identifies the spherical Hecke algebra with symmetric Laurent polynomials.
    Used to connect Hecke algebra structure to Hall-Littlewood polynomials.

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Pith. "Pith review of Non-Archimedean GUE corners and Hecke modules." pith.science (2026). https://pith.science/paper/6SUY5CQ6

@misc{pith2026241205999,
  author       = {Pith},
  title        = {Pith review of: Non-Archimedean GUE corners and Hecke modules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6SUY5CQ6}},
  note         = {Machine review of arXiv:2412.05999}
}
abstract

We compute the joint distribution of singular numbers for all principal corners of a $p$-adic Hermitian (resp. alternating) matrix with additive Haar distribution, the non-archimedean analogue of the GUE (resp. aGUE) corners process. In the alternating case we find that it is a Hall-Littlewood process, explaining -- and recovering as a corollary -- results of Fulman-Kaplan. In the Hermitian case we obtain a `marginal distribution' of a formal Hall-Littlewood process with both positive and negative transition `probabilities'. The proofs relate natural random matrix operations to structural results of Hironaka and Hironaka-Sato on modules over the spherical Hecke algebra, yielding other probabilistic statements of independent interest along the way.

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Reference graph

Works this paper leans on

50 extracted references · 43 canonical work pages · cited by 3 Pith papers

  1. [1]

    Gaussian unitary ensemble in random lozenge tilings

    Amol Aggarwal and Vadim Gorin. Gaussian unitary ensemble in random lozenge tilings. Probability Theory and Related Fields , 184(3):1139--1166, 2022

    work page 2022

  2. [2]

    Fluctuations of - J acobi product processes

    Andrew Ahn. Fluctuations of - J acobi product processes. Probability Theory and Related Fields , 183(1-2):57--123, 2022

    work page 2022

  3. [3]

    Infinite p -adic random matrices and ergodic decomposition of p -adic H ua measures

    Theodoros Assiotis. Infinite p -adic random matrices and ergodic decomposition of p -adic H ua measures. Transactions of the American Mathematical Society , 375(3):1745--1766, 2022

    work page 2022

  4. [4]

    Modeling the distribution of ranks, S elmer groups, and S hafarevich-- T ate groups of elliptic curves

    Manjul Bhargava, Daniel M Kane, Hendrik W Lenstra, Bjorn Poonen, and Eric Rains. Modeling the distribution of ranks, S elmer groups, and S hafarevich-- T ate groups of elliptic curves. Cambridge Journal of Mathematics , 3(3):275--321, 2015

    work page 2015

  5. [5]

    Macdonald processes

    Alexei Borodin and Ivan Corwin. Macdonald processes. Probability Theory and Related Fields , 158(1-2):225--400, 2014

    work page 2014

  6. [6]

    Observables of Macdonald processes

    Alexei Borodin, Ivan Corwin, Vadim Gorin, and Shamil Shakirov. Observables of Macdonald processes . Transactions of the American Mathematical Society , 368(3):1517--1558, 2016

    work page 2016

  7. [7]

    Product matrix processes as limits of random plane partitions

    Alexei Borodin, Vadim Gorin, and Eugene Strahov. Product matrix processes as limits of random plane partitions. International Mathematics Research Notices , 2018

    work page 2018

  8. [8]

    Ergodic measures on spaces of infinite matrices over non-archimedean locally compact fields

    Alexander I Bufetov and Yanqi Qiu. Ergodic measures on spaces of infinite matrices over non-archimedean locally compact fields. Compositio Mathematica , 153(12):2482--2533, 2017

    work page 2017

Show all 50 references
  1. [9]

    Spherical functions on symmetric spaces of friedberg-jacquet type

    Murilo Corato-Zanarella. Spherical functions on symmetric spaces of friedberg-jacquet type. arXiv preprint arXiv:2311.00148 , 2023

    2023

  2. [10]

    Infinite-dimensional groups over finite fields and H all- L ittlewood symmetric functions

    Cesar Cuenca and Grigori Olshanski. Infinite-dimensional groups over finite fields and H all- L ittlewood symmetric functions. Advances in Mathematics , 395:108087, 2022

    2022

  3. [11]

    Heuristics on T ate- S hafarevitch groups of elliptic curves defined over Q

    Christophe Delaunay. Heuristics on T ate- S hafarevitch groups of elliptic curves defined over Q . Experimental Mathematics , 10(2):191--196, 2001

    2001

  4. [12]

    On the evaluation of certain p-adic integrals, s \'e m

    Jan Denef. On the evaluation of certain p-adic integrals, s \'e m. de th \'e orie des nombres, paris 1983/84. Progress in Math. , 59:25--47, 1983

    1983

  5. [13]

