{"id":"f53edddc-0736-4eb2-83b5-72ba1bec9fa5","arxiv_id":"2412.05999","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The joint distribution of singular numbers of all principal corners of a Haar-random p-adic Hermitian or alternating matrix is a (formal) Hall-Littlewood process.","lead":"This paper computes the joint distribution of singular numbers of all principal corners of random p-adic Hermitian and alternating matrices, the non-Archimedean analogue of the GUE corners process. It shows these distributions are Hall-Littlewood processes, connecting p-adic random matrix theory to the Macdonald process framework and giving a conceptual explanation for previously mysterious formulas.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Hermitian Hecke-module normalizations in Theorem 4.3 and the finite-field rank product in Eq. (5.29) are the load-bearing links from Haar corners to Hall-Littlewood processes; they need an independent check before acceptance.","rationale":"The reader's weakest assumption correctly identified the dependence on the Hironaka and Hironaka-Sato module-structure theorems. I agree that this is the most load-bearing unverified premise: Theorem 1.2 is derived from Theorem 1.4, and Theorem 1.4 is derived from Theorem 3.3 and Theorem 4.3 by matching normalizations and volume factors. Because the paper does not reproduce those external proofs, and the Hermitian map includes a non-obvious sign and an explicit correction to the z-variables, a normalization error there would silently invalidate the central formulas. In addition, the proof of Corollary 5.5 contains a displayed finite-field rank-count product whose exponents appear to run in the wrong direction; this is precisely the step that turns the corners of invertible matrices into i.i.d. Haar-entry matrices and then feeds Theorem 1.2. The paper's small worked examples check only very special cases and do not rule out such an error. A direct small-q, small-n enumeration against formulas (1.5) and (1.6) is cheap and would settle both issues conclusively. For these reasons I do not reject the paper, but I would not move from ACCEPT to a higher-confidence unconditional acceptance until the normalization and the finite-field product are verified or corrected.","tokens_in":48315,"tokens_out":21685,"duration_ms":208075,"concrete_test":"Run a computational cross-check for q=2 and n=2, and if feasible q=3. Enumerate all Hermitian (resp. alternating) matrices in o/p^M with M large enough that every singular number appearing with positive probability is less than M, weight each lift by the Haar measure of its ball, compute the joint distribution of all corner singular numbers, and compare term-by-term with (1.6) (resp. (1.5)). Separately, recompute a non-Pieri structure coefficient cher in (1.14)-(4.11) from the coset definition Gher in (4.1), for example with mu=(1,1), nu=(1,0), and compare it with the formula in Corollary 4.4. This will isolate whether any mismatch comes from the Theorem 4.3 normalization or from the finite-field product in (5.29).","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central formula of Theorem 1.2 is obtained by composing Theorem 1.4, which translates the Hironaka/Hironaka-Sato spherical-function module structure into symmetric-function identities. In the Hermitian case this translation is delicate: Theorem 4.3 and Corollary 4.4 carry a stated factor-of-two correction to Hironaka's z-variables [Hir88a], and the volume factors (4.13) enter with signed q-powers. If either the q-power or the sign normalization is off, Theorem 1.4(2) and hence formulas (1.6), (1.8), and (1.11) inherit an error. The examples in Section 4 check only a Pieri-type coefficient and do not exercise the full module map. A second, internal red flag appears in (5.29): for an l x (m-n) matrix over F_{q^2}, the full-row-rank probability is the product over i=0,...,l-1 of (1 - q^{2n-2m+2i}), with increasing exponents, whereas (5.29) displays decreasing exponents 2n-2m-2i. If that product is not a typo, the proof of Corollary 5.5, which supplies the base marginal distribution used to build the corners process in Theorem 1.2, does not go through as written. Both issues are local and checkable, but they sit exactly where probabilistic statements are derived from the algebra.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":48609,"tokens_out":49477,"duration_ms":385325,"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":[{"comment":"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.","section":"Section 4, display (4.14)."},{"comment":"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.","section":"Section 5, display (5.29)."},{"comment":"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.","section":"Theorem 4.3 and footnote 6."}],"minor_comments":[{"comment":"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.","section":"Theorem 1.4(2)."},{"comment":"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.","section":"Abstract."},{"comment":"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)'.","section":"Corollary 4.4, display (4.11)."},{"comment":"In Definition 4.1, the phrase 'the convolution of the Hecke algebra over the symplectic Hecke module' should read 'Hermitian Hecke module'.","section":"Definition 4.1."},{"comment":"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}'.","section":"Example 4.5(2) and proof of Theorem 1.4."},{"comment":"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.","section":"Footnote 6."}],"recommendation":"major_revision","confidential_remarks":"To the editor: my independent check of the internal algebra finds sign inconsistencies in the Hermitian normalization chain (4.11)–(4.14) and an exponent error in the full-rank probability product of (5.29). Because these steps feed Theorem 1.2(2) through Corollary 5.5 and Theorem 1.3(2), I could not recommend acceptance in the present form; the errors appear to be fixable typos or normalization slips rather than conceptual failures, since the final formulas are anchored by the literature ([FK19], [Lee23]) and by the small-case checks. However, the authors should be asked to re-derive the sign bookkeeping of Section 4 and to verify the module map (4.7) on generic signatures before publication. The paper is a good fit for the journal, and the requested revisions are well within the manuscript's reach."