we-solve (beta)

Math-2907 · mathematics · Probability

A universal joint critical SK equilibrium and quench autocorrelation law

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
At SK critical $\beta =1$, do Gaussian and Rademacher couplings yield the same joint limiting continuous law for the precisely rescaled equilibrium autocorrelation $A_n(t)$ and quench autocorrelation $B_n(s,t)$, on positive-time compact sets? Ask for positive-time entrance densities $h_s$ selected by Gaussian dynamics, $h_{s+t}=S_t$ $h_s$, $h_s$ $Pi\to \delta _0$ as $s\downarrow 0$, $h_s\to 1$ in L1 as s$\to$infinity, and $B(s,t)\to A(t)$ locally uniformly in probability as waiting time s$\to$infinity.

Short English statement

At SK critical $\beta =1$, do Gaussian and Rademacher couplings yield the same joint limiting continuous law for the precisely rescaled equilibrium autocorrelation $A_n(t)$ and quench autocorrelation $B_n(s,t)$, on positive-time compact sets? Ask for positive-time entrance densities $h_s$ selected by Gaussian dynamics, $h_{s+t}=S_t$ $h_s$, $h_s$ $Pi\to \delta _0$ as $s\downarrow 0$, $h_s\to 1$ in L1 as s$\to$infinity, and $B(s,t)\to A(t)$ locally uniformly in probability as waiting time s$\to$infinity.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2907-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2907

The quench-universality manuscript claims the joint Gaussian/sign-coupling limit for the equilibrium process $A$ and quench process $B$, together with the entrance-law and long-waiting-time assertions in the question. The stationary functional-universality companion covers only $A$. The full classification uses the quench theorem, which supplies the clauses absent from the stationary companion.

A universal joint critical SK equilibrium and quench autocorrelation law

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: Critical stationary SK autocorrelation is claimed to have a common random functional limit for Gaussian and sign couplings after time scaling by $n^{2/3}$ and amplitude by $n^{1/3}$; The critical SK quench from independent fair spins is claimed to share a functional scaling law for Gaussian and sign couplings and to relax to the stationary limit at long waiting times. Together they give a claimed affirmative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
For independent standard Gaussian or fair Rademacher $J_{ij}$, $i<j$, with $J_{ji}=J_{ij},J_{ii}=0$, use the zero-field SK measure \[ \pi_{n,\beta}^J(x)=Z^{-1}\exp\left(\frac\beta{\sqrt n}\sum_{i<j}J_{ij}x_ix_j\right),\qquad x\in\{-1,1\}^n. \] Continuous heat-bath dynamics refreshes every site at rate one; a discrete step refreshes one uniformly selected site, including attempts that retain its spin. At $\beta=1$, write $P_t^J$ for the rate-one-per-site semigroup, and let \[ C_{n,J}(t)=n^{-1}\sum_i\langle x_i,P_t^Jx_i\rangle_\pi,\qquad A_{n,J}(s)=n^{1/3}C_{n,J}(sn^{2/3}). \] There exists a common probability law $Q_{\mathrm{crit}}$ on $C_{\mathrm{loc}}((0,\infty))$ such that $\operatorname{Law}(A_{n,J})\Longrightarrow Q_{\mathrm{crit}}$ for both interaction distributions. This law is nondegenerate and supported on nonzero, nonnegative, nonincreasing continuous functions tending to zero at infinity. The topology is uniform convergence on compact positive-time intervals.
Result 2.
For independent standard Gaussian or fair Rademacher $J_{ij}$, $i<j$, with $J_{ji}=J_{ij},J_{ii}=0$, use the zero-field SK measure \[ \pi_{n,\beta}^J(x)=Z^{-1}\exp\left(\frac\beta{\sqrt n}\sum_{i<j}J_{ij}x_ix_j\right),\qquad x\in\{-1,1\}^n. \] Continuous heat-bath dynamics refreshes every site at rate one; a discrete step refreshes one uniformly selected site, including attempts that retain its spin. At $\beta=1$, start the quench from uniform independent spins $\nu_n$, independent of $J$. With $T_n=n^{2/3}$ and heat-bath semigroup $P_t^J$, define \[ A_{n,J}(t)=n^{-2/3}\sum_i\langle x_i,P_{tT_n}^Jx_i\rangle_\pi,\quad B_{n,J}(s,t)=n^{-2/3}\sum_i\mathbb E_{\nu_n}^J[x_i(sT_n)x_i((s+t)T_n)]. \] To specify the common limit, let $(g_a)_{a\ge1}$ be the increasing GOE edge process, normalized by $n^{2/3}(2-\lambda_a)\Longrightarrow g_a$. For $b+g_1>0$, set $\ell_a=(g_a+b)^{-1}$ and \[ D_g(b)=\lim_{L\to\infty}\left[\sqrt L-\sum_a\left((g_a+b)^{-1}-(g_a+L)^{-1}\right)\right]. \] The defining limit for $D_g(b)$ is in probability over the edge process; an almost-sure version is fixed along a deterministic sequence of $L$. For independent $Z_a\sim N(0,\ell_a)$, let $\Pi_g$ be their law conditioned, by continuous densities, on $\sum_a(Z_a^2-\ell_a)=D_g(b)$; it is independent of the admissible $b$. Let $H_g$ generate the closed cylinder-gradient form $\int\sum_a|\partial_aF|^2\,d\Pi_g$, and set $S_t^g=e^{-tc_*H_g}$ with the common positive microscopic coefficient $c_*$. There are measurable densities $h_s^g\in L^2(\Pi_g)$ with \[ h_s^g\ge0,\quad\Pi_g[h_s^g]=1,\quad h_{s+t}^g=S_t^gh_s^g,\quad h_s^g\Pi_g\Longrightarrow\delta_0\ (s\downarrow0),\quad \|h_s^g-1\|_{L^1(\Pi_g)}\to0\ (s\to\infty). \] Here the initial convergence uses the product topology. The locally uniformly convergent series \[ A_g(t)=\sum_a\int x_aS_t^gx_a\,d\Pi_g,\qquad B_g(s,t)=\sum_a\int h_s^gx_aS_t^gx_a\,d\Pi_g \] define continuous limits, and for either interaction law \[ (A_{n,J},B_{n,J})\Longrightarrow(A_g,B_g) \quad\text{in }C_{\mathrm{loc}}((0,\infty))\times C_{\mathrm{loc}}((0,\infty)^2). \] For every $0<a<b<\infty$, $\sup_{a\le t\le b}|B_g(s,t)-A_g(t)|\to0$ in probability as $s\to\infty$.

Outline of the argument

A conditioned spectral-edge measure and quantitative observation estimates control variance at the cap scale. Microscopic relaxation identifies the effective coefficient, resolvent-trace convergence yields the Gaussian process limit, and coupling comparison transports it to sign disorder.

Bottom-edge Gaussian estimates control warming on the critical clock. Positive-time limits select an entrance law from the uniform initial state, while smooth truncation of dynamical histories permits disorder replacement; comparison with the stationary edge input identifies the eventual relaxation.

Formal verification

For family 227, the selected statements include the high-temperature gap for $\beta<1$, cutoff ratios only for $\beta<1/2$, critical lower mixing scales with a subexponential upper bound, and low-temperature barriers. They do not provide the full critical exponent, the full cutoff location, or the newer autocorrelation limits. 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 →