Perturbed Beta Corners Process

Jiaming Xu; Leonid Petrov

Perturbed Beta Corners Process

T0 review · 0 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Adding a fixed diagonal matrix to a GOE/GUE/GSE-type random matrix leaves the zero-temperature crystallized eigenvalue array unchanged, while a perturbation growing linearly with beta deforms the crystal lattice; in both cases the fluctuati

desk verdict Genuinely new crystallization result for a natural beta-corners deformation; mostly rigorous and self-contained, with a couple of proof details that should be tightened before publication. read the letter →

arxiv 2607.27810 v1 pith:A5NX4UWC submitted 2026-07-30 math.PR math-phmath.MP

Leonid Petrov , Jiaming Xu This is my paper · ORCID

classification math.PRmath-phmath.MP MSC 60B2060F0533C67

keywords β-cornersprocessexternalsourcemultivariateBesselfunctionscrystallizationdiscreteGaussianfreefieldzero-temperaturelimitGelfand–Tsetlinpolytopeshiftedderivative

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

The paper introduces a perturbed version of the beta-corners process—the interlacing array of eigenvalues of nested corners of a Gaussian random matrix—obtained by adding a fixed diagonal matrix $A$, and extends the definition from the classical $\beta=1,2,4$ to all $\beta>0$ by analytic continuation through multivariate Bessel functions. Its central goal is to determine what happens to the array in the zero-temperature limit $\beta\to\infty$. With a fixed perturbation the external source is uniformly bounded while repulsion grows like $\beta$, so the array still crystallizes on the unperturbed lattice of polynomial-derivative roots and the fluctuations remain the same discrete Gaussian free field. In the second regime, where $a_i = (\beta/2) a_i$, the external source competes with the repulsion at leading order; the paper proves the array crystallizes on a deformed lattice, the unique global minimizer of an explicit effective Hamiltonian, with fluctuations given by a deformed discrete Gaussian free field whose covariance is the Hessian of that Hamiltonian. A single spike in the last coordinate yields a fully explicit deformed lattice: apply the shifted derivative $D_c$ and then iterated ordinary derivatives, discarding one spurious root.

What carries the argument

The central objects are the multivariate Bessel function $B_a(z; \beta)$—the $\beta$-deformation of the Harish-Chandra–Itzykson–Zuber integral that defines the perturbed density for all $\beta>0$ and cancels from the joint law, leaving the log-gas with effective Hamiltonian $H_a(Y) = \sum_k \left[\sum_{i<j} \log(y_i^k - y_j^k) - \frac{1}{2} \sum_{p,q} \log|y_p^k - y_q^{k+1}| - \frac{a_k - a_{k+1}}{2} \sum_i y_i^k\right]$—and the Gelfand–Tsetlin polytope on which $H_a$ lives. The proof of strict convexity of $H_a$ (via Lemma 5.4, a sign-definite quadratic-form identity) and its blow-up at the boundary give the unique minimizer $\bar Y$, the deformed lattice, whose equations couple all levels. The Hessian of $H_a$, independent of the perturba

What would settle it

For $N=2$, $\beta=2$, write the exact joint density of the two levels of $A+G$ (GUE) and compute the $\beta\to\infty$ limit of the level-one eigenvalue; compare it to the explicit root of the shifted-derivative equation from Proposition 5.14—this is a direct numerical or symbolic check of the deformed-lattice prediction in the classical case.

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Extended reading notes

Core claim

Under the scaled perturbation $a_i = (\beta/2) a_i$ with a fixed top row $z$, the perturbed beta-corners process crystallizes: the rescaled fluctuations $\sqrt{\beta}( y_i^k(\beta) - \bar y_i^k )$ converge jointly, as $\beta$ tends to infinity, to the deformed discrete Gaussian free field. The limiting field is the centered Gaussian vector with covariance given by the inverse Hessian of the effective Hamiltonian $H_a$ evaluated at the unique minimizer $\bar Y$ of $H_a$ in the Gelfand–Tsetlin polytope; equivalently, its density is proportional to $\exp\left(\frac{1}{2} \sum \zeta_i^k H_{(i,k),(j,l)} \zeta_j^l\right)$ with $H$ the Hessian. The deformed lattice $\bar Y$ solves the coupled optimality equations expressing balance between same-

Load-bearing premise

For general $\beta > 0$ the entire model is a postulate: the perturbed density is defined through multivariate Bessel functions by analytic continuation from $\beta=1,2,4$ rather than derived from a matrix model, so the positivity of the Bessel function and the existence of an interlacing probability law are load-bearing.

