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On the limit of random hives with GUE boundary conditions

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Pith's one-line read The paper proves that random hives with independent GUE boundary conditions converge in probability to a single continuum hive, with the pointwise limit given by a variational formula over asymptotic height functions.

desk verdict Real progress on an open limit problem; two repair-level gaps (boundary cases and a covariance bound) keep the theorem from being proven as stated. read the letter →

arxiv 2502.06414 v1 pith:L7K6ZJAM submitted 2025-02-10 math.PR

Hariharan Narayanan This is my paper · ORCID
classification math.PR MSC 60B2060F05
keywords randomhivesGUEmatricesHornproblemlozengetilingsasymptoticheightfunctionssurfacetensioncontinuumlimit
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

Random hives encode the possible spectra of sums of Hermitian matrices through discrete concave functions on a triangle. This paper treats hives whose boundary data on two sides are the eigenvalues of two independent scaled GUE matrices and whose third side carries the Vandermonde weight from the randomized Horn problem. It proves that, after rescaling by $n^{-2}$, the value at any fixed relative location converges in probability to a single deterministic limit, not merely along a subsequence. The limit is identified as $\sup_{f^{\sharp} \in AHT^{\infty}_v} S_v(f^{\sharp})$, a variational supremum over asymptotic height functions of lozenge tilings of a scaled excavation hexagon. If correct, this gives a single explicit continuum limit for random hives and settles the convergence question left open by the earlier concentration result.

What carries the argument

The load-bearing construction is the excavation hexagon $\mathcal{H}^n_v$ together with the octahedron recurrence: the hive value $h_n(v)$ is exactly the maximum weight of a lozenge tiling of this hexagon, with lozenge weights read from two independent GUE minor processes. The limiting variational data are the asymptotic height function pairs in $AHT^{\infty}_v$ and the functional $S_v = S_{v,\diamond}+S_{v,\Delta}+S_{v,7}$, whose bulk term integrates a surface tension $\sigma_\diamond(\rho,\partial f)$ over the hexagon. The upper bound decomposes each height function into dyadic nearly-linear patches using quantitative differentiation and controls the fluctuation of each patch with the patch-variance estimate, while the lower bound patches local optimal matchings together, so the upper and lower estimates meet at the supremum.

What would settle it

Fix an interior point $v$, simulate two independent scaled GUE minor processes up to large $n$, compute $h_n(v_n)$ as the maximum lozenge weight of the excavation hexagon via the octahedron recurrence, and compare $n^{-2}h_n(v_n)$ with the predicted supremum $\sup_{f^{\sharp} \in AHT^{\infty}_v} S_v(f^{\sharp})$. If the rescaled values do not converge, or converge to a number different from that supremum for some $v$, the central claim is wrong.

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Extended reading notes

Core claim

Let $\lambda_n$ and $\mu_n$ be the eigenvalues of two independent scaled GUE matrices, and let $a_n$ be sampled from the normalized Lebesgue measure on the augmented hive polytope with those boundary data. The main theorem states that if $v_n/n \to v$, then $\lim_{n\to\infty} n^{-2} \mathbb{E}_n h_n(v_n) = \sup_{f^{\sharp} \in AHT^{\infty}_v} S_v(f^{\sharp})$, and as a corollary $n^{-2}h_n(v_n)$ converges in probability to the same value. The content is that the continuum hive is unique: the earlier subsequential limit and vanishing variance are upgraded to an exact limit, and the limiting value is the value of a specific variational problem over pairs of asymptotic height functions on the excavation hexagon.

Load-bearing premise

The upper bound requires that the maximum lozenge weight on a patch of side $m$ has variance at most $O(m^4/\log^C m)$ for every fixed $C$; the paper obtains this from a covariance decay bound for interlacing gaps of the GUE minor process quoted from an unpublished preprint, together with log-concavity. If that patch-level variance does not decay, the variational upper bound on the expected hive value fails.

