{"id":"c1f3ef9d-1d74-45eb-af61-412839e3d361","arxiv_id":"2607.27810","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Adding a deterministic diagonal perturbation to the Gaussian beta corners process is defined for all beta>0, and its zero-temperature limit is shown to crystallize either on the known unperturbed lattice or, for linearly growing perturbations, on a new deformed lattice with Gaussian free field fluct","lead":"This paper extends the random-matrix 'corners process' to a perturbed version where a fixed diagonal matrix is added, and proves that at very low temperature the eigenvalues crystallize onto deterministic lattices. For large perturbations it finds a new deformed lattice and Gaussian fluctuations around it, which matters because it connects random matrix theory to exactly solvable statistical mechanics.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the crystallization/CLT argument is internally consistent. The main residual risk is verification depth of Lemma 5.4, not a demonstrated flaw.","rationale":"The reader flagged the Bessel/analytic-continuation postulate as the weakest assumption. I find that concern mostly non-load-bearing for the β→∞ theorems: in the fixed-top-row conditional density used for the asymptotic analysis, the Bessel factor B_a(z;β) cancels between the perturbed GβE density (3.6) and the perturbed links (3.11), leaving the explicit kernel (5.20). For classical β this kernel is derived from matrix models, and for general β it can be taken as the definition of the model; the crystallization and CLT statements are about that kernel, so the analytic continuation is a modeling convention rather than a hidden assumption in the proof. The genuinely load-bearing piece is the convexity Proposition 5.5. Its proof depends on Lemma 5.4, which I could not fault but which is intricate and not independently verified. The other flagged issues (the large-s part of Lemma 5.9, the omitted computation in Remark 5.15, AI-assisted proof details) are minor or peripheral. Since I found no concrete error, I do not change the reader's conditional acceptance; I recommend keeping CONDITIONAL and adding an independent check of Lemma 5.4 before full acceptance.","tokens_in":31412,"tokens_out":25746,"duration_ms":218464,"concrete_test":"Independently verify Lemma 5.4. Concretely: (a) run a randomized numerical search for n=2,...,10 sampling strictly ordered t and arbitrary u and checking inequality (5.10); and (b) re-derive identity (5.18) by differentiating the orthogonality relations (5.14)/(5.16) with a computer algebra system, confirming the nonnegative right-hand side for general n. If (a) finds any violation or (b) produces a sign mismatch, Proposition 5.5 and hence Theorem 5.12 are unsupported; if both pass, the convexity step is sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim Theorem 5.12 follows from a Laplace expansion of the exact conditional density (5.20), where the multivariate Bessel factors have cancelled; for fixed top row the β→∞ asymptotics depend only on the explicit kernel R(Y)e^{-βH_a(Y)}. Thus the analytic-continuation postulate for non-classical β is not what carries Theorem 5.12. The actual load-bearing step is strict convexity of H_a (Proposition 5.5), which rests entirely on Lemma 5.4. That lemma is a lengthy, AI-assisted algebraic identity; if a sign or normalization error slipped into the orthogonal-matrix argument, Proposition 5.5, the tail bound Lemma 5.9, and Theorem 5.12 would all collapse. I checked the proof carefully and found no error, so this is a concern about verification depth rather than a demonstrated flaw. The manuscript itself flags an omitted computation in Remark 5.15; that is a gap in Section 5.3's optional sharp result, not in the main theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":31599,"tokens_out":19888,"duration_ms":163196,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"§5.1, Lemma 5.4"},{"comment":"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.","section":"§5.3, Remark 5.15"},{"comment":"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).","section":"§3.4, Definition 3.4"}],"recommendation":"minor_revision","confidential_remarks":"The paper is strong and I recommend minor revision. Two points to weigh: (1) Lemma 5.4 is long and the manuscript states it was produced with AI assistance; although I checked it carefully and found no error, a second independent review of this lemma would be prudent. (2) Remark 5.15 contains an explicitly omitted computation that should either be supplied or downgraded. The general-β definition is a postulate but is handled honestly and does not affect the main theorems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nSoft 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.\n\nOn 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.\n\nThis 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.","headline":"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.","tokens_in":32146,"tokens_out":3597,"would_cite":true,"duration_ms":35866,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","60F05","33C67"],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["β-corners process","external source","multivariate Bessel functions","crystallization","discrete Gaussian free field","zero-temperature limit","Gelfand–Tsetlin polytope","shifted derivative"],"falsifier":"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.","tokens_in":31219,"feed_emoji":"❄️","tokens_out":8884,"duration_ms":73320,"temperature":0.7,"texified_at":"2026-08-05T21:51:44.554353+00:00","pith_summary":"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.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":3571,"prompt_tokens":906,"completion_tokens":2665,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":906,"completion_tokens_details":{"reasoning_tokens":1765}},"feed_headline":"Zero-temperature spikes bend the random-matrix crystal","feed_subtitle":"Fixed perturbations vanish; growing ones deform the frozen lattice, and fluctuations stay a Gaussian free field.","key_machinery":"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","core_discovery":"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-","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Growing perturbations bend the matrix crystal lattice","Deformed lattice emerges in beta-corners crystallization","Spiked perturbations reshape the frozen interlacing array","Crystal lattice curves under a single spike perturbation","Perturbed corners freeze on a deformed lattice"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Growing perturbations bend the matrix crystal lattice","Deformed lattice emerges in beta-corners crystallization","Spiked perturbations reshape the frozen interlacing array","Crystal lattice curves under a single spike perturbation","Perturbed corners freeze on a deformed lattice"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000179,"raw_usage":{"total_tokens":1214,"prompt_tokens":898,"completion_tokens":316,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":642,"completion_tokens_details":{"reasoning_tokens":246}},"tokens_in":642,"tokens_out":316,"duration_ms":3840,"temperature":1.0,"reasoning_tokens":246,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T00:57:59.758258+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}