Editorial extensions

If this is right

For beta=1,2,4, where the model is a genuine matrix-additive GOE/GUE/GSE corner process, the crystallization and CLT are rigorous consequences of the definitions and can be tested by numerical simulation of finite-beta corners of A+G.
A constant perturbation (a_1 = ... = a_N) reduces the deformed lattice and the deformed dGFF to the unperturbed ones, so the scaling regime contains a universality statement: a common drift does not change the zero-temperature limit.
In the single-spike case the deformed lattice is algorithmic: factor the top-row polynomial, apply D_{a_{N-1}-a_N}, discard the spurious root, and form lower levels by iterated derivatives; this gives an explicit closed form where general optimality equations do not.
The Gaussian tail bound yields exponential concentration of the whole interlacing array at scale beta^{-1/2}, uniformly in beta, so the LLN holds at an exponentially fast rate.
The zero-temperature up transitions are governed by Rodrigues-type operators—e^{x^2} d/dx(e^{-x^2} ... ) in the Hermite scaling and shifted derivatives in the scaled regime—linking the crystallization to classical orthogonal-polynomial raising relations.

Reading between the lines

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

Beyond the paper: the same variational-convexity route should produce crystallization theorems for other confined beta-ensembles with external fields, e.g., a perturbed Jacobi corners process, with the deformed lattice given by the analogous optimality equations and the fluctuation field again a dGFF on it.
Beyond the paper: the threshold a_N ~ beta^{-1/2} where the two regimes cross (Remark 6.3) hints at a BBP-type edge transition for the whole array at zero temperature; the present work does not pursue N->infinity or spiked-edge scaling limits.
Beyond the paper: because the dGFF is defined from a Hessian at the minimizer, one might conjecture a broader universality—any perturbation strong enough to shift the lattice leaves the dGFF nature of fluctuations unchanged, only the lattice changes; this is in the spirit of, but not proven by, the paper.
Beyond the paper: the spurious root appearing in the shifted-derivative construction may carry information about the edge eigenvalue in the scaled regime, and the zero-temperature up-transition formulas could seed finite-beta edge asymptotics for spiked beta ensembles.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here is my take. The paper has one genuinely new result: the linearly growing perturbation regime, where the external source competes with repulsion and the lattice deforms. The deformed lattice, the coupled optimality equations, and the dGFF fluctuations are not in GM20 or anywhere else I know. Everything else — the fixed-perturbation crystallization, the matrix-model derivation for beta=1,2,4, and the Bessel construction for general beta — is a clean, correct extension of known technology, but the scaled regime is the contribution that makes the paper worth reading.

The paper is in good shape overall. It is largely self-contained. The derivation in Section 2 is explicit and easy to follow. The key cancellation of the Bessel factors in the conditional density is handled carefully, and the paper is honest that for general beta the construction is a postulate rather than a matrix model. The strict convexity of the effective Hamiltonian is a strong and non-obvious result; Lemma 5.4 is very nice. Once convexity is in hand, the Laplace argument is standard and works.

Soft spots, in proportion: Lemma 5.9's tail bound is the weakest part. The claim that for large s the event is empty unless s^2 = O(beta) needs a precise argument; as written it is a sketch. Remark 5.15 says a direct computation is omitted; either include it or clearly mark the claim as optional, since as stated it is a dangling assertion. The positivity of the Bessel function requires a distinct top row, so ties are excluded; that is a minor caveat. The general-beta extension is analytic continuation, with no matrix model, but the paper says this openly and, importantly, it does not carry the main theorem because the Bessel factors cancel.

