The $n^{1/6}$ free-energy fluctuation scale of the low-temperature Sherrington-Kirkpatrick model, with a limiting law

Companion (fluctuation scale): for every fixed $\beta>1$, $\mathrm{sd}(F_n)=n^{1/6+o(1)}$ and $|F_n-\mathbb EF_n|$ is typically between $n^{1/6-\eta}$ and $n^{1/6+\eta}$. Principal (limiting law): $\mathrm{Var}(F_n)/n^{1/3}$ converges to a constant in $(0,\infty)$, every moment of the $n^{-1/6}$-rescaled completed free energy converges with a polynomial rate, exponential moments are uniformly bounded, and $(F_n-\mathbb EF_n)/\mathrm{sd}(F_n)$ converges along all $n$ to a nondegenerate law with mean 0 and variance 1, uniquely determined by its moments. The law is not identified with a named distribution and no formula for the variance constant is given.

Companion (fluctuation scale): for every fixed $\beta>1$, $\mathrm{sd}(F_n)=n^{1/6+o(1)}$ and $|F_n-\mathbb EF_n|$ is typically between $n^{1/6-\eta}$ and $n^{1/6+\eta}$. Principal (limiting law): $\mathrm{Var}(F_n)/n^{1/3}$ converges to a constant in $(0,\infty)$, every moment of the $n^{-1/6}$-rescaled completed free energy converges with a polynomial rate, exponential moments are uniformly bounded, and $(F_n-\mathbb EF_n)/\mathrm{sd}(F_n)$ converges along all $n$ to a nondegenerate law with mean 0 and variance 1, uniquely determined by its moments. The law is not identified with a named distribution and no formula for the variance constant is given.