Editorial extensions

If this is right

  • At each fixed relative location $v$, the scaled hive value $n^{-2}h_n(v_n)$ converges in probability to the explicit number $\sup_{f^{\sharp} \in AHT^{\infty}_v} S_v(f^{\sharp})$.
  • The earlier concentration result is upgraded from vanishing variance and subsequential convergence to full convergence to a single continuum hive.
  • The randomized Horn problem for two independent scaled GUE matrices acquires a well-defined asymptotic law, so numerical studies can compare against a unique limit.
  • The variational formula gives a computable route to the limiting value, since $S_v$ is an integral of a surface tension over the bulk plus explicit equator and hexagon terms.

Reading between the lines

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

  • The same proof structure should yield a unique continuum limit for any ensemble of boundary data whose minor process is log-concave and has the analogous interlacing-gap covariance decay, not only GUE.
  • The pointwise variational formula is strong evidence that the rate-function minimizer in the large-deviation principle for augmented hives is unique, since any minimizer must attain this supremum at every point.
  • One testable extension is to compute the left-hand side numerically by direct simulation of the octahedron recurrence and compare it with an independent numerical evaluation of the variational supremum; agreement at moderate $n$ would support the formula, and disagreement would pinpoint the step to examine.
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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

2 major / 7 minor

Summary. This paper studies the augmented hives arising from the spectra of two independent scaled GUE matrices, the model that encodes the Horn probability measure for the sum of two random Hermitian matrices. The main theorem (Theorem 8.5) states that for such a hive h_n, at any vertex v_n with v_n/n → v, the normalized expectation n^{-2} E_n h_n(v_n) converges to sup_{f^♯ ∈ AHT^∞_v} S_v(f^♯), where S_v is an explicit functional of asymptotic height functions on the infinite excavation hexagon 7^∞_v; Corollary 8.6 upgrades this to convergence in probability. This would settle the uniqueness of the continuum limit left open by the author's earlier joint work with Sheffield and Tao [30], which established only subsequential convergence and vanishing variance. The proof scheme is to express the hive value as the maximum lozenge weight on an excavation hexagon, then bound this maximum from above (Sections 8.2–8.4) via a quantitative-differentiation decomposition into dyadic triangles, local variance estimates (Proposition 7.1), and the surface tension σ_⋄, and from below (Section 8.5) by explicit patching of height functions of prescribed tilt near a candidate maximizer, with separate treatment of the equator and the hexagon weight in Lemmas 8.1 and 8.2. The argument draws on fixed-index universality results for the GUE minor process quoted from Tao's preprint [40] (Theorems A.3–A.5) and on Fefferman's quantitative differentiation theorem, whose proof is reproduced in Appendix B.

Significance. If Theorem 8.5 is proved, it is a significant advance for the theory of random hives and for the randomized Horn problem: it converts the subsequential compactness of [30] into genuine convergence in probability to a single continuum limit, and it identifies the limit pointwise by a variational formula that is explicit (a supremum of integrals of the surface tension σ_⋄, itself defined from fixed-index GUE minor patches and lozenge tilings) and falsifiable by the existing simulations of [11,12]. The paper has real structural strengths: the two-sided bounds are organized around quantitative differentiation (Fefferman's theorem, proved in Appendix B), the fixed-index convergence of the GUE minor process is supported by a self-contained argument in Appendix A.4, and the surface-tension calculus of Section 11 (subadditivity, convexity, continuity, and ψ-independence) is developed in unusual detail. The paper also makes a parameter-free statement in the end: the auxiliary cutoff ψ(m) = (log m)^{-1/10} of Definition 14 is shown in Lemma 11.4 not to affect the limiting value.