On the central proof: the strict convexity lemma is load-bearing. I read the proof and found no error, but it is intricate and AI-assisted, so I would want an independent referee to verify it. That is a request for verification depth, not a demonstrated flaw. The central argument holds up.

This paper deserves a serious referee. I would send it to peer review. With the tail bound tightened and the omitted computation either supplied or explicitly deferred, publication should be straightforward. I would cite it and bring it to a random-matrix reading group.

Referee Report

0 major / 3 minor

Summary. This paper introduces a perturbed β-corners process: for β=1,2,4 it is the corners process of A+G with A=diag(a_i) and G a GOE/GUE/GSE matrix, and for general β>0 it is defined through multivariate Bessel generating functions. The main results concern β→∞ at fixed N. With fixed perturbation, the process crystallizes on the usual Gorin–Marcus polynomial-derivative lattice with the same dGFF fluctuations (Theorem 4.6). With a linearly growing perturbation ai=(β/2)ai, the limiting lattice is deformed and is the unique minimizer of an effective Hamiltonian H_a; the LLN (Theorem 5.8) and CLT to a deformed dGFF (Theorem 5.12) are proved. For a single spike in the last coordinate, the deformed lattice is built from a shifted derivative. The paper also analyzes zero-temperature up transitions (Section 6).

Significance. If correct, the paper provides the first treatment of an external source in the crystallization limit of β-corners processes at fixed N, producing a nontrivial deformed lattice and a deformed dGFF. The proof strategy is sound: the matrix derivation for classical β is explicit; the multivariate Bessel factors cancel in the conditional density (5.20), so the scaled-regime asymptotics depend only on the exact log-gas density; the strict convexity of H_a is proved via Lemma 5.4, which I checked without finding an error; and the Laplace asymptotic argument is standard. The single-spike shifted-derivative solution is elegant. The main limitation is that the general-β definition is a Bessel-function postulate rather than a matrix-model construction, but the paper is transparent about this and the postulate does not affect the zero-temperature results. The remaining risk is the verification depth of Lemma 5.4, not a demonstrated flaw.

minor comments (3)

[§5.1, Lemma 5.4] This lemma is load-bearing for Proposition 5.5 and hence for the uniqueness of the deformed lattice, the tail bound, and the CLT. The proof is correct as far as I was able to check, but it is extremely compressed and the paper states that proof details were AI-assisted. For the published version, I strongly recommend adding a structural outline or a computer-algebra certificate for the key identity (5.10), and explicitly verifying the positivity/orthogonality claims around (5.16)–(5.18). This is a verifiability request, not an objection to correctness.
[§5.3, Remark 5.15] The statement that the natural top-down shifted-derivative recursion fails for all other perturbation profiles is asserted with 'the latter may be verified by a direct computation, which we omit.' Since Section 5.3 advertises a sharp decoupling result and this remark rules out a natural generalization, the computation should be included or the claim should be marked as a conjecture. This gap is outside the main theorems and does not affect the central argument.
[§3.4, Definition 3.4] For non-classical β the perturbed corners process is defined by analytic continuation through multivariate Bessel functions rather than by a matrix model; the paper is transparent about this, and the Bessel factors cancel in the conditional density (5.20), so the zero-temperature theorems are unaffected. Still, the introduction and abstract should state more explicitly that the general-β construction is a Bessel-function definition and requires a strictly ordered top row for positivity of B_a(z;β).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the crystallization and CLT are derived from the explicit effective Hamiltonian, not from fitted parameters or load-bearing self-citations.

