we-solve (beta)

Math-2873 · mathematics · Probability

Joint metric, spectrum and heat-trace limit for FK–Ising maps

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For critical spherical FK–Ising maps with the manuscript graph and stationary-measure conventions, does $(M_n,a_n$ $d_n,\mu _n)$, the complete ordered spectrum (n $\lambda _n,j)_{j\geq 0}$, and the heat trace converge jointly to the sqrt(3)-quantum sphere, its closed Liouville Dirichlet-form spectrum, and its heat trace? Require compact resolvent, constants as the one-dimensional kernel, and locally uniform heat-trace convergence for every positive-time interval.

Short English statement

For critical spherical FK–Ising maps with the manuscript graph and stationary-measure conventions, does $(M_n,a_n$ $d_n,\mu _n)$, the complete ordered spectrum (n $\lambda _n,j)_{j\geq 0}$, and the heat trace converge jointly to the sqrt(3)-quantum sphere, its closed Liouville Dirichlet-form spectrum, and its heat trace? Require compact resolvent, constants as the one-dimensional kernel, and locally uniform heat-trace convergence for every positive-time interval.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2873-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2873

For critical FK--Ising spherical maps, the authors claim joint convergence of ordered eigenvalues and positive-time heat traces to the Liouville Brownian spectrum, with time multiplied by the number of edges. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Joint metric, spectrum and heat-trace limit for FK–Ising maps

OpenAI's claim

OpenAI presents a proof in the selected manuscript. For critical FK--Ising spherical maps, the authors claim joint convergence of ordered eigenvalues and positive-time heat traces to the Liouville Brownian spectrum, with time multiplied by the number of edges.

Mathematical statement of the claimed result

Result 1.
For rooted FK--Ising map/subset pairs with $n$ edges and weight $2^{\ell/2}$, let $d_n$ be all-edge graph distance, $\mu_n(v)=\deg(v)/(2n)$, and $L_n=P_n-I$ the rate-one simple-walk generator. Let $\lambda_{n,j}$ be the eigenvalues of $-L_n$, extended by infinity beyond the vertex count. On the ordinary unit-area $\sqrt3$-LQG sphere, close \[ \mathcal E_h(f,f)=\tfrac12\int_{S^2}|\nabla f|^2\,d\mathrm{vol} \] on $L^2(\mu_h)$; the claim gives compact resolvent, one-dimensional constant kernel, and finite $H_h(t)=\sum_je^{-t\Lambda_j(h)}$ for $t>0$. For deterministic spatial normalization $a_n$, it asserts \[ ((V_n,a_nd_n,\mu_n),(n\lambda_{n,j})_{j\ge0},H_n) \Longrightarrow((S^2,D_h,\mu_h),(\Lambda_j(h))_{j\ge0},H_h), \] where $H_n(t)=\sum_je^{-nt\lambda_{n,j}}$. The respective topologies are Gromov--Hausdorff--Prokhorov, coordinatewise spectral convergence, and uniform convergence on compact positive-time intervals. All limits use the same sphere. The deterministic scale is fixed by \[ a_n=\exp\{m(\log\operatorname{diam}(S,D_h))-m(\log\max(1,\operatorname{diam}(V_n,d_n)))\}. \] For a real random variable $Y$, $m(Y)$ is the unique real $r$ solving $\mathbb E\arctan(Y-r)=0$; both expectations here use unconditional laws.

Outline of the argument

The argument takes coupled conformal and metric limits as inputs and proves conductivity one for the unnormalized graph energy. Local recovery and convergence of centered inverse operators identify the limiting Dirichlet form and its compact resolvent, which then yield eigenvalue and heat-trace convergence.

Formal verification

For family 211, the selected Lean theorem is the $q>4$ Brownian tree metric-measure limit, with degree measure and the $n^{-1/2}$ distance scale. The subcritical, critical-Liouville, walk, and spectral companions are not included. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

Go to contribution →