major comments (2)
  1. [§9, Lemma 9.1, used by Prop. 7.1 and Claim 4] The bound ∥M∥_op = O(m²/log^C m) for the 4m²×4m² covariance matrix of interlacing gaps in a (2m+1)×(2m+1) patch is the load-bearing estimate for the upper bound, but the proof ('This follows from Theorem A.5, taking the parameter A in it to be sufficiently large') is ineffective. Theorem A.5 is one-dimensional and stated for fixed m: it bounds the variance of Σ_{l=1}^m a_l g̃_{i+l} for consecutive interlacing gaps along a single level of the minor process, and its right-hand side is a constant for fixed m, with no growing-m content. It therefore cannot control even a single row of the patch whose length is 2m+1, and it says nothing about covariances between gaps in different rows, which make up most of the 4m²×4m² matrix; the cross-row structure is precisely what a spectral-norm bound must control. The trivial bound on ∥M∥_op is O(m²), and without the polylogarithmic saving the variance estimate in the proof of Claim 4 (§8.4) gives O(n²) instead of o_ε(n²), so Lemma 8.7 and Lemma 8.9 are not established. The same Proposition 7.1 is used in the proof of Lemma 11.10, so the gap also propagates into Lemma 11.11, Proposition 7.5 and Proposition 8.4. I agree with the stress-test assessment: a genuinely two-dimensional covariance bound for fixed-index interlacing gaps (for example, from the determinantal kernel of the GUE minor process or from correlation decay in the bead model of [4]) must be supplied; the quoted Theorem A.5 cannot yield it.
  2. [Theorem 8.5, lower bound via §8.5, Lemma 8.11] Theorem 8.5 is asserted for every vertex limit v ∈ T, but the lower-bound Lemma 8.11 is proved only under the hypothesis that both trapezoids of the infinite excavation hexagon 7^∞_v contain Euclidean balls of positive radius, and this hypothesis is not stated in Theorem 8.5 or in Lemma 8.12, which appeals to Lemma 8.11 in the proof of the theorem. The hypothesis fails on the boundary of T: for v = (0, β) with 0 < β < 1 the vertices of Definition 6 give A = F and C = D, so both trapezoids collapse to line segments, and Claim 5's bound d(L_{T,ḡ}, ∂K) > cε², together with the inclusion HT^n_v(f^{♯,δ}, ε) ⊃ HT^n_v(ḡ_n, εC0) and the subsequent patching, have no geometric slack. Thus the pointwise formula is unproved for boundary points, which are included in the statement of the main theorem. The fix may be local (approximate boundary v by interior points and use continuity of both sides in v, or restrict Theorem 8.5 to the interior), but as written the theorem claims more than the proof establishes.
minor comments (7)
  1. [§8.4, proof of Claim 3] The choice of the constant C0 is left implicit: the line that 2^{-k}C0 → ∞ as ε → 0 is only true if C0 is taken to grow with k (e.g., C0 > 2^{2k}, which is available since 2^k < C0^{1/2} is imposed); this dependence should be stated explicitly, since in the ε → 0 asymptotics the minimal allowed C0 is astronomically large.
  2. [§8.4, passage from (8.6) to Lemma 8.9] The passage from the bound (8.6) for a fixed f^♯ and ε to the final display of Lemma 8.9 (limsup_{n→∞} E n^{-2} h_n(v_n) ≤ sup_{f^♯ ∈ AHT^∞_v} S_v(f^♯)) interchanges a supremum with an ε → 0 limit; this requires an explicit finite ε-covering of the compact space AHT^∞_v by tubes HT^n_v(f^♯_i, εn) with the o_ε(1) errors uniform over the finitely many centers.
  3. [§8.5, Lemma 8.11, last step] The last step identifies the integral against ∂f^{♯,δ} with S_{v,⋄}(f^♯) − o_ε(1), which uses δ = ε/10 → 0 together with convexity of σ_⋄(ρ,·) in the second argument (giving S_{v,⋄}(f^{♯,δ}) ≥ S_{v,⋄}(f^♯) − C_{|ρ|}δ); the text merely cites Lemma 8.3, which addresses mesh approximation rather than the δ-removal, so the step should be justified explicitly.
  4. [§11, proof of Lemma 11.2] The proof states that 'choosing m1 = ⌊m3/m0⌋, we see that m1m0(1 + C′ψ(m0)) = m3'; for arbitrary m3 this is not an identity, and the subadditivity inequality in Lemma 11.1 B is only stated for exact multiples with the specified border term. The rounding at both ends must be absorbed into the error terms before the argument feeds into Corollary 11.3 and Lemma 11.4.
  5. [§11, final paragraph of Lemma 11.4] The claimed independence of σ_⋄(ρ, a_∞) from the choice of ψ is proved in a compressed paragraph: the constants in the patching inequality are unspecified, the divisibility assumptions on m1 and m2 are not stated, and the displayed condition 'm2ψ2(m2) > m1ψ1(m1)' appears to swap the roles of ψ1 and ψ2 relative to the contradiction in (11.3). Since this paragraph establishes the parameter-freeness of the limit claimed in the main theorem, it should be expanded into a complete argument.
  6. [§5, Theorem 5.1 and Observation 2] The two fixed-index convergence statements are dimensionally inconsistent: the square-patch case is stated as taking values in R^{2m} with the vector 1_{2m}, while the line-segment case is stated in R^{2m+1} with 1_{2m+1}; the number of interlacing gaps in a (2m+1)×(2m+1) box should be fixed consistently, and the notation M|_{⌈ℓx⌉+2m} versus M|_{⌈ℓy⌉+l_m} should be defined with the same conventions.
  7. [§1.4, normalization around (1.10)–(1.11)] The normalization in §1.4 contains typos that make the scaling convention hard to verify: 'the the distribution of the eigenvalues of A + B are the pushforward', 'Cnσ−n2' (presumably C_n σ^{−n²}), and the expression 'A√σ2n' are all unclear as printed; these should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the variational limit is computed from independent local surface-tension data, not from the hive value being predicted.

