A Uniform Random-Lattice Tail Bound for the SVP Kissing-Profile Parameter

Yutong Zhang; Yaoran Yang

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arXiv · 2605.21966

A Uniform Random-Lattice Tail Bound for the SVP Kissing-Profile Parameter

Yutong Zhang·Yaoran Yang

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Abstract

A recent SICOMP paper on classical and quantum algorithms for the shortest vector problem introduced a lattice-dependent parameter $\gamma(L)$, bounded universally in the exponential sense by $2^{0.402n+o(n)}$, and conjectured that this parameter is $2^{o(n)}$ for most lattices. We prove the Haar--Siegel random-lattice version in a stronger, dimension-uniform form. Let $X_n=\operatorname{SL}_n(\R)/\operatorname{SL}_n(\Z)$, let $\mu_n$ be its invariant probability measure, and let $\gamma(L)=\sup_{r\ge1} N_L(r\lambda_1(L))/r^n$, where $N_L(R)$ counts nonzero vectors of $L$ of Euclidean norm at most $R$. For every $n\ge3$ and every $T>0$, \[ \mu_n\{L\in X_n:\gamma(L)>T\}\le C T^{-1} \] with an absolute constant $C$. Consequently, for every sequence $a_n\to\infty$, $\gamma(L_n)\le a_n$ with $\mu_n$-probability tending to one; in particular $\gamma(L_n)=2^{o(n)}$ with high probability. In the product model of independent Haar--Siegel lattices, $\gamma(L_n)\le \exp(\sqrt n)$ eventually almost surely. The proof uses Rogers's second-moment estimate only through a dyadic self-normalization argument around the random scale $\lambda_1(L)$.

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Yutong Zhang, Yaoran Yang. 2026-05-21. A Uniform Random-Lattice Tail Bound for the SVP Kissing-Profile Parameter. https://arxiv.org/abs/2605.21966

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