full rationale

The central derivation chain is self-contained. Theorem 5.12 is obtained from the exact conditional density (5.20), pβ(Y|z) ∝ R(Y) exp(−β H_a(Y)), where the multivariate Bessel factors have canceled by the definition of the perturbed links. The deformed lattice Ybar is defined as the minimizer of the explicit Hamiltonian H_a (Definition 5.1, Eq. 5.3), and its existence and uniqueness are proved from the boundary blow-up (Lemma 5.3) and strict convexity (Proposition 5.5, Lemma 5.4), not assumed or imported from the authors' prior work. The deformed dGFF covariance (Definition 5.10) is the Hessian of the same explicit H_a at Ybar; the CLT is a Laplace expansion of the exact density, with the tail bound Lemma 5.9 upgrading local convergence. This is a theorem about an explicit rate function, not a tautology: the Gaussian limit is computed from H_a, not imposed as an input. The fixed-perturbation result (Theorem 4.6) reduces transparently to the independent external result [GM20] via the bounded reweighting h(Y) (4.6)–(4.8). There are self-citations — Proposition 3.2 cites [GXZ24], a paper with an overlapping author, for the general-β density formula — but this supports the definitional analytic continuation of the model, and for β = 1,2,4 the same density is independently derived in Section 2 and matched in Section 3.5. The main zero-temperature results do not reduce to this citation. The paper explicitly states that for general β there is no underlying matrix model; that is a declared modeling assumption, not a circular step. Remark 5.15 omits a direct computation for a negative claim, and the AI statement acknowledges AI-assisted proof details; these are verification gaps, not circularity. No prediction is equivalent by construction to a fitted input, and no uniqueness or ansatz is imported from self-citations to force the conclusion.

Assumptions & free parameters 2 free parameters · 4 assumptions · 2 invented entities

The paper is a mathematical construction, so the only 'invented entities' are definitions. The free parameters are the perturbation parameters, which are inputs. The main axiom is the validity of the Bessel-function analytic continuation for all beta>0.

free parameters (2)

perturbation a_i = fixed real numbers
The entire family of perturbed processes has N real parameters; the asymptotic results are uniform in a in compact sets. They are input, not fitted.
scaling factor a_i in the scaled regime = fixed real numbers
The regime a_i = beta/2 a_i is chosen to match the competition between source and repulsion; the a_i are parameters of the model.

assumptions (4)

standard math Multivariate Bessel functions have positivity B_a(z)>0 for distinct z and the duality B_a(z)=B_z(a).
Section 3.1 relies on this to define the perturbed density and links; the paper cites [OO97], [HO97], [Opd93]. Assumed as standard.
domain assumption The beta-addition operation via Bessel generating functions is well-defined and coincides with matrix addition for beta=1,2,4.
Section 3.2 uses the definition of c = a ⊞_β g via multiplication of Bessel generating functions. This is an established approach [GM20],[BGCG22], but for general beta it is a postulate.
standard math The asymptotic Laplace method applies: after establishing strict convexity and boundary blow-up, LLN and CLT follow from [Won01, Ch.9].
Section 5.2 uses a Laplace-type argument. Assumed as standard.
domain assumption The interlacing is strict and the top row z has distinct coordinates.
The density requires strict interlacing; the top row being distinct is needed for the normalization constant. The paper notes the top row under GbetaE is distinct a.s.

invented entities (2)

Perturbed beta-corners process for all beta>0
purpose: Deformation of the classical beta-corners process; the main object of study.
This is a new mathematical object defined in Definition 3.4 via Bessel functions. It has no independent empirical evidence outside the paper, but it is a mathematical definition, not a physical new entity.
Deformed lattice \bar{Y} (deformed infinite-corners lattice) independent evidence
purpose: Ground state of the scaled perturbation regime; encodes where the array freezes.
The deformed lattice is a deterministic object with explicit optimality equations (5.2). It is a new mathematical entity, but it has an independent handle: it is uniquely characterized as the minimizer of H_a and is explicitly computable in the single-spike case. Not an invented physical entity.