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about 2412.05999. First, it actually delivers the conceptual unification it promises: the non-archimedean alternating GUE corners process is literally a Hall-Littlewood process, and the Hermitian corners process is the marginal of a formal Hall-Littlewood process with signed weights. Second, the load-bearing step is an external black box: the Hecke module structure theorems of Hironaka and Hironaka-Sato, which the authors translate into symmetric function identities but do not fully reproduce.\n\nWhat's new is real. The joint distribution of singular numbers of all principal corners is new in both symmetry classes; the alternating case recovers Fulman-Kaplan as a corollary and explains why Hall-Littlewood polynomials appear there. The Hermitian result, with its Ennola-duality-esque t=-1/q parameter and signed transitions, is a genuinely novel object. The proofs are detailed and, given the stated module-structure theorems, mostly self-contained. The combinatorial development of the modified Littlewood-Richardson coefficients is clean and the product convolution Theorem 2.13 is a nice tool.\n\nThe soft spots, in proportion: first, the extraction of Theorems 3.3 and 4.3 is delicate. In the Hermitian case the authors apply a factor-of-two correction to Hironaka's z-variables and put a sign in the module map; the Section 4 examples only exercise a Pieri-type coefficient, not the full module homomorphism. If that normalization is off, formulas (1.6), (1.8), (1.11) all inherit an error. I cannot rule that out without checking Hironaka's papers carefully. A referee should do that. Second, equation (5.29) displays a product of full-rank probabilities with decreasing exponents, whereas the standard formula for an l x (m-n) matrix over F_{q^2} gives increasing exponents, product_{i=0}^{l-1}(1-q^{2n-2m+2i}). This is almost certainly a typographical sign error—the subsequent limit m→∞ works regardless—but it sits in the proof of Corollary 5.5, which supplies the base marginal distribution, so it should be fixed and the step verified. Minor: the paper assumes characteristic zero and unramified Hermitian extension, and Appendix A honestly says positive characteristic is not checked. There are a few typos like 'B in GL_{2n}' in Theorem 1.4(2) where GL_n is meant.\n\nBottom line: the central argument is sound as far as I can tell, the concerns are local and checkable, and the paper has enough new content and clarity to deserve a serious referee.\n\nRecommendation: send it to referees. One referee should be asked to verify the module structure extraction (Theorems 3.3, 4.3) against Hironaka/Hironaka-Sato and the rank-probability identity (5.29).","headline":"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.","tokens_in":49142,"tokens_out":4682,"would_cite":true,"duration_ms":36901,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","05E05","22E50","11S80"],"pacs":[],"model":"deepseek-v4-flash","headline":"Random p-adic alternating and Hermitian matrices have corner singular-number spectra given explicitly by Hall-Littlewood processes.","keywords":["non-archimedean GUE","Hall-Littlewood process","spherical Hecke algebra","p-adic random matrices","corners process","alternating matrices","Hermitian matrices","Macdonald processes"],"falsifier":"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).","tokens_in":48101,"feed_emoji":"🎲","tokens_out":7955,"duration_ms":73950,"temperature":0.7,"pith_summary":"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.","feed_headline":"p-adic GUE corners are Hall-Littlewood processes","feed_subtitle":"Exact joint laws for every principal corner, recovering known formulas and opening the way to asymptotics.","key_machinery":"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).","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the alternating Hecke-module identification used as Theorem 3.3: orbit indicators map to Hall-Littlewood polynomials with parameter $q^{-2}$.","marker":"[HS88]"},{"why":"Supplies the Hermitian Hecke-module identification used as Theorem 4.3, including the orbit-volume formulas that enter the probability computation.","marker":"[Hir99]"},{"why":"Establishes the spherical-function computations for Hermitian forms that Theorem 4.3 extracts and rephrases.","marker":"[Hir88a]"},{"why":"Its Theorem 3.2 gives the single-matrix alternating distribution recovered as a corollary of the process in Theorem 1.2.","marker":"[FK19]"},{"why":"Provides the explicit probability formula and finite-field counts used to verify and recover the alternating and Hermitian marginals.","marker":"[BKL+15]"},{"why":"Defines the Macdonald-process framework in which the Hall-Littlewood processes of Theorem 1.2 sit.","marker":"[BC14]"},{"why":"Supplies the Hall-Littlewood polynomial theory, principal specializations, and Hecke-ring background used throughout the proofs.","marker":"[Mac98]"},{"why":"Gives the no-symmetry analogue and the corner-process techniques that the alternating case reduces to.","marker":"[VP21]"}],"fun_headline_variants":["Hall-Littlewood processes from p-adic corners","p-adic GUE corners: Hall-Littlewood law","Corners of p-adic matrices follow Hall-Littlewood","Exact corner laws in p-adic GUE","Hecke modules reveal p-adic corner laws"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Hall-Littlewood processes from p-adic corners","p-adic GUE corners: Hall-Littlewood law","Corners of p-adic matrices follow Hall-Littlewood","Exact corner laws in p-adic GUE","Hecke modules reveal p-adic corner laws"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000203,"raw_usage":{"total_tokens":1373,"prompt_tokens":917,"completion_tokens":456,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":533,"completion_tokens_details":{"reasoning_tokens":377}},"tokens_in":533,"tokens_out":456,"duration_ms":3887,"temperature":1.0,"reasoning_tokens":377,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T20:06:26.932333+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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).","supporting_citations":[],"review_version":1}