Thermal Concentration and Poisson--Dirichlet Edge Statistics for Random--Lattice Gibbs Ensembles

We study Gibbs measures on high--dimensional Haar--random unimodular lattices, where the energy of a lattice vector is its squared Euclidean norm.

The random lattice is viewed as quenched geometric disorder, and $c>0$ denotes the scaled inverse temperature.

We first analyze the edge window of vectors whose length is within the factor $e^{a/n}$ of the shortest length, with fixed $a$ as $n\to\infty$.

For the full sign--class Gibbs ensemble, we prove a Poisson point process limit theorem for the Gibbs mass of this window.

The mass vanishes in probability for $0<c\le1$, while for $c>1$ it has a nontrivial Poisson limit, and the ranked Gibbs weights converge to the Poisson--Dirichlet distribution with parameter $1/c$.

We then pass to a primitive--direction Gibbs ensemble and consider a fixed approximation factor $\gamma>1$.

For this modified ensemble, we prove a weighted moment formula and a quenched thermal concentration result in the high--temperature range $0<c<1$.

This yields the primitive fixed--factor visibility curve $c=\gamma^{-2}$ for approximate shortest directions.

More precisely, the primitive Gibbs mass of the fixed--factor window tends to zero for $c<\gamma^{-2}$, to one for $\gamma^{-2}<c<1$, and to $1/2$ at the critical boundary $c=\gamma^{-2}$.

Thus the fixed--factor theorem is a visibility statement for an idealized primitive target measure, not for the original full lattice Gibbs measure.

The results provide a random--lattice thermodynamic reference model for Gibbs targets related to approximate shortest vectors, without implying an efficient algorithm for the shortest vector problem.