full rationale

The paper's main theorem derives a limit for n^{-2} E_n h_n(v_n) and expresses it as sup_{f^sharp in AHT^infty_v} S_v(f^sharp). The functional S_v is built from sigma_diamond, defined in Definitions 13-17 via independent GUE-minor patches and lozenge tilings, and from sigma_Delta and S_{v,7}, which are boundary and hexagon terms. None of these ingredients is defined in terms of h_n(v_n) or the target limit; the only role of the hive is the variational maximization over asymptotic height functions. The auxiliary psi(m) = (log m)^{-1/10} is chosen by hand, but Lemma 11.4 explicitly proves the limit is independent of the specific choice of psi, so this is not a fitted parameter. The proof builds on [30] and quotes Tao's unpublished Theorem A.5, but those are prior external inputs; the paper's contribution is the upper and lower bounds in Sections 8.4-8.5. The concern that Lemma 9.1 may not follow from Theorem A.5 is a correctness gap rather than circularity: it concerns an insufficient proof step, not a definitional or self-referential reduction. Consequently no step is circular.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new particles, forces, or conserved quantities. It uses existing objects such as the marked bead model (Boutillier), hives, and lozenge tilings. The only hand-chosen quantity is the auxiliary sequence psi(m), shown to be asymptotically irrelevant, so the ledger contains no fitted or invented entities.

free parameters (1)
  • psi(m) = (log m)^{-1/10}
    An auxiliary scale in the definition of sigma_{m,diamond} (Definitions 14-15). It is chosen by hand, not fitted to data, and Lemma 11.4 shows the final surface tension is independent of the specific choice satisfying the conditions in Observation 3.
assumptions (6)
  • domain assumption GUE minor process converges to the Boutillier bead model in constant-size squares around bulk energies
    Used in Section 5 and Theorem 5.1 via [1, Corollary 1.4]; the bead model is the local limit of interlacing GUE eigenvalues.
  • standard math Fefferman quantitative differentiation (Theorem 7.6)
    Appendix B provides a proof; used in Lemma 8.7 to decompose height functions into GOOD and BAD dyadic cubes with controlled L2 approximation error.
  • standard math Prekopa's theorem on log-concavity of marginal densities (Theorem 4.1)
    Used repeatedly to assert log-concavity of the GUE minor process, derived marginal weights, and the hexagon weight in Lemma 8.2.
  • standard math Thurston's height function extension criteria (Propositions 10.1 and 10.2)
    Used in the proofs of Propositions 7.2 and 7.3, which provide the patching of height functions required in the surface-tension estimates.
  • domain assumption Mesoscopic CLT of Landon-Sosoe [24, Theorem 1.4] gives convergence of E[n^{-2} wt'(7_n^v)]
    Used in Lemma 8.2 to assert that the normalized hexagon weight converges in expectation, which is a component of S_v in Definition 21.
  • domain assumption Theorem A.5 (Tao): interlacing gap statistics variance bound in fixed-index patches
    Quoted from the unpublished preprint [40]; underpins Lemma 9.1 and Proposition 7.1, the key local variance decay used in the upper-bound proof.