    The rationality of the poincar \'e series associated to the p-adic points on a variety

    Jan Denef. The rationality of the poincar \'e series associated to the p-adic points on a variety. Inventiones mathematicae , 77:1--23, 1984

    1984

  6. [14]

    Six-vertex models and the GUE-corners process

    Evgeni Dimitrov. Six-vertex models and the GUE-corners process . International Mathematics Research Notices , 2020(6):1794--1881, 2020

    2020

  7. [15]

    On the characters of the finite unitary groups

    Veikko Ennola. On the characters of the finite unitary groups. Annales Fennici Mathematici , (323), 1963

    1963

  8. [16]

    Elementary divisors and determinants of random matrices over a local field

    Steven N Evans. Elementary divisors and determinants of random matrices over a local field. Stochastic processes and their applications , 102(1):89--102, 2002

    2002

  9. [17]

    The Anti-Symmetric GUE Minor Process

    Peter J Forrester and Eric Nordenstam. The Anti-Symmetric GUE Minor Process . Moscow Mathematical Journal , 9(4):749--774, 2009

    2009

  10. [18]

    Hall- L ittlewood polynomials and C ohen- L enstra heuristics for J acobians of random graphs

    Jason Fulman. Hall- L ittlewood polynomials and C ohen- L enstra heuristics for J acobians of random graphs . Annals of Combinatorics , 20(1):115--124, 2016

    2016

  11. [19]

    Random partitions and Cohen--Lenstra heuristics

    Jason Fulman and Nathan Kaplan. Random partitions and Cohen--Lenstra heuristics . Annals of Combinatorics , 23:295--315, 2019

    2019

  12. [20]

    Probability in the classical groups over finite fields: symmetric functions, stochastic algorithms, and cycle indices

    Jason Edward Fulman. Probability in the classical groups over finite fields: symmetric functions, stochastic algorithms, and cycle indices . Harvard University, 1997

    1997

  13. [21]

    Crystallization of random matrix orbits

    Vadim Gorin and Adam W Marcus. Crystallization of random matrix orbits. International Mathematics Research Notices , 2020(3):883--913, 2020

    2020

  14. [22]

    Gaussian fluctuations for products of random matrices

    Vadim Gorin and Yi Sun. Gaussian fluctuations for products of random matrices. American Journal of Mathematics , 144(2):287--393, 2022

    2022

  15. [23]

    Random sorting networks: Edge limit

    Vadim Gorin and Jiaming Xu. Random sorting networks: Edge limit . In Annales de l'Institut Henri Poincare (B) Probabilites et statistiques , volume 60, pages 1025--1047. Institut Henri Poincar \'e , 2024

    2024

  16. [24]

    Spherical functions of hermitian and symmetric forms i

    Yumiko Hironaka. Spherical functions of hermitian and symmetric forms i. Japanese journal of mathematics. New series , 14(1):203--223, 1988

    1988

  17. [25]

    Spherical functions of hermitian and symmetric forms iii

    Yumiko Hironaka. Spherical functions of hermitian and symmetric forms iii. Tohoku Mathematical Journal, Second Series , 40(4):651--671, 1988

    1988

  18. [26]

    Spherical functions of hermitian and symmetric forms ii

    Yumiko Hironaka. Spherical functions of hermitian and symmetric forms ii. Japanese journal of mathematics. New series , 15(1):15--51, 1989

    1989

  19. [27]

    Spherical functions and local densities on hermitian forms dedicated to professor ichiro satake on his seventieth birthday

    Yumiko Hironaka. Spherical functions and local densities on hermitian forms dedicated to professor ichiro satake on his seventieth birthday. Journal of the Mathematical Society of Japan , 51(3):553--581, 1999

    1999

  20. [28]

    Spherical functions and local densities of alternating forms

    Yumiko Hironaka and Fumihiro Sato. Spherical functions and local densities of alternating forms. Amer. J. Math. , 110(3):473--512, 1988

    1988

  21. [29]

    Eigenvalues of GUE minors

    Kurt Johansson and Eric Nordenstam. Eigenvalues of GUE minors . Electronic Journal of Probability [electronic only] , 11:1342--1371, 2006

    2006

  22. [30]

    Generalized Gelfand-Graev representations and Ennola duality

    Noriaki Kawanaka. Generalized Gelfand-Graev representations and Ennola duality . Algebraic groups and related topics (Kyoto/Nagoya, 1983) , 6:175--206, 1985

    1983

  23. [31]

    New combinatorial formula for modified H all- L ittlewood polynomials

    Anatol N Kirillov. New combinatorial formula for modified H all- L ittlewood polynomials. arXiv preprint math/9803006 , 1998

    1998 arXiv

  24. [32]