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Pith. "Pith review of Perturbed Beta Corners Process." pith.science (2026). https://pith.science/paper/A5NX4UWC

@misc{pith2026260727810,
  author       = {Pith},
  title        = {Pith review of: Perturbed Beta Corners Process},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A5NX4UWC}},
  note         = {Machine review of arXiv:2607.27810}
}

abstract

We introduce and study the perturbed $\beta$-corners process, a deformation of the classical $\beta$-corners process. The latter is a probability measure on interlacing arrays of real numbers that, for the classical values $\beta=1,2,4$, describes the joint distribution of eigenvalues of principal submatrices of a Gaussian random matrix with the corresponding GOE/GUE/GSE symmetry. For classical $\beta$, the perturbed process arises from adding a deterministic diagonal matrix $A=diag(a_1,...,a_N)$ to a GOE/GUE/GSE matrix. The eigenvalues of the resulting random matrix depend symmetrically on $a_1,...,a_N$, but this symmetry does not extend to the whole corners process. We extend the construction to all $\beta>0$ via multivariate Bessel functions, and analyze the crystallization ($\beta\to\infty$) of the resulting interlacing array, proving a law of large numbers and a central limit theorem in two regimes. For fixed perturbation, the eigenvalue repulsion dominates, and the array crystallizes on the same lattice of polynomial-derivative roots, with the same discrete Gaussian free field fluctuations, as in the unperturbed case treated by Gorin and Marcus (arXiv:1706.07393). New phenomena appear in the second regime, where the $a_i$'s grow linearly in $\beta$. Then the external source competes with the repulsion at leading order. Here the array freezes on a deformed lattice characterized by a coupled system of optimality equations. The fluctuations are governed by the same discrete Gaussian free field, now attached to the deformed lattice. The deformed lattice equations decouple in the special case of a single spike in the last coordinate. Then the deformed lattice is obtained explicitly by applying one shifted derivative $D_c f = f' + cf$ followed by iterated ordinary derivatives.

Figures

Figures reproduced from arXiv: 2607.27810 by the authors.

**Figure 1.** Cutting the upper left (N − 1) × (N − 1) corner. The eigenvalues µ of the corner interlace with the eigenvalues λ of the matrix. The corners process describes the joint distribution of eigenvalues of nested principal submatrices ( [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

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Works this paper leans on

2 extracted references · 2 linked inside Pith

[1]

Adler, P

[AvMW13] M. Adler, P. van Moerbeke, and D. Wang,Random matrix minor processes related to percolation theory, Random Matrices Theory Appl.2(2013), no. 04, 1350008. arXiv:1301.7017 [math.PR].↑3 [And91] G. W. Anderson,A short proof of Selberg’s generalized beta formula, Forum Math.3(1991), 415–417. ↑11 [BBP05] J. Baik, G. Ben Arous, and S. Péché,Phase transi...

arXiv 2013
[913]

Gorin and M

arXiv:1706.07393 [math.PR].↑1, 2, 3, 4, 8, 9, 15, 16, 23, 28 [GS15] V. Gorin and M. Shkolnikov,Multilevel Dyson Brownian motions via Jack polynomials, Probab. Theory Relat. Fields163(2015), no. 3-4, 413–463. arXiv:1401.5595 [math.PR].↑2, 8, 10 [GXZ24] V. Gorin, J. Xu, and L. Zhang,Airyβ line ensemble and its Laplace transform, arXiv preprint (2024). arXiv...

arXiv 2015

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Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Adler, P

[AvMW13] M. Adler, P. van Moerbeke, and D. Wang,Random matrix minor processes related to percolation theory, Random Matrices Theory Appl.2(2013), no. 04, 1350008. arXiv:1301.7017 [math.PR].↑3 [And91] G. W. Anderson,A short proof of Selberg’s generalized beta formula, Forum Math.3(1991), 415–417. ↑11 [BBP05] J. Baik, G. Ben Arous, and S. Péché,Phase transi...

arXiv 2013

[2] [913]

Gorin and M

arXiv:1706.07393 [math.PR].↑1, 2, 3, 4, 8, 9, 15, 16, 23, 28 [GS15] V. Gorin and M. Shkolnikov,Multilevel Dyson Brownian motions via Jack polynomials, Probab. Theory Relat. Fields163(2015), no. 3-4, 413–463. arXiv:1401.5595 [math.PR].↑2, 8, 10 [GXZ24] V. Gorin, J. Xu, and L. Zhang,Airyβ line ensemble and its Laplace transform, arXiv preprint (2024). arXiv...

arXiv 2015