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Pith. "Pith review of On the limit of random hives with GUE boundary conditions." pith.science (2026). https://pith.science/paper/L7K6ZJAM

@misc{pith2026250206414,
  author       = {Pith},
  title        = {Pith review of: On the limit of random hives with GUE boundary conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L7K6ZJAM}},
  note         = {Machine review of arXiv:2502.06414}
}
abstract

We show that hives chosen at random with independent GUE boundary conditions on two sides, weighted by a Vandermonde factor depending on the third side (which is necessary in the context of the randomized Horn problem), when normalized so that the eigenvalues at the edge are asymptotically constant, converge in probability to a continuum hive as $n \rightarrow \infty.$ It had previously been shown in joint work with Sheffield and Tao \cite{NST} that the variance of these scaled random hives tends to $0$ and consequently, from compactness, that they converge in probability subsequentially. In the present paper, building on \cite{NST}, we prove convergence in probability to a single continuum hive, without having to pass to a subsequence. We moreover show that the value at a given point $v$ of this continuum hive equals the supremum of a certain functional acting on asymptotic height functions of lozenge tilings.

Figures

Figures reproduced from arXiv: 2502.06414 by the authors.

Figure 1.1
Figure 1.1. The mean and variance of a random hive corresponding to two GUE distributions [PITH_FULL_IMAGE:figures/full_fig_p003_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Values taken at interior vertices in the hive model satisfy rhombus inequalities as [PITH_FULL_IMAGE:figures/full_fig_p004_1_2.png] view at source ↗
Figure 1.3
Figure 1.3. An n = 4 Gelfand–Tsetlin pattern. Each number λi,j in the pattern is greater than or equal to numbers immediately to the northeast or southeast of the pattern; in particular, every row of the pattern is decreasing. Note that such patterns are sometimes depicted as inverted pyramids instead of pyramids in the literature [PITH_FULL_IMAGE:figures/full_fig_p008_1_3.png] view at source ↗
Figures from the paper (23 more)
Figure 1.4
Figure 1.4. Figure 1.4: A Gelfand–Tsetlin pattern with boundary diag(15 [PITH_FULL_IMAGE:figures/full_fig_p008_1_4.png]
Figure 1.5
Figure 1.5. Figure 1.5: A schematic depiction of an augmented hive in [PITH_FULL_IMAGE:figures/full_fig_p009_1_5.png]
Figure 1.6
Figure 1.6. Figure 1.6: A visual depiction of the the second order discrete operators ∆ [PITH_FULL_IMAGE:figures/full_fig_p011_1_6.png]
Figure 2.2
Figure 2.2. Figure 2.2: A quadruple of the form (iii) that crosses the diagonal separating [PITH_FULL_IMAGE:figures/full_fig_p013_2_2.png]
Figure 2.1
Figure 2.1. Figure 2.1: The standard lozenge tiling of a hexagon [PITH_FULL_IMAGE:figures/full_fig_p015_2_1.png]
Figure 2.2
Figure 2.2. Figure 2.2: A typical lozenge tiling of 7(3,2), n = 6. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_2_2.png]
Figure 2.3
Figure 2.3. Figure 2.3: The hive associated with the Gelfand–Tsetlin pattern in Figure 1.3 and some [PITH_FULL_IMAGE:figures/full_fig_p016_2_3.png]
Figure 2.4
Figure 2.4. Figure 2.4: The weights of the green, blue, and red lozenges are, respectively, [PITH_FULL_IMAGE:figures/full_fig_p016_2_4.png]
Figure 2.5
Figure 2.5. Figure 2.5: A schematic depiction from [30] of the octahedron recurrence that transforms one pair (k, k′ ) of hives into another (h, h′ ). The hives h, h′ , k, k′ have been shifted to lie on triangles T, T′ , U, U′ respectively. and a blue lozenge ⋄ that has labels λi,j and λi−1…