    Universality of the cokernels of random ��-adic hermitian matrices

    Jungin Lee. Universality of the cokernels of random ��-adic hermitian matrices. Transactions of the American Mathematical Society , 376(12):8699--8732, 2023

    2023

  25. [33]

    Spherical functions on a group of p-adic type

    Ian Grant Macdonald. Spherical functions on a group of p-adic type. Publ. Ramanujan Inst. , 2, 1971

    1971

  26. [34]

    Symmetric functions and Hall polynomials

    Ian Grant Macdonald. Symmetric functions and Hall polynomials . Oxford university press, 1998

    1998

  27. [35]

    A phase transition for the cokernels of random band matrices over the p-adic integers

    Andr \'a s M \'e sz \'a ros. A phase transition for the cokernels of random band matrices over the p-adic integers. arXiv preprint arXiv:2408.13037 , 2024

    2024 arXiv

  28. [36]

    GUE corners limit of q -distributed lozenge tilings

    Sevak Mkrtchyan and Leonid Petrov. GUE corners limit of q -distributed lozenge tilings . Electronic Journal of Probability , 22(none):1 -- 24, 2017

    2017

  29. [37]

    Rank fluctuations of matrix products and a moment method for growing groups

    Hoi H Nguyen and Roger Van Peski. Rank fluctuations of matrix products and a moment method for growing groups. arXiv preprint arXiv:2409.03099 , 2024

    2024 arXiv

  30. [38]

    Universality for cokernels of random matrix products

    Hoi H Nguyen and Roger Van Peski. Universality for cokernels of random matrix products. Advances in Mathematics , 438, 2024

    2024

  31. [39]

    Local and global universality of random matrix cokernels

    Hoi H Nguyen and Melanie Matchett Wood. Local and global universality of random matrix cokernels. arXiv preprint arXiv:2210.08526 , 2022

    2022 arXiv

  32. [40]

    The birth of a random matrix

    Andrei Yur'evich Okounkov and Nikolai Yur'evich Reshetikhin. The birth of a random matrix. Moscow Mathematical Journal , 6(3):553--566, 2006

    2006

  33. [41]

    Ergodic unitarily invariant measures on the space of infinite H ermitian matrices

    Grigori Olshanski and Anatoli Vershik. Ergodic unitarily invariant measures on the space of infinite H ermitian matrices. In Contemporary mathematical physics , volume 175 of Amer. Math. Soc. Transl. Ser. 2 , pages 137--175. Amer. Math. Soc., Providence, RI, 1996

    1996

  34. [42]

    Theory of spherical functions on reductive algebraic groups over p -adic fields

    Ichir \^o Satake. Theory of spherical functions on reductive algebraic groups over p -adic fields. Publications Math \'e matiques de l'IH \'E S , 18:5--69, 1963

    1963

  35. [43]

    On functional equations of zeta distributions

    Fumihiro Sato. On functional equations of zeta distributions. In Automorphic forms and geometry of arithmetic varieties , pages 465--508. Elsevier, 1989

    1989

  36. [44]

    Galleries, Hall-Littlewood polynomials, and structure constants of the spherical Hecke algebra

    Christoph Schwer. Galleries, Hall-Littlewood polynomials, and structure constants of the spherical Hecke algebra . International Mathematics Research Notices , 2006(9):75395--75395, 2006

    2006

  37. [45]

    Limits and fluctuations of p -adic random matrix products

    Roger Van Peski. Limits and fluctuations of p -adic random matrix products. Selecta Mathematica , 27(5):1--71, 2021

    2021

  38. [46]

    Hall– L ittlewood polynomials, boundaries, and p -adic random matrices

    Roger Van Peski. Hall– L ittlewood polynomials, boundaries, and p -adic random matrices. International Mathematics Research Notices , 2023(13):11217--11275, 2022

    2023

  39. [47]

    Reflecting Poisson walks and dynamical universality in p -adic random matrix theory

    Roger Van Peski. Reflecting Poisson walks and dynamical universality in p -adic random matrix theory . arXiv preprint arXiv:2312.11702 , 2023

    2023

  40. [48]

    What is a p -adic Dyson B rownian motion? arXiv preprint arXiv:2309.02865 , 2023

    Roger Van Peski. What is a p -adic Dyson B rownian motion? arXiv preprint arXiv:2309.02865 , 2023

    2023 arXiv

  41. [49]

    Local limits in p -adic random matrix theory

    Roger Van Peski. Local limits in p -adic random matrix theory. Proceedings of the London Mathematical Society , 129(3):e12626, 2024

    2024

  42. [50]

    Probability theory for random groups arising in number theory

    Melanie Matchett Wood. Probability theory for random groups arising in number theory. arXiv preprint arXiv:2301.09687 , 2023

    2023 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.