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$\beta$-ensembles

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $B$ be standard Brownian motion on $[0,\infty)$ and $\beta>0$. Define the Dirichlet stochastic Airy operator $A_\beta=-d^2/dx^2+x+(2/\sqrt\beta)B^{\prime}$ through the closure of the quadratic form \[q_\beta[f]=\int_0^\infty(|f^{\prime}|^2+x|f|^2)\,dx+\frac2{\sqrt\beta}\int_0^\infty|f|^2\,dB\] initially on smooth compactly supported $f$ with $f(0)=0$; the last term can equivalently be read by integration by parts against the continuous Brownian path. Use the source's sign convention $H_\beta=-A_\beta$, whose eigenvalues decrease as $\lambda_1>\lambda_2>\cdots$. For $s\in\mathbb R$, the Airy kernel on $L^2(s,\infty)$ is $K_{\rm Ai}(x,y)=[\operatorname{Ai}(x)\operatorname{Ai}^{\prime}(y)-\operatorname{Ai}^{\prime}(x)\operatorname{Ai}(y)]/(x-y)$, continuously extended at $x=y$; $\operatorname{Ai}$ is the standard decaying Airy solution.

Statement recorded in the supplied source.
Use $H_\beta$ to obtain distributional information on its eigenvalues for general $\beta>0$, and in particular derive directly from $H_2$ the Tracy--Widom identity \[\mathbb P\{\lambda_1(B,2)\le s\}=\det(I-K_{\rm Ai})_{L^2(s,\infty)}.\] Here the determinant is the Fredholm determinant of the Airy integral operator. [S2]

Short English statement

Develop eigenvalue statistics for general beta through the stochastic Airy operator and obtain the beta-two Tracy--Widom formula directly from that operator.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-89-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Phys-89

$\beta$-ensembles

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $B$ be standard Brownian motion on $[0,\infty)$ and $\beta>0$. Define the Dirichlet stochastic Airy operator $A_\beta=-d^2/dx^2+x+(2/\sqrt\beta)B^{\prime}$ through the closure of the quadratic form \[q_\beta[f]=\int_0^\infty(|f^{\prime}|^2+x|f|^2)\,dx+\frac2{\sqrt\beta}\int_0^\infty|f|^2\,dB\] initially on smooth compactly supported $f$ with $f(0)=0$; the last term can equivalently be read by integration by parts against the continuous Brownian path. Use the source's sign convention $H_\beta=-A_\beta$, whose eigenvalues decrease as $\lambda_1>\lambda_2>\cdots$. For $s\in\mathbb R$, the Airy kernel on $L^2(s,\infty)$ is $K_{\rm Ai}(x,y)=[\operatorname{Ai}(x)\operatorname{Ai}^{\prime}(y)-\operatorname{Ai}^{\prime}(x)\operatorname{Ai}(y)]/(x-y)$, continuously extended at $x=y$; $\operatorname{Ai}$ is the standard decaying Airy solution.

Statement recorded in the supplied source.
Use $H_\beta$ to obtain distributional information on its eigenvalues for general $\beta>0$, and in particular derive directly from $H_2$ the Tracy--Widom identity \[\mathbb P\{\lambda_1(B,2)\le s\}=\det(I-K_{\rm Ai})_{L^2(s,\infty)}.\] Here the determinant is the Fredholm determinant of the Airy integral operator. [S2]

Short English statement

Develop eigenvalue statistics for general beta through the stochastic Airy operator and obtain the beta-two Tracy--Widom formula directly from that operator.

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. A one-test-function Gaussian upper bound for the top stochastic Airy eigenvalue distribution and its scale family. The beta-two Fredholm identity and general joint laws remain unresolved.
Sources and literature check.
Read the supplied complete definition, catalogue wording, scope qualifications and source list for this canonical ID. The original-source citations are inherited as supplied references; no fresh primary-source or novelty search was performed for this attempt. No current open-status claim is inferred from this calculation.
A direct variational probability bound.
Let $f$ be a real smooth compactly supported function in $(0,\infty)$, normalized by $\int f^2=1$. Set $a_f=\int(f'^2+xf^2)$ and $v_f=\int f^4>0$. For the stochastic Airy quadratic form in the statement, its value on this deterministic test function is a Gaussian random variable: \[q_\beta[f]=a_f+(2/\sqrt\beta)Z_f,\qquad Z_f\sim N(0,v_f).\] Indeed the stochastic integral of a deterministic integrand is Gaussian, with variance the integral of its square; the same value agrees with the pathwise integration-by-parts definition. Under the stated self-adjoint form realization the lowest eigenvalue $\Lambda_0$ of $A_\beta$ is at most $q_\beta[f]$, and $\lambda_1=-\Lambda_0$. Therefore \[\mathbb P\{\lambda_1\le s\}\le \Phi\left(\frac{(a_f+s)\sqrt\beta}{2\sqrt{v_f}}\right),\] where $\Phi$ is the standard Gaussian distribution function. The inclusion used here is $\{\Lambda_0\ge-s\}\subseteq\{q_\beta[f]\ge-s\}$, which fixes the sign.
Scale test.
For a fixed smooth real $h$ supported in $(0,1)$ with $\int h^2=1$, take $f_L(x)=L^{-1/2}h(x/L)$, $L>0$. If $A=\int h'^2$, $B=\int yh(y)^2\,dy$, and $C=\int h^4$, then $a_{f_L}=A/L^2+BL$ and $v_{f_L}=C/L$. Thus an explicit family of upper bounds results by minimizing \[\Phi\left(\frac{\sqrt\beta}{2\sqrt C}(A L^{-3/2}+B L^{3/2}+sL^{1/2})\right) \] over $L>0$. No independence between different choices of $L$ is needed, since each is a separate upper bound.
Remaining obstruction.
The Rayleigh principle gives only one side and one trial state at a time. Recovering the full distribution requires controlling the infimum over all random admissible test functions. No Fredholm determinant or beta-two identity follows from replacing that random infimum by this deterministic one-parameter family. The calculation is an elementary variational bound, not a derivation of the source's requested identity or general eigenvalue laws.
Verification and adversarial audit.
Checked H=-A sign, event inclusion, normalized scaling, Gaussian variance and beta>0; no unjustified independence or exchange of infimum and probability. This was an LLM adversarial audit, not independent human verification; no theorem prover was used. Prompts A then B applied to this ID: Checked H=-A sign, event inclusion, normalized scaling, Gaussian variance and beta>0; no unjustified independence or exchange of infimum and probability. Scope remains limited: Control the random variational infimum sufficiently sharply to identify the entire distribution and its Fredholm determinant.

Coordinator checked the event inclusion and sign in the Rayleigh Gaussian bound and the three scaling powers. A stochastic infimum is not replaced by a deterministic trial family. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.
Python 3 and local shell used to read supplied files and serialize this manuscript; mathematics derived symbolically in the text, with no numerical experiment or formal proof assistant. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.
Final status and first remaining gap.
UNRESOLVED. Control the random variational infimum sufficiently sharply to identify the entire distribution and its Fredholm determinant.

Completion Estimate

COMPLETION: 0% The calculation is an elementary benchmark or conditional reduction; it establishes no claimed new portion of the complete source target.

Sources

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