Figure 2.6
Figure 2.6. Figure 2.6: A random lozenge tiling for n = 4, [11] 3 Height functions of lozenge tilings Let ˆi and ˆj denote the unit vectors along the x and y axes respectively (adjusted to be unit vectors in the triangular lattice which we will identify with Z 2 ; see [PITH_FULL_IMAGE:figu…
Figure 2.7
Figure 2.7. Figure 2.7: A random lozenge Tiling for n = 6, [11] [PITH_FULL_IMAGE:figures/full_fig_p018_2_7.png]
Figure 2.8
Figure 2.8. Figure 2.8: A random lozenge Tiling for n = 10, [11] [PITH_FULL_IMAGE:figures/full_fig_p018_2_8.png]
Figure 2.9
Figure 2.9. Figure 2.9: A random lozenge tiling of 7 20 (10,10), n = 20, [11]. the following be called unit vectors in “positive directions”: {ˆi, −ˆj, −ˆi + ˆj}. Let positive multiples of these vectors be called positive vectors. Let R be a domain whose boundary ∂R is a piecewise linear no…
Figure 2.10
Figure 2.10. Figure 2.10: A random lozenge tiling for n = 50, [11] [PITH_FULL_IMAGE:figures/full_fig_p019_2_10.png]
Figure 2.11
Figure 2.11. Figure 2.11: A random lozenge tiling for n = 100, [11] piece is along a positive direction. For two points u, v ∈ R, let the asymmetric distance function dR(u, v) be defined as (see page 10 of [13] for further discussion) the minimal total length over all positively oriented pat…
Figure 2.12
Figure 2.12. Figure 2.12: A random lozenge tiling of 7 100 (50,50), n = 100, [11] segments of ∂R have endpoints in the triangular lattice L, we call R a lattice domain. If R is also simply connected, we call R a simply connected lattice domain. Definition 8 (Height function for a simply conn…
Figure 3.1
Figure 3.1. Figure 3.1: A typical lozenge tiling of 7 6 (3,3), with the height functions f6,up and f6,lo on the upper and lower trapezoids. More generally, in 7 n (a,b) the westmost cor￾ner in the upper trapezoid gets the height 2b − a while the same point, viewed as belonging to the lower …
Figure 3.3
Figure 3.3. Figure 3.3: The convex set K that the gradients of an asymptotic height function must belong to. Theorem 4.1 (Pr´ekopa). Let f(x, y) be a function of R n ⊕ R m where x ∈ R n and and y ∈ R m. Suppose that f is log-concave in R n+m and let A be a convex subset of R m. Then the fun…
Figure 8.1
Figure 8.1. Figure 8.1: Diagram accompanying the calculation of the gradient of [PITH_FULL_IMAGE:figures/full_fig_p037_8_1.png]
Figure 8.1
Figure 8.1. Figure 8.1: An explicit calculation shows for x in the upper trapezoid (the situation when x is in the lower trapezoid is analogous) that f ‡ (x) = x + z(2a − b − 2x) a + b − x = x + 2z − 3bz a + b − x . When x < a − ϵ, the directional derivative ∂xf ‡ (x) < 1 − ϵ, and the direc…
Figure 10.1
Figure 10.1. Figure 10.1: Values are specified on the boundary of an annular region. [PITH_FULL_IMAGE:figures/full_fig_p041_10_1.png]
Figure 11.1
Figure 11.1. Figure 11.1: Within each “thin-strip” lattice domain bounded by transverse lines, there is [PITH_FULL_IMAGE:figures/full_fig_p046_11_1.png]
Figure 11.2
Figure 11.2. Figure 11.2: Lengths of interleaving pieces in the washboard in the proof of Lemma 11.6. [PITH_FULL_IMAGE:figures/full_fig_p047_11_2.png]

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Hives from deformed GUE minor processes

    math.PR 2026-07 unverdicted novelty 6.5 of 10

    Deformed GUE minor processes plus the octahedron recurrence yield random hives within O(n log n) KL divergence of a GUE hive law under a matching condition on